Research Article Helmholtz Theorem for Nondifferentiable Hamiltonian ...346]Pierret_Torres.pdf · Research Article Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleHelmholtz Theorem for Nondifferentiable Hamiltonian Systemsin the Framework of Cressonrsquos Quantum Calculus
Freacutedeacuteric Pierret1 and Delfim F M Torres2
1 Institut de Mecanique Celeste et de Calcul des Ephemerides Observatoire de Paris 75014 Paris France2Center for Research and Development in Mathematics and Applications (CIDMA) Department of MathematicsUniversity of Aveiro 3810-193 Aveiro Portugal
Correspondence should be addressed to Delfim F M Torres delfimuapt
Received 23 January 2016 Accepted 28 April 2016
Academic Editor Taher S Hassan
Copyright copy 2016 F Pierret and D F M TorresThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
We derive the Helmholtz theorem for nondifferentiable Hamiltonian systems in the framework of Cressonrsquos quantum calculusPrecisely we give a theorem characterizing nondifferentiable equations admitting a Hamiltonian formulation Moreover in theaffirmative case we give the associated Hamiltonian
1 Introduction
Several types of quantum calculus are available in the lit-erature including Jacksonrsquos quantum calculus [1 2] Hahnrsquosquantum calculus [3ndash5] the time-scale 119902-calculus [6 7] thepower quantum calculus [8] and the symmetric quantumcalculus [9ndash11] Cresson introduced in 2005 his quantum cal-culus on a set of Holder functions [12]This calculus attractedattention due to its applications in physics and the calculusof variations and has been further developed by several dif-ferent authors (see [13ndash16] and references therein) Cressonrsquoscalculus of 2005 [12] presents however some difficulties andin 2011 Cresson and Greff improved it [17 18] Indeed thequantum calculus of [12] let a free parameter which is presentin all the computations Such parameter is certainly difficultto interpretThe new calculus of [17 18] bypasses the problemby considering a quantity that is free of extra parameters andreduces to the classical derivative for differentiable functionsIt is this new version of 2011 that we consider here with abrief review of it being given in Section 2 Along the textby Cressonrsquos calculus we mean this quantum version of 2011[17 18] For the state of the art on the quantum calculusof variations we refer the reader to the recent book [19]With respect to Cressonrsquos approach the quantum calculusof variations is still in its infancy see [13 17 18 20ndash22] In[17] nondifferentiable Euler-Lagrange equations are used in
the study of PDEs Euler-Lagrange equations for variationalfunctionals with Lagrangians containing multiple quantumderivatives depending on a parameter or containing higher-order quantum derivatives are studied in [20] Variationalproblems with constraints with one and more than oneindependent variable of first and higher-order type are inves-tigated in [21] Recently problems of the calculus of variationsand optimal control with time delay were considered [22]In [18] a Noether type theorem is proved but only withthe momentum term This result is further extended in [23]by considering invariance transformations that also changethe time variable thus obtaining not only the generalizedmomentum term of [18] but also a new energy term In [13]nondifferentiable variational problems with a free terminalpoint with or without constraints of first and higher-orderare investigated Here we continue to develop Cressonrsquosquantum calculus in obtaining a result for Hamiltoniansystems and by considering the so-called inverse problem ofthe calculus of variations
A classical problem in analysis is the well-known Helm-holtzrsquos inverse problem of the calculus of variations find anecessary and sufficient condition under which a (system of)differential equation(s) can be written as an Euler-Lagrangeor a Hamiltonian equation and in the affirmative case findall possible Lagrangian or Hamiltonian formulations Thiscondition is usually called the Helmholtz condition The
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 8073023 8 pageshttpdxdoiorg10115520168073023
2 Discrete Dynamics in Nature and Society
Lagrangian Helmholtz problem has been studied and solvedby Douglas [24] Mayer [25] and Hirsch [26 27]TheHamil-tonian Helmholtz problem has been studied and solved upto our knowledge by Santilli in his book [28] Generalizationof this problem in the discrete calculus of variations frame-work has been done in [29 30] in the discrete Lagrangiancase In the case of time-scale calculus that is a mixingbetween continuous and discrete subintervals of time see [31]for a necessary condition for a dynamic integrodifferentialequation to be an Euler-Lagrange equation on time scalesFor the Hamiltonian case it has been done for the discretecalculus of variations in [32] using the framework of [33]and in [34] using a discrete embedding procedure derivedin [35] In the case of time-scale calculus it has been donein [36] for the Stratonovich stochastic calculus see [37] Herewe give the Helmholtz theorem for Hamiltonian systemsin the case of nondifferentiable Hamiltonian systems in theframework of Cressonrsquos quantum calculus By definition thenondifferentiable calculus extends the differentiable calculusSuch as in the discrete time-scale and stochastic cases werecover the same conditions of existence of a Hamiltonianstructure
The paper is organized as follows In Section 2 we givesome generalities and notions about the nondifferentiablecalculus introduced in [17] the so-called Cressonrsquos quantumcalculus In Section 3 we remind definitions and resultsabout classical and nondifferentiableHamiltonian systems InSection 4 we give a brief survey of the classical HelmholtzHamiltonian problem and then we prove the main resultof this papermdashthe nondifferentiable Hamiltonian Helmholtztheorem Finally we give two applications of our results inSection 5 and we end in Section 6 with conclusions andfuture work
2 Cressonrsquos Quantum Calculus
We briefly review the necessary concepts and results of thequantum calculus [17]
21 Definitions Let X119889 denote the set R119889 or C119889 119889 isin N andlet 119868 be an open set in R with [119886 119887] sub 119868 119886 lt 119887 We denote byF(119868X119889) the set of functions 119891 119868 rarr X119889 and by C0(119868X119889)the subset of functions ofF(119868X119889) which are continuous
Definition 1 (Holderian functions [17]) Let 119891 isin C0(119868R119889)Let 119905 isin 119868 Function 119891 is said to be 120572-Holderian 0 lt 120572 lt 1 atpoint 119905 if there exist positive constants 120598 gt 0 and 119888 gt 0 suchthat |119905 minus 119905
1015840| ⩽ 120598 implies 119891(119905) minus 119891(119905
1015840) ⩽ 119888|119905 minus 119905
1015840|120572 for all 1199051015840 isin 119868
where sdot is a norm on R119889
The set of Holderian functions of Holder exponent 120572 forsome 120572 is denoted by 119867
120572(119868R119889) The quantum derivative is
defined as follows
Definition 2 (the 120598-left and 120598-right quantum derivatives [17])Let 119891 isin C0(119868R119889) For all 120598 gt 0 the 120598-left and 120598-right
quantum derivatives of 119891 denoted respectively by 119889minus
120598119891 and
119889+
120598119891 are defined by
119889minus
120598119891 (119905) =
119891 (119905) minus 119891 (119905 minus 120598)
120598
119889+
120598119891 (119905) =
119891 (119905 + 120598) minus 119891 (119905)
120598
(1)
Remark 3 The 120598-left and 120598-right quantum derivatives of acontinuous function 119891 correspond to the classical derivativeof the 120598-mean function 119891
120590
120598defined by
119891120590
120598(119905) =
120590
120598int
119905+120590120598
119905
119891 (119904) 119889119904 120590 = plusmn (2)
The next operator generalizes the classical derivative
Definition 4 (the 120598-scale derivative [17]) Let 119891 isin C0(119868R119889)For all 120598 gt 0 the 120598-scale derivative of 119891 denoted by ◻120598119891◻119905is defined by
◻120598119891
◻119905=
1
2[(119889+
120598119891 + 119889minus
120598119891) + 119894120583 (119889
+
120598119891 minus 119889minus
120598119891)] (3)
where 119894 is the imaginary unit and 120583 isin minus1 1 0 minus119894 119894
Remark 5 If 119891 is differentiable then one can take the limit ofthe scale derivative when 120598 goes to zero We then obtain theclassical derivative 119889119891119889119905 of 119891
We also need to extend the scale derivative to complexvalued functions
Definition 6 (see [17]) Let 119891 isin C0(119868C119889) be a continuouscomplex valued function For all 120598 gt 0 the 120598-scale derivativeof 119891 denoted by ◻120598119891◻119905 is defined by
◻120598119891
◻119905=
◻120598Re (119891)◻119905
+ 119894◻120598 Im (119891)
◻119905 (4)
where Re(119891) and Im(119891) denote the real and imaginary partof 119891 respectively
