ROUTH REDUCTION AS AN EXAMPLE OF LAGRANGIAN REDUCTION KATARZYNA GRABOWSKA , PAWET URBANSKI arlkv : 1708.0976%1
ROUTH REDUCTION AS AN EXAMPLE
OF LAGRANGIAN REDUCTION
KATARZYNA GRABOWSKA,
PAWET URBANSKI
arlkv : 1708.0976%1
CALCULATIONS IN COORDINATES
R"+m
> ( q ,x , qi , .x ) ->L( q ,× , .q , .x)eR2L
cyclic variables : q= 0
212R( q ,×,p , .x)= poi - L(q ,x,oj,x
.
) Jq=0
edward ]°nnpwu+np=¥q=0gifts ftp.t?j.3I=o
1831-1907 IP= a Ra( x. x. ) = aq.ca) - LA ,q .ca ) , .x )
IRouthian depends only on ( xii )
WHAT IS GEOMETRIC PICTURE BEHIND THESE CALCULATIONS ?
I E. Marsden ,TSRatiu , Jsoneurle , F. Cantrijn , B. Langerock ,T . Metsdag ,E. Garcia . Toriano Andre 's
,] . Vankevschaver
, Mcrampin . . .
WHY DO IT AGAIN ?
D .
BQ I do ,T*T*Q - TT*Q - T*TQ
0¥pj'cdHA*a))=D=xaYdLGaD# ° "T*Q TQ
a } + }I¥5u"itionEASY V INTERESTING
RELATION
? ??_?
??_?
? ?? ?
?µHats ? ? ?
TOOLS :
→ SYMPLECTIC GEOMETRY : SYMPLECTIC REDUCTIONS,
LAGRANGIAN
SUBMANIFOLDS,
GENERATING OBJECTS :
GENERATING FUNCTION FUNCTION ON
A SUBMANIFOLD
f :Q→R QaCf→R
df(Q)cT*Q { yeT*Q : VJETC ( df ,v>=< y ,v>]cT*QT LAGRANGIAN 7
SUBMANIFOLD
itGENERATING C={ MEM : drfcm) - 0 }FAMILY fM → R { yeT*Q : g*y(m)=dfCm) }etQ
SYMPLECTIC RELATIONS ARE LAGRANGIAN
SUBMANIFOLDS 11
g :B - Pz
graph (g) c(Pz×Pz , Wz - 02 )¥2,13
ARE COTANGENT BUNDLES
WE CAN GENERATE THE GRAPH
COMPOSING SYMPLECTIC RELATIONS
MEANS ADDING GENERATING OBJECTS
EXAMPLET * to # THTHQ SYMPLECTOMORPHISM
L :TQ→R GENERATED BY
DLCTQ )cT*TQ TQ×T*Q > TQ×aT*Q > ( rfp )1→ - ( p ,O ) ERT
LAGRANGIAN
SUBMANIFOLDWHAT GENERATES Va( OILCTQD ?
TQ×aT*Q scrip ) - LCV ) - ( p ,v ) e R* * Q
TOOLS :
→ GEOMETRY OF AFFINE VALUES : AV - BUNDLES,
AFFINE PHASE
SPACES
AFFINE PHASE SPACE :
QXR( T.it ) ~ ( 9. g) ⇐ > d(q . g) (g) =O
tgff-§¥r("
'"
drcq ) :=[ ( r ,qD
PZ={ OITCQ ) ] ← AFFINE BUNDLE OVER QSYMPLECTIC
/ [ MODELED ON T*QMANIFOLD
Wz=Z*Wa2 :P2→T*Q
g-FZ -7 TQ 15 A BUNDLE
ATIYAH ALGEBROID FZ = T #OF AFFINE VALUES
n
,VECTOR BUNDLE OVER Q
ROUTH REDUCTION RELATION
Q - CONFIGURATION MANIFOLD
X - VECTOR FIELD ON Q - SYMMETRY
L :TQ→lR dtX.lyGEOMETRIC VERSION
OF A LAGRANGIAN WITHCYCLIC VARIABLE
YIN FACT
,WHAT IS IMPORTANT IS THE DISTRIBUTION
0×=< 04×7IT IS ONE DIMENSIONAL DISTRIBUTION
,DOUBLE
VECTOR SUBBUNDLE OF
TTQ
Lead7¥aTQ
ROUTH REDUCTION RELATION VIA HAMILTONIAN MECHANICS
8-+ * t * a
#TT*Q ¥
T*TQy)o, ,T*QyaTQ-7112 ( p ,v ) HLA )
- < p ,v >✓
to
T¥Q
→ IN HAMILTONIAN MECHANICS SYMMETRY is GIVEN BY
dt*X - COTANGENT LIFT OF XTHAMILTONIAN
VECTOR FIELD FOR T*Q2pl-)i×(p)=(p , X(Ta( p ) ) )→ PHASE TRAJECTORIES LIE IN LEVEL SETS OF I×
c×={ PET * Q : i×(p)=d } ~ WISOTROPIC SUBMANIFOLD , REDUCTIONWITH RESPECT TO Ca = DIVIDING BYT AFFINE SUBBUNDLE OFT*Q→Q LIFTED ( R ,t ) ACTION
WTH MODEL Co → Q
ROUTH REDUCTION RELATION VIA HAMILTONIAN MECHANICS
if =L MANIFOLD OF+ * Qsca -7 Pa Q→M=Q/p TRAJECTORIES
-\
/ OF X) gympllAFFINE BUNDLE OVER M =CT1c MANIFOLDMODELED ON T*M REDUCED AFFINE PHASE
SPACE
BQ XQT*T*Q c- TT*Q - T*TQ
µt*g HalB
✓
TQ
txpa c- TB - ? ? ?( np.ec , , . .eu , ,oµ gene ... ,gx . , . - aBY A FUNCTION EQUAL ZERO ON
T*Q×P.