In Definition 4 the 120598-scale derivative depends on 120598which is a free parameter related to the smoothing orderof the function This brings many difficulties in applicationsto physics when one is interested in particular equationsthat do not depend on an extra parameter To solve theseproblems the authors of [17] introduced a procedure toextract information independent of 120598 but related with themean behavior of the function
Definition 7 (see [17]) Let C0conv(119868 times ]0 1]R119889) sube C0(119868 times ]0
1]R119889) be such that for any function119891 isin C0conv(119868 times ]0 1]R119889)
the lim120598rarr0119891(119905 120598) exists for any 119905 isin 119868 We denote by 119864 a com-plementary space ofC0conv(119868 times ]0 1]R119889) inC0(119868 times ]0 1]R119889)We define the projection map 120587 by
120587 C0
conv (119868 times ]0 1] R119889) oplus 119864 997888rarr C
Remark 10 For 119891 isin C1(119868R119889) and 119892 isin C1(119868R119889) oneobtains from (8) the classical Leibniz rule (119891 sdot 119892)
1015840= 1198911015840sdot 119892 +
119891 sdot 1198921015840
Definition 11 We denote by C1◻the set of continuous func-
tions 119902 isin C0([119886 119887]R119889) such that ◻119902◻119905 isin C0(119868R119889)
Theorem 12 (the quantum version of the fundamental theo-rem of calculus [17]) Let 119891 isin C1
◻([119886 119887]R119889) be such that
lim120598rarr0
int
119887
119886
(◻120598119891
◻119905)
119864
(119905) 119889119905 = 0 (9)
Then
int
119887
119886
◻119891
◻119905(119905) 119889119905 = 119891 (119887) minus 119891 (119886) (10)
22 Nondifferentiable Calculus of Variations In [17] thecalculus of variations with quantum derivatives is introducedand respective Euler-Lagrange equations derived without thedependence of 120598
Definition 13 An admissible Lagrangian 119871 is a continuousfunction 119871 R times R119889 times C119889 rarr C such that 119871(119905 119909 V) is holo-morphic with respect to V and differentiable with respect to119909 Moreover 119871(119905 119909 V) isin R when V isin R119889 119871(119905 119909 V) isin C whenV isin C119889
An admissible Lagrangian function 119871 RtimesRdtimesC119889 rarr C
defines a functional onC1(119868R119889) denoted by
The nondifferentiable embedding procedure allows us todefine a natural extension of the classical Euler-Lagrangeequation in the nondifferentiable context
Definition 15 (see [17]) The nondifferentiable LagrangianfunctionalL◻ associated withL is given by
119867120572(119868R119889) with 120572 + 120573 gt 1 A 119867
120573
0-variation of 119902 is a function
of the form 119902 + ℎ where ℎ isin 119867120573
119874 We denote by 119863L◻(119902)(ℎ)
the quantity
lim120598rarr0
L◻ (119902 + 120598ℎ) minusL◻ (119902)
120598(14)
if there exists the so-called Frechet derivative ofL◻ at point119902 in direction ℎ
Definition 16 (nondifferentiable extremals) A 119867120573
0-extremal
curve of the functionalL◻ is a curve 119902 isin 119867120572(119868R119889) satisfying
119863L◻(119902)(ℎ) = 0 for any ℎ isin 119867120573
0
Theorem 17 (nondifferentiable Euler-Lagrange equations[17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be an admissibleLagrangian of class C2 We assume that 120574 isin 119867
120572(119868R119889) such
that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that 119871(119905 120574(119905)
◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin 119867120573
0(119868R119889)
A curve 120574 satisfying the nondifferentiable Euler-Lagrangeequation
◻
◻119905[120597119871
120597V(119905 120574 (119905)
◻120574 (119905)
◻119905)] =
120597119871
120597119909(119905 120574 (119905)
◻120574 (119905)
◻119905) (15)
is an extremal curve of functional (13)
3 Reminder about Hamiltonian Systems
We now recall the main concepts and results of both classicaland Cressonrsquos nondifferentiable Hamiltonian systems
31 Classical Hamiltonian Systems Let 119871 be an admissibleLagrangian function If 119871 satisfies the so-called Legendreproperty then we can associate to 119871 a Hamiltonian functiondenoted by119867
4 Discrete Dynamics in Nature and Society
Definition 18 Let 119871 be an admissible Lagrangian functionTheLagrangian119871 is said to satisfy the Legendre property if themapping V 997891rarr (120597119871120597V)(119905 119909 V) is invertible for any (119905 119902 V) isin
119868 timesR119889 times C119889
If we introduce a new variable
119901 =120597119871
120597V(119905 119902 V) (16)
and 119871 satisfies the Legendre property then we can find afunction 119891 such that
V = 119891 (119905 119902 119901) (17)
Using this notation we have the following definition
Definition 19 Let 119871 be an admissible Lagrangian functionsatisfying the Legendre property The Hamiltonian function119867 associated with 119871 is given by
Theorem 20 (Hamiltonrsquos least-action principle) The curve(119902 119901) isin C(119868R119889)timesC(119868C119889) is an extremal of the Hamiltonianfunctional
denotes the symplectic matrix with 119868119889 being the identitymatrix on R119889
32 NondifferentiableHamiltonian Systems Thenondifferen-tiable embedding induces a change in the phase space withrespect to the classical case As a consequence we have towork with variables (119909 119901) that belong to R119889 times C119889 and notonly to R119889 timesR119889 as usual
Definition 21 (nondifferentiable embedding of Hamiltoniansystems [17]) The nondifferentiable embedded Hamiltoniansystem (20) is given by
The nondifferentiable calculus of variations allows us toderive the extremals forH◻
Theorem 22 (nondifferentiable Hamiltonrsquos least-action prin-ciple [17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be anadmissible C2-Lagrangian We assume that 120574 isin 119867
120572(119868R119889)
such that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that
119871(119905 120574(119905) ◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin
119867120573
0(119868R119889) Let 119867 be the corresponding Hamiltonian defined
by (18) A curve 120574 997891rarr (119905 119902(119905) 119901(119905)) isin 119868 times R119889 times C119889 solution ofthe nondifferentiable Hamiltonian system (23) is an extremalof functional (24) over the space of variations119881 = 119867
120573
0(119868R119889)times
119867120573
0(119868C119889)
4 Nondifferentiable Helmholtz Problem
In this section we solve the inverse problem of the nondif-ferentiable calculus of variations in the Hamiltonian case Wefirst recall the usual way to derive the Helmholtz conditionsfollowing the presentation made by Santilli [28] Two mainderivations are available
(i) The first is related to the characterization of Hamilto-nian systems via the symplectic two-differential formand the fact that by duality the associated one-differential form to a Hamiltonian vector field isclosedmdashthe so-called integrability conditions
(ii) The second uses the characterization of Hamiltoniansystems via the self-adjointness of the Frechet deriva-tive of the differential operator associated with theequationmdashthe so-called Helmholtz conditions
Of course we have coincidence of the two procedures in theclassical case As there is no analogous of differential formin the framework of Cressonrsquos quantum calculus we followthe second way to obtain the nondifferentiable analogue of
Discrete Dynamics in Nature and Society 5
the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way
41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N
411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by
for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by
412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form
119889
119889119905(119902
119901) = (
119883119902 (119902 119901)
119883119901 (119902 119901)) (27)
where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902
and 119901 The associated differential operator is written as
119874119883 (119902 119901) = (
minus 119883119902 (119902 119901)
minus 119883119901 (119902 119901)) (28)
A natural notion of adjoint for a differential operator is thendefined as follows
Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast
119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by
⟨119860 sdot 119891 119892⟩1198712119869
= ⟨119860lowast
119869sdot 119892 119891⟩
1198712 119869 (29)
An operator 119860 will be called self-adjoint if 119860 = 119860lowast
119869with
respect to the 1198712 symplectic scalar product
413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])
be a vector field defined by 119883(119902 119901)⊤
= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871
2 symplecticscalar product In this case the Hamiltonian is given by
The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)
Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤
=
(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if
By definition we obtain the expression of the adjoint 119863119874lowast
◻119883
of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar
product
In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions
Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions
We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems
Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)
⊤=
(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by
Proof If 119883 is Hamiltonian then there exists a function 119867
R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian
We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]
Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867
120572(119868C119889) be
such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867
120572(119868C119889)
51 The Linear Case Let us consider the discrete nondiffer-entiable system
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 + 120575119901
(45)
where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 minus 120572119901
(46)
Using formula (40) we compute explicitly the Hamiltonianwhich is given by
52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby
=119901
119898
= minus1198801015840(119902)
(48)
with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by
119867(119902 119901) =1
21198981199012+ 119880 (119902) (49)
Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by
◻119902
◻119905=
119901
119898
◻119901
◻119905= minus1198801015840(119902)
(50)
The Hamiltonian Helmholtz conditions are clearly satisfied
Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)
6 Conclusion
We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions
References
[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002
[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012
[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013
8 Discrete Dynamics in Nature and Society
[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010
[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001
[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011
[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012
[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013
[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013
[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013
[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005
[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015
[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009
[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010
[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009
[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011
[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011
[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014
[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010
[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011
[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013
[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014
[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941
[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896
[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897
[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898
[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978
[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013
[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004
[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014
[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999
[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203
[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117
[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015
[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015
[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978
Lagrangian Helmholtz problem has been studied and solvedby Douglas [24] Mayer [25] and Hirsch [26 27]TheHamil-tonian Helmholtz problem has been studied and solved upto our knowledge by Santilli in his book [28] Generalizationof this problem in the discrete calculus of variations frame-work has been done in [29 30] in the discrete Lagrangiancase In the case of time-scale calculus that is a mixingbetween continuous and discrete subintervals of time see [31]for a necessary condition for a dynamic integrodifferentialequation to be an Euler-Lagrange equation on time scalesFor the Hamiltonian case it has been done for the discretecalculus of variations in [32] using the framework of [33]and in [34] using a discrete embedding procedure derivedin [35] In the case of time-scale calculus it has been donein [36] for the Stratonovich stochastic calculus see [37] Herewe give the Helmholtz theorem for Hamiltonian systemsin the case of nondifferentiable Hamiltonian systems in theframework of Cressonrsquos quantum calculus By definition thenondifferentiable calculus extends the differentiable calculusSuch as in the discrete time-scale and stochastic cases werecover the same conditions of existence of a Hamiltonianstructure
The paper is organized as follows In Section 2 we givesome generalities and notions about the nondifferentiablecalculus introduced in [17] the so-called Cressonrsquos quantumcalculus In Section 3 we remind definitions and resultsabout classical and nondifferentiableHamiltonian systems InSection 4 we give a brief survey of the classical HelmholtzHamiltonian problem and then we prove the main resultof this papermdashthe nondifferentiable Hamiltonian Helmholtztheorem Finally we give two applications of our results inSection 5 and we end in Section 6 with conclusions andfuture work
2 Cressonrsquos Quantum Calculus
We briefly review the necessary concepts and results of thequantum calculus [17]
21 Definitions Let X119889 denote the set R119889 or C119889 119889 isin N andlet 119868 be an open set in R with [119886 119887] sub 119868 119886 lt 119887 We denote byF(119868X119889) the set of functions 119891 119868 rarr X119889 and by C0(119868X119889)the subset of functions ofF(119868X119889) which are continuous
Definition 1 (Holderian functions [17]) Let 119891 isin C0(119868R119889)Let 119905 isin 119868 Function 119891 is said to be 120572-Holderian 0 lt 120572 lt 1 atpoint 119905 if there exist positive constants 120598 gt 0 and 119888 gt 0 suchthat |119905 minus 119905
1015840| ⩽ 120598 implies 119891(119905) minus 119891(119905
1015840) ⩽ 119888|119905 minus 119905
1015840|120572 for all 1199051015840 isin 119868
where sdot is a norm on R119889
The set of Holderian functions of Holder exponent 120572 forsome 120572 is denoted by 119867
120572(119868R119889) The quantum derivative is
defined as follows
Definition 2 (the 120598-left and 120598-right quantum derivatives [17])Let 119891 isin C0(119868R119889) For all 120598 gt 0 the 120598-left and 120598-right
quantum derivatives of 119891 denoted respectively by 119889minus
120598119891 and
119889+
120598119891 are defined by
119889minus
120598119891 (119905) =
119891 (119905) minus 119891 (119905 minus 120598)
120598
119889+
120598119891 (119905) =
119891 (119905 + 120598) minus 119891 (119905)
120598
(1)
Remark 3 The 120598-left and 120598-right quantum derivatives of acontinuous function 119891 correspond to the classical derivativeof the 120598-mean function 119891
120590
120598defined by
119891120590
120598(119905) =
120590
120598int
119905+120590120598
119905
119891 (119904) 119889119904 120590 = plusmn (2)
The next operator generalizes the classical derivative
Definition 4 (the 120598-scale derivative [17]) Let 119891 isin C0(119868R119889)For all 120598 gt 0 the 120598-scale derivative of 119891 denoted by ◻120598119891◻119905is defined by
◻120598119891
◻119905=
1
2[(119889+
120598119891 + 119889minus
120598119891) + 119894120583 (119889
+
120598119891 minus 119889minus
120598119891)] (3)
where 119894 is the imaginary unit and 120583 isin minus1 1 0 minus119894 119894
Remark 5 If 119891 is differentiable then one can take the limit ofthe scale derivative when 120598 goes to zero We then obtain theclassical derivative 119889119891119889119905 of 119891
We also need to extend the scale derivative to complexvalued functions
Definition 6 (see [17]) Let 119891 isin C0(119868C119889) be a continuouscomplex valued function For all 120598 gt 0 the 120598-scale derivativeof 119891 denoted by ◻120598119891◻119905 is defined by
◻120598119891
◻119905=
◻120598Re (119891)◻119905
+ 119894◻120598 Im (119891)
◻119905 (4)
where Re(119891) and Im(119891) denote the real and imaginary partof 119891 respectively
In Definition 4 the 120598-scale derivative depends on 120598which is a free parameter related to the smoothing orderof the function This brings many difficulties in applicationsto physics when one is interested in particular equationsthat do not depend on an extra parameter To solve theseproblems the authors of [17] introduced a procedure toextract information independent of 120598 but related with themean behavior of the function
Definition 7 (see [17]) Let C0conv(119868 times ]0 1]R119889) sube C0(119868 times ]0
1]R119889) be such that for any function119891 isin C0conv(119868 times ]0 1]R119889)
the lim120598rarr0119891(119905 120598) exists for any 119905 isin 119868 We denote by 119864 a com-plementary space ofC0conv(119868 times ]0 1]R119889) inC0(119868 times ]0 1]R119889)We define the projection map 120587 by
120587 C0
conv (119868 times ]0 1] R119889) oplus 119864 997888rarr C
Remark 10 For 119891 isin C1(119868R119889) and 119892 isin C1(119868R119889) oneobtains from (8) the classical Leibniz rule (119891 sdot 119892)
1015840= 1198911015840sdot 119892 +
119891 sdot 1198921015840
Definition 11 We denote by C1◻the set of continuous func-