> Caxpa f. ( Pitt > LG ) - < p ,v )Pa
ROUTH REDUCTION RELATION VIA HAMILTONIAN MECHANICS
HERE COMES AV - GEOMETRY !
Q×R : ( q ,v)~ ( yscq ) , ntsx )R ' 9 ' "
y |=nQFLOW Ot X Ys
Z&e= Q×lR/~ Z&→M[ qiiftt :=[q ,n+t]=[ys( g) irtttsa ]
BEPZ , Lv
BTxp c- TB - PFZa ×
WHAT is it ? ?
FZ &= TZ%pp) ATIYAH ALGEBROID
TZ,
→ TM AV - BUNDLEg-
FUNDAMENTAL VECTOR
FIELD OF R - ACTIONWETZ
. [w]ts= [ wts } ] ON Z×
PFZ.
- AFFINE DIFFERENTIALS OF SECTIONS OF FZ,
→ TM
ROUTHIAN is NOT A FUNCTION ONTM BUT A SECTIONOF AN AV - BUNDLE
AFFINE ANALOG OF A MAP T*QXaTQ 3 ( p ,v ) - ) ( p ,v > EIR
PZ, XMTM a ( p ,v ) 1- < p ,v > e FZ ,
>
2,
ftp=dr( m ) < pie > =[ Trcv ) ]
mm
✓
B a
T*pZa-
TPZ. - PFZ
,
-Txtpzaepfza
SYMTPLECTOMORPHISMGENERATED BY A SECTION OVER A
SUBMANIFOLD
Pzaxtmo Pzxmtm - FZ , ( p . ,w ) - > < Pa ,w >
C&×aTQ → R
TM×mCx×aTQ
- FZ×
| ( Rr ) - La - < p ,v > |v .P tm ( w ,P,o)1→ Le )
- < p ,v > + Cpa ,w >a
TTGENERATING FAMILY
WITH PARAMETERS PIV. .
BQ XQT*T*Q c- TT*Q - T*TQ
to t ? ?B aTxpz- TPZ. - PFZ×
QUESTIONS :
2.
15 IT POSSIBLE to g|µpL , ,=y TMXMCYQTQ
- FZ×
y Fm ( w ,p,o)l→LH- < p ,v > + Cpa ,w >
YES, BUT NOT ALWAYS TO ONE ROUTHIAN
SECTION → EXAMPLE
2. 15 IT POSSIBLE TO GO DIRECTLY FROM T*TQ TO PFZ× ?^
,YES
,BUT YO HAVE TO INCLUDE VALUES
OF GENERATING OBJECTS
↳LAGRANGIAN REDUCTION
ANSWER TO QUESTION 1 WITH AN EXAMPLE :
TQ -
FZ×TM×mG×aTQ - FZ×/ → / vi→L(v ) - < p ,v > + < p . ,w ). vtm ( w ,p,o)t
> La ) - < pie > + Cpa ,w > TM
TQ 20h > WETM
FORMULA DOES NOT DEPEND ON P ,ANY PEG OVER to ( v ) IS OK !
EXAMPLE :
Q=R2a(×,y ) L :TQ > ( x ,y ,x.
,j ) 1- x. y.
-
y'
er M=R⇒( y )
X=2*f PARAMETER
ROUTHIAN FAMILY READS ( y , .x , g.) 1- x. y.
-
y'
. xx.
ER
T*Tm > { ( yijioyb ) : j=& , a= -2g }TT*Mo{ ( yip , jip ) : j=x , p= -2g } Xn(y,p)=&3J
- 2yFp
h(y,p)=Lp+y2
ANSWER TO QUESTION 2
RTHE RELATION T*TQ - DPFZ
,15 THE AFFINE PHASE LIFT OF THE
REDUCED TANGENT RELATION F(Q×lR ) → 72£ OF THE PROJECTION
Q×1R→Z:QXIR - Za ( q ,r)1→[q ,v]={ ( yscq ) , Ntsd ) } AV - BUNDLE MORPHISMTsaFCQXIRFTQHR - FZ , ( v ,t)i→[yt]={ ( Tyscv )+sX( q ) , ttxs ) ]AV - BUNDLE MORPHISM
ROUTH REDUCTION R IS PF }× . IT IS GENERATED BYNONTRIVIAL SECTION OF CERTAIN AV - BUNDLE
T*T*Q-D T* PZ,
PHASE LIFT GENERATED
T*Q3G→ PZ , ✓ By FUNCTON ZERO ON CLXMPZL
FINAL COMMENT ON ( NOT REALLY ) A GENERALISATION
LAGRANGIAN WITH SYMMETRY of,
X.L=O
DTX -3 OECDTX ) DLCTQ) c 0×05
weather.INT?hNesaur..3YIsEEutoINt0tva#
↳ONE CAN SHOW THAT IT DEFINES X UP TO
MULTIPLICATION BY NUMBERS
NO DISTINGUISHED PARAMETERIZATION OF {Cd } BY x. . .