tions 119902 isin C0([119886 119887]R119889) such that ◻119902◻119905 isin C0(119868R119889)
Theorem 12 (the quantum version of the fundamental theo-rem of calculus [17]) Let 119891 isin C1
◻([119886 119887]R119889) be such that
lim120598rarr0
int
119887
119886
(◻120598119891
◻119905)
119864
(119905) 119889119905 = 0 (9)
Then
int
119887
119886
◻119891
◻119905(119905) 119889119905 = 119891 (119887) minus 119891 (119886) (10)
22 Nondifferentiable Calculus of Variations In [17] thecalculus of variations with quantum derivatives is introducedand respective Euler-Lagrange equations derived without thedependence of 120598
Definition 13 An admissible Lagrangian 119871 is a continuousfunction 119871 R times R119889 times C119889 rarr C such that 119871(119905 119909 V) is holo-morphic with respect to V and differentiable with respect to119909 Moreover 119871(119905 119909 V) isin R when V isin R119889 119871(119905 119909 V) isin C whenV isin C119889
An admissible Lagrangian function 119871 RtimesRdtimesC119889 rarr C
defines a functional onC1(119868R119889) denoted by
The nondifferentiable embedding procedure allows us todefine a natural extension of the classical Euler-Lagrangeequation in the nondifferentiable context
Definition 15 (see [17]) The nondifferentiable LagrangianfunctionalL◻ associated withL is given by
119867120572(119868R119889) with 120572 + 120573 gt 1 A 119867
120573
0-variation of 119902 is a function
of the form 119902 + ℎ where ℎ isin 119867120573
119874 We denote by 119863L◻(119902)(ℎ)
the quantity
lim120598rarr0
L◻ (119902 + 120598ℎ) minusL◻ (119902)
120598(14)
if there exists the so-called Frechet derivative ofL◻ at point119902 in direction ℎ
Definition 16 (nondifferentiable extremals) A 119867120573
0-extremal
curve of the functionalL◻ is a curve 119902 isin 119867120572(119868R119889) satisfying
119863L◻(119902)(ℎ) = 0 for any ℎ isin 119867120573
0
Theorem 17 (nondifferentiable Euler-Lagrange equations[17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be an admissibleLagrangian of class C2 We assume that 120574 isin 119867
120572(119868R119889) such
that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that 119871(119905 120574(119905)
◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin 119867120573
0(119868R119889)
A curve 120574 satisfying the nondifferentiable Euler-Lagrangeequation
◻
◻119905[120597119871
120597V(119905 120574 (119905)
◻120574 (119905)
◻119905)] =
120597119871
120597119909(119905 120574 (119905)
◻120574 (119905)
◻119905) (15)
is an extremal curve of functional (13)
3 Reminder about Hamiltonian Systems
We now recall the main concepts and results of both classicaland Cressonrsquos nondifferentiable Hamiltonian systems
31 Classical Hamiltonian Systems Let 119871 be an admissibleLagrangian function If 119871 satisfies the so-called Legendreproperty then we can associate to 119871 a Hamiltonian functiondenoted by119867
4 Discrete Dynamics in Nature and Society
Definition 18 Let 119871 be an admissible Lagrangian functionTheLagrangian119871 is said to satisfy the Legendre property if themapping V 997891rarr (120597119871120597V)(119905 119909 V) is invertible for any (119905 119902 V) isin
119868 timesR119889 times C119889
If we introduce a new variable
119901 =120597119871
120597V(119905 119902 V) (16)
and 119871 satisfies the Legendre property then we can find afunction 119891 such that
V = 119891 (119905 119902 119901) (17)
Using this notation we have the following definition
Definition 19 Let 119871 be an admissible Lagrangian functionsatisfying the Legendre property The Hamiltonian function119867 associated with 119871 is given by
Theorem 20 (Hamiltonrsquos least-action principle) The curve(119902 119901) isin C(119868R119889)timesC(119868C119889) is an extremal of the Hamiltonianfunctional
denotes the symplectic matrix with 119868119889 being the identitymatrix on R119889
32 NondifferentiableHamiltonian Systems Thenondifferen-tiable embedding induces a change in the phase space withrespect to the classical case As a consequence we have towork with variables (119909 119901) that belong to R119889 times C119889 and notonly to R119889 timesR119889 as usual
Definition 21 (nondifferentiable embedding of Hamiltoniansystems [17]) The nondifferentiable embedded Hamiltoniansystem (20) is given by
The nondifferentiable calculus of variations allows us toderive the extremals forH◻
Theorem 22 (nondifferentiable Hamiltonrsquos least-action prin-ciple [17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be anadmissible C2-Lagrangian We assume that 120574 isin 119867
120572(119868R119889)
such that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that
119871(119905 120574(119905) ◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin
119867120573
0(119868R119889) Let 119867 be the corresponding Hamiltonian defined
by (18) A curve 120574 997891rarr (119905 119902(119905) 119901(119905)) isin 119868 times R119889 times C119889 solution ofthe nondifferentiable Hamiltonian system (23) is an extremalof functional (24) over the space of variations119881 = 119867
120573
0(119868R119889)times
119867120573
0(119868C119889)
4 Nondifferentiable Helmholtz Problem
In this section we solve the inverse problem of the nondif-ferentiable calculus of variations in the Hamiltonian case Wefirst recall the usual way to derive the Helmholtz conditionsfollowing the presentation made by Santilli [28] Two mainderivations are available
(i) The first is related to the characterization of Hamilto-nian systems via the symplectic two-differential formand the fact that by duality the associated one-differential form to a Hamiltonian vector field isclosedmdashthe so-called integrability conditions
(ii) The second uses the characterization of Hamiltoniansystems via the self-adjointness of the Frechet deriva-tive of the differential operator associated with theequationmdashthe so-called Helmholtz conditions
Of course we have coincidence of the two procedures in theclassical case As there is no analogous of differential formin the framework of Cressonrsquos quantum calculus we followthe second way to obtain the nondifferentiable analogue of
Discrete Dynamics in Nature and Society 5
the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way
41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N
411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by
for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by
412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form
119889
119889119905(119902
119901) = (
119883119902 (119902 119901)
119883119901 (119902 119901)) (27)
where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902
and 119901 The associated differential operator is written as
119874119883 (119902 119901) = (
minus 119883119902 (119902 119901)
minus 119883119901 (119902 119901)) (28)
A natural notion of adjoint for a differential operator is thendefined as follows
Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast
119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by
⟨119860 sdot 119891 119892⟩1198712119869
= ⟨119860lowast
119869sdot 119892 119891⟩
1198712 119869 (29)
An operator 119860 will be called self-adjoint if 119860 = 119860lowast
119869with
respect to the 1198712 symplectic scalar product
413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])
be a vector field defined by 119883(119902 119901)⊤
= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871
2 symplecticscalar product In this case the Hamiltonian is given by
The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)
Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤
=
(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if
By definition we obtain the expression of the adjoint 119863119874lowast
◻119883
of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar
product
In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions
Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions
We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems
Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)
⊤=
(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by
Proof If 119883 is Hamiltonian then there exists a function 119867
R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian
We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]
Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867
120572(119868C119889) be
such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867
120572(119868C119889)
51 The Linear Case Let us consider the discrete nondiffer-entiable system
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 + 120575119901
(45)
where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 minus 120572119901
(46)
Using formula (40) we compute explicitly the Hamiltonianwhich is given by
52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby
=119901
119898
= minus1198801015840(119902)
(48)
with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by
119867(119902 119901) =1
21198981199012+ 119880 (119902) (49)
Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by
◻119902
◻119905=
119901
119898
◻119901
◻119905= minus1198801015840(119902)
(50)
The Hamiltonian Helmholtz conditions are clearly satisfied
Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)
6 Conclusion
We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions
References
[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002
[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012
[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013
8 Discrete Dynamics in Nature and Society
[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010
[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001
[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011
[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012
[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013
[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013
[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013
[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005
[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015
[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009
[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010
[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009
[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011
[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011
[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014
[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010
[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011
[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013
[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014
[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941
[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896
[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897
[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898
[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978
[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013
[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004
[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014
[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999
[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203
[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117
[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015
[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015
[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978
Remark 10 For 119891 isin C1(119868R119889) and 119892 isin C1(119868R119889) oneobtains from (8) the classical Leibniz rule (119891 sdot 119892)
1015840= 1198911015840sdot 119892 +
119891 sdot 1198921015840
Definition 11 We denote by C1◻the set of continuous func-
tions 119902 isin C0([119886 119887]R119889) such that ◻119902◻119905 isin C0(119868R119889)
Theorem 12 (the quantum version of the fundamental theo-rem of calculus [17]) Let 119891 isin C1
◻([119886 119887]R119889) be such that
lim120598rarr0
int
119887
119886
(◻120598119891
◻119905)
119864
(119905) 119889119905 = 0 (9)
Then
int
119887
119886
◻119891
◻119905(119905) 119889119905 = 119891 (119887) minus 119891 (119886) (10)
22 Nondifferentiable Calculus of Variations In [17] thecalculus of variations with quantum derivatives is introducedand respective Euler-Lagrange equations derived without thedependence of 120598
Definition 13 An admissible Lagrangian 119871 is a continuousfunction 119871 R times R119889 times C119889 rarr C such that 119871(119905 119909 V) is holo-morphic with respect to V and differentiable with respect to119909 Moreover 119871(119905 119909 V) isin R when V isin R119889 119871(119905 119909 V) isin C whenV isin C119889
An admissible Lagrangian function 119871 RtimesRdtimesC119889 rarr C
defines a functional onC1(119868R119889) denoted by
The nondifferentiable embedding procedure allows us todefine a natural extension of the classical Euler-Lagrangeequation in the nondifferentiable context
Definition 15 (see [17]) The nondifferentiable LagrangianfunctionalL◻ associated withL is given by
119867120572(119868R119889) with 120572 + 120573 gt 1 A 119867
120573
0-variation of 119902 is a function
of the form 119902 + ℎ where ℎ isin 119867120573
119874 We denote by 119863L◻(119902)(ℎ)
the quantity
lim120598rarr0
L◻ (119902 + 120598ℎ) minusL◻ (119902)
120598(14)
if there exists the so-called Frechet derivative ofL◻ at point119902 in direction ℎ
Definition 16 (nondifferentiable extremals) A 119867120573
0-extremal
curve of the functionalL◻ is a curve 119902 isin 119867120572(119868R119889) satisfying
119863L◻(119902)(ℎ) = 0 for any ℎ isin 119867120573
0
Theorem 17 (nondifferentiable Euler-Lagrange equations[17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be an admissibleLagrangian of class C2 We assume that 120574 isin 119867
120572(119868R119889) such
that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that 119871(119905 120574(119905)
◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin 119867120573
0(119868R119889)
A curve 120574 satisfying the nondifferentiable Euler-Lagrangeequation
◻
◻119905[120597119871
120597V(119905 120574 (119905)
◻120574 (119905)
◻119905)] =
120597119871
120597119909(119905 120574 (119905)
◻120574 (119905)
◻119905) (15)
is an extremal curve of functional (13)
3 Reminder about Hamiltonian Systems
We now recall the main concepts and results of both classicaland Cressonrsquos nondifferentiable Hamiltonian systems
31 Classical Hamiltonian Systems Let 119871 be an admissibleLagrangian function If 119871 satisfies the so-called Legendreproperty then we can associate to 119871 a Hamiltonian functiondenoted by119867
4 Discrete Dynamics in Nature and Society
Definition 18 Let 119871 be an admissible Lagrangian functionTheLagrangian119871 is said to satisfy the Legendre property if themapping V 997891rarr (120597119871120597V)(119905 119909 V) is invertible for any (119905 119902 V) isin
119868 timesR119889 times C119889
If we introduce a new variable
119901 =120597119871
120597V(119905 119902 V) (16)
and 119871 satisfies the Legendre property then we can find afunction 119891 such that
V = 119891 (119905 119902 119901) (17)
Using this notation we have the following definition
Definition 19 Let 119871 be an admissible Lagrangian functionsatisfying the Legendre property The Hamiltonian function119867 associated with 119871 is given by
Theorem 20 (Hamiltonrsquos least-action principle) The curve(119902 119901) isin C(119868R119889)timesC(119868C119889) is an extremal of the Hamiltonianfunctional
denotes the symplectic matrix with 119868119889 being the identitymatrix on R119889
32 NondifferentiableHamiltonian Systems Thenondifferen-tiable embedding induces a change in the phase space withrespect to the classical case As a consequence we have towork with variables (119909 119901) that belong to R119889 times C119889 and notonly to R119889 timesR119889 as usual
Definition 21 (nondifferentiable embedding of Hamiltoniansystems [17]) The nondifferentiable embedded Hamiltoniansystem (20) is given by
The nondifferentiable calculus of variations allows us toderive the extremals forH◻
Theorem 22 (nondifferentiable Hamiltonrsquos least-action prin-ciple [17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be anadmissible C2-Lagrangian We assume that 120574 isin 119867
120572(119868R119889)
such that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that
119871(119905 120574(119905) ◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin
119867120573
0(119868R119889) Let 119867 be the corresponding Hamiltonian defined
by (18) A curve 120574 997891rarr (119905 119902(119905) 119901(119905)) isin 119868 times R119889 times C119889 solution ofthe nondifferentiable Hamiltonian system (23) is an extremalof functional (24) over the space of variations119881 = 119867
120573
0(119868R119889)times
119867120573
0(119868C119889)
4 Nondifferentiable Helmholtz Problem
In this section we solve the inverse problem of the nondif-ferentiable calculus of variations in the Hamiltonian case Wefirst recall the usual way to derive the Helmholtz conditionsfollowing the presentation made by Santilli [28] Two mainderivations are available
(i) The first is related to the characterization of Hamilto-nian systems via the symplectic two-differential formand the fact that by duality the associated one-differential form to a Hamiltonian vector field isclosedmdashthe so-called integrability conditions
(ii) The second uses the characterization of Hamiltoniansystems via the self-adjointness of the Frechet deriva-tive of the differential operator associated with theequationmdashthe so-called Helmholtz conditions
Of course we have coincidence of the two procedures in theclassical case As there is no analogous of differential formin the framework of Cressonrsquos quantum calculus we followthe second way to obtain the nondifferentiable analogue of
Discrete Dynamics in Nature and Society 5
the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way
41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N
411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by
for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by
412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form
119889
119889119905(119902
119901) = (
119883119902 (119902 119901)
119883119901 (119902 119901)) (27)
where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902
and 119901 The associated differential operator is written as
119874119883 (119902 119901) = (
minus 119883119902 (119902 119901)
minus 119883119901 (119902 119901)) (28)
A natural notion of adjoint for a differential operator is thendefined as follows
Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast
119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by
⟨119860 sdot 119891 119892⟩1198712119869
= ⟨119860lowast
119869sdot 119892 119891⟩
1198712 119869 (29)
An operator 119860 will be called self-adjoint if 119860 = 119860lowast
119869with
respect to the 1198712 symplectic scalar product
413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])
be a vector field defined by 119883(119902 119901)⊤
= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871
2 symplecticscalar product In this case the Hamiltonian is given by
The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)
Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤
=
(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if
By definition we obtain the expression of the adjoint 119863119874lowast
◻119883
of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar
product
In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions
Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions
We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems
Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)
⊤=
(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by
Proof If 119883 is Hamiltonian then there exists a function 119867
R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian
We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]
Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867
120572(119868C119889) be
such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867
120572(119868C119889)
51 The Linear Case Let us consider the discrete nondiffer-entiable system
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 + 120575119901
(45)
where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 minus 120572119901
(46)
Using formula (40) we compute explicitly the Hamiltonianwhich is given by
52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby
=119901
119898
= minus1198801015840(119902)
(48)
with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by
119867(119902 119901) =1
21198981199012+ 119880 (119902) (49)
Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by
◻119902
◻119905=
119901
119898
◻119901
◻119905= minus1198801015840(119902)
(50)
The Hamiltonian Helmholtz conditions are clearly satisfied
Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)
6 Conclusion
We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions
References
[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002
[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012
[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013
8 Discrete Dynamics in Nature and Society
[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010
[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001
[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011
[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012
[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013
[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013
[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013
[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005
[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015
[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009
[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010
[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009
[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011
[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011
[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014
[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010
[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011
[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013
[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014
[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941
[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896
[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897
[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898
[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978
[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013
[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004
[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014
[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999
[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203
[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117
[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015
[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015
[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978
Definition 18 Let 119871 be an admissible Lagrangian functionTheLagrangian119871 is said to satisfy the Legendre property if themapping V 997891rarr (120597119871120597V)(119905 119909 V) is invertible for any (119905 119902 V) isin
119868 timesR119889 times C119889
If we introduce a new variable
119901 =120597119871
120597V(119905 119902 V) (16)
and 119871 satisfies the Legendre property then we can find afunction 119891 such that
V = 119891 (119905 119902 119901) (17)
Using this notation we have the following definition
Definition 19 Let 119871 be an admissible Lagrangian functionsatisfying the Legendre property The Hamiltonian function119867 associated with 119871 is given by
Theorem 20 (Hamiltonrsquos least-action principle) The curve(119902 119901) isin C(119868R119889)timesC(119868C119889) is an extremal of the Hamiltonianfunctional
denotes the symplectic matrix with 119868119889 being the identitymatrix on R119889
32 NondifferentiableHamiltonian Systems Thenondifferen-tiable embedding induces a change in the phase space withrespect to the classical case As a consequence we have towork with variables (119909 119901) that belong to R119889 times C119889 and notonly to R119889 timesR119889 as usual
Definition 21 (nondifferentiable embedding of Hamiltoniansystems [17]) The nondifferentiable embedded Hamiltoniansystem (20) is given by
The nondifferentiable calculus of variations allows us toderive the extremals forH◻
Theorem 22 (nondifferentiable Hamiltonrsquos least-action prin-ciple [17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be anadmissible C2-Lagrangian We assume that 120574 isin 119867
120572(119868R119889)
such that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that
119871(119905 120574(119905) ◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin
119867120573
0(119868R119889) Let 119867 be the corresponding Hamiltonian defined
by (18) A curve 120574 997891rarr (119905 119902(119905) 119901(119905)) isin 119868 times R119889 times C119889 solution ofthe nondifferentiable Hamiltonian system (23) is an extremalof functional (24) over the space of variations119881 = 119867
120573
0(119868R119889)times
119867120573
0(119868C119889)
4 Nondifferentiable Helmholtz Problem
In this section we solve the inverse problem of the nondif-ferentiable calculus of variations in the Hamiltonian case Wefirst recall the usual way to derive the Helmholtz conditionsfollowing the presentation made by Santilli [28] Two mainderivations are available
(i) The first is related to the characterization of Hamilto-nian systems via the symplectic two-differential formand the fact that by duality the associated one-differential form to a Hamiltonian vector field isclosedmdashthe so-called integrability conditions
(ii) The second uses the characterization of Hamiltoniansystems via the self-adjointness of the Frechet deriva-tive of the differential operator associated with theequationmdashthe so-called Helmholtz conditions
Of course we have coincidence of the two procedures in theclassical case As there is no analogous of differential formin the framework of Cressonrsquos quantum calculus we followthe second way to obtain the nondifferentiable analogue of
Discrete Dynamics in Nature and Society 5
the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way
41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N
411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by
for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by
412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form
119889
119889119905(119902
119901) = (
119883119902 (119902 119901)
119883119901 (119902 119901)) (27)
where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902
and 119901 The associated differential operator is written as
119874119883 (119902 119901) = (
minus 119883119902 (119902 119901)
minus 119883119901 (119902 119901)) (28)
A natural notion of adjoint for a differential operator is thendefined as follows
Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast
119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by
⟨119860 sdot 119891 119892⟩1198712119869
= ⟨119860lowast
119869sdot 119892 119891⟩
1198712 119869 (29)
An operator 119860 will be called self-adjoint if 119860 = 119860lowast
119869with
respect to the 1198712 symplectic scalar product
413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])
be a vector field defined by 119883(119902 119901)⊤
= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871
2 symplecticscalar product In this case the Hamiltonian is given by
The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)
Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤
=
(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if
By definition we obtain the expression of the adjoint 119863119874lowast
◻119883
of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar
product
In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions
Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions
We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems
Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)
⊤=
(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by
Proof If 119883 is Hamiltonian then there exists a function 119867
R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian
We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]
Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867
120572(119868C119889) be
such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867
120572(119868C119889)
51 The Linear Case Let us consider the discrete nondiffer-entiable system
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 + 120575119901
(45)
where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 minus 120572119901
(46)
Using formula (40) we compute explicitly the Hamiltonianwhich is given by
52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby
=119901
119898
= minus1198801015840(119902)
(48)
with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by
119867(119902 119901) =1
21198981199012+ 119880 (119902) (49)
Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by
◻119902
◻119905=
119901
119898
◻119901
◻119905= minus1198801015840(119902)
(50)
The Hamiltonian Helmholtz conditions are clearly satisfied
Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)
6 Conclusion
We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions
References
[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002
[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012
[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013
8 Discrete Dynamics in Nature and Society
[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010
[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001
[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011
[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012
[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013
[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013
[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013
[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005
[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015
[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009
[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010
[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009
[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011
[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011
[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014
[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010
[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011
[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013
[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014
[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941
[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896
[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897
[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898
[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978
[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013
[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004
[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014
[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999
[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203
[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117
[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015
[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015
[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978
for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by
412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form
119889
119889119905(119902
119901) = (
119883119902 (119902 119901)
119883119901 (119902 119901)) (27)
where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902
and 119901 The associated differential operator is written as
119874119883 (119902 119901) = (
minus 119883119902 (119902 119901)
minus 119883119901 (119902 119901)) (28)
A natural notion of adjoint for a differential operator is thendefined as follows
Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast
119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by
⟨119860 sdot 119891 119892⟩1198712119869
= ⟨119860lowast
119869sdot 119892 119891⟩
1198712 119869 (29)
An operator 119860 will be called self-adjoint if 119860 = 119860lowast
119869with
respect to the 1198712 symplectic scalar product
413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])
be a vector field defined by 119883(119902 119901)⊤
= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871
2 symplecticscalar product In this case the Hamiltonian is given by
The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)
Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤
=
(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if
By definition we obtain the expression of the adjoint 119863119874lowast
◻119883
of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar
product
In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions
Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions
We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems
Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)
⊤=
(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by
Proof If 119883 is Hamiltonian then there exists a function 119867
R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian
We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]
Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867
120572(119868C119889) be
such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867
120572(119868C119889)
51 The Linear Case Let us consider the discrete nondiffer-entiable system
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 + 120575119901
(45)
where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 minus 120572119901
(46)
Using formula (40) we compute explicitly the Hamiltonianwhich is given by
52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby
=119901
119898
= minus1198801015840(119902)
(48)
with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by
119867(119902 119901) =1
21198981199012+ 119880 (119902) (49)
Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by
◻119902
◻119905=
119901
119898
◻119901
◻119905= minus1198801015840(119902)
(50)
The Hamiltonian Helmholtz conditions are clearly satisfied
Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)
6 Conclusion
We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions
References
[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002
[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012
[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013
8 Discrete Dynamics in Nature and Society
[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010
[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001
[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011
[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012
[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013
[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013
[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013
[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005
[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015
[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009
[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010
[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009
[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011
[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011
[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014
[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010
[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011
[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013
[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014
[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941
[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896
[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897
[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898
[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978
[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013
[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004
[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014
[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999
[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203
[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117
[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015
[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015
[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978
By definition we obtain the expression of the adjoint 119863119874lowast
◻119883
of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar
product
In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions
Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions
We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems
Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)
⊤=
(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by
Proof If 119883 is Hamiltonian then there exists a function 119867
R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian
We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]
Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867
120572(119868C119889) be
such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867
120572(119868C119889)
51 The Linear Case Let us consider the discrete nondiffer-entiable system
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 + 120575119901
(45)
where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 minus 120572119901
(46)
Using formula (40) we compute explicitly the Hamiltonianwhich is given by
52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby
=119901
119898
= minus1198801015840(119902)
(48)
with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by
119867(119902 119901) =1
21198981199012+ 119880 (119902) (49)
Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by
◻119902
◻119905=
119901
119898
◻119901
◻119905= minus1198801015840(119902)
(50)
The Hamiltonian Helmholtz conditions are clearly satisfied
Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)
6 Conclusion
We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions
References
[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002
[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012
[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013
8 Discrete Dynamics in Nature and Society
[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010
[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001
[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011
[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012
[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013
[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013
[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013
[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005
[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015
[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009
[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010
[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009
[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011
[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011
[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014
[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010
[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011
[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013
[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014
[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941
[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896
[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897
[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898
[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978
[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013
[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004
[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014
[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999
[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203
[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117
[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015
[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015
[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978
We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]
Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867
120572(119868C119889) be
such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867
120572(119868C119889)
51 The Linear Case Let us consider the discrete nondiffer-entiable system
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 + 120575119901
(45)
where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform
◻119902
◻119905= 120572119902 + 120573119901
◻119901
◻119905= 120574119902 minus 120572119901
(46)
Using formula (40) we compute explicitly the Hamiltonianwhich is given by
52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby
=119901
119898
= minus1198801015840(119902)
(48)
with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by
119867(119902 119901) =1
21198981199012+ 119880 (119902) (49)
Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by
◻119902
◻119905=
119901
119898
◻119901
◻119905= minus1198801015840(119902)
(50)
The Hamiltonian Helmholtz conditions are clearly satisfied
Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)
6 Conclusion
We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions
References
[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002
[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012
[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013
8 Discrete Dynamics in Nature and Society
[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010
[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001
[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011
[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012
[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013
[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013
[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013
[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005
[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015
[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009
[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010
[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009
[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011
[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011
[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014
[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010
[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011
[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013
[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014
[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941
[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896
[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897
[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898
[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978
[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013
[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004
[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014
[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999
[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203
[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117
[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015
[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015
[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978
[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010
[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001
[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011
[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012
[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013
[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013
[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013
[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005
[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015
[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009
[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010
[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009
[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011
[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011
[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014
[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010
[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011
[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013
[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014
[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941
[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896
[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897
[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898
[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978
[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013
[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004
[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014
[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999
[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203
[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117
[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015
[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015
[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978