Manifolds, Vector Fields & Flows - Drexel Universityhgk22/courses/MEM636_638/Lecture2.pdf · Manifolds, Vector Fields & Flows Harry G. Kwatny Department of Mechanical Engineering

Manifolds, Vector Fields & Flows

Harry G. KwatnyDepartment of Mechanical Engineering & Mechanics

Drexel University

Outline

From Flat to Curved State Space – Why?ManifoldsMaximum rank conditionRegular manifoldsTangent space & tangent bundleDifferential mapVector fields & flowsLie bracket

From Flat to Curved State Space

Global models may require it: Mechanical systems spatial rotationElectric power systems DAE description

Correct local approximations at least require acknowledging itEven if not required it may have conceptual benefits Computation often involves flat local approximations – but this may not be necessary (e.g. quaternionsvs. Euler angles)

Representations of Surfaces

Explicit ImplicitParametric

( )y g x=( , ) 0f x y =

1 2( ), ( ),x h s y h s s U R= = ∈ ⊂

2

2 2

11 0

cos , sin , [0,2 )

y xx yx s y s s π

= ± −

+ − == = ∈

x

y

Examples: Parametrically defined manifolds

cos sinsin sin [0, 2 ), [0, )

cos

t uf t u t u

uπ π

⎡ ⎤⎢ ⎥= ∈ ∈⎢ ⎥⎢ ⎥⎣ ⎦

cos (3 cos )sin (3 cos ) [0,2 ), [0, 2 )

sin

t uf t u t u

uπ π

+⎡ ⎤⎢ ⎥= + ∈ ∈⎢ ⎥⎢ ⎥⎣ ⎦

Definition - Manifold

( )

An is a set together with a countablecollection of subsets and one-to-one mappings onto open

subsets of , : , with the following properties: the pair , is

-dimensional man old

cal

if

i

mi i i

i i

MU M

R U VU

m

ϕϕ

⊂

→

1

led a the coordinate chartes cover , on the overlap of any pair of charts the composite map is a smooth function

: ( ) ( )

if and

coordinate

are distinct poin

cha

t

t

s

r

j i i i j j i j

i j

M

f U U U U

p U q U

φ φ φ φ−= ∩ → ∩

∈ ∈

( ) ( )( ) ( )1 1

of , then there are neighborhoods

of and of such thati i j j

i j

M

W p V U q V

W U

ϕ ϕ

ϕ ϕ− −

∈ ∈

∩ = ∅

Definition: Manifold

1−= ijf ϕϕ

jϕ

iϕ

iU jU

iV

jV

Example: Planet Earth

Example: CircleThe unit circle S1 = {(x,y) | x2+y2=1} can be viewed as a one dimensional manifold with two coordinate charts. Define the charts U1=S1-{(-1,0)} and U2=S1-{(1,0)}. Now we define the coordinate maps by projection as shown in the figure.

{(x,y) | x2+y2=1}

(-1,0)

(x,y) (x,y)

(1,0)

{(x,y) | x2+y2=1}

R1 R1

Submanifold & ImmersionDefinition: Let F: Rm→ Rn be a smooth map The rank of F at x0∈ Rm is the rank of the Jacobian DxF at x0. F is of maximal rank on S⊂ Rm if the rank of Fis maximal for each x0∈S.

Definition: A (smooth) submanifold of Rn is a subset M⊂Rn, together with a smooth one-to-one map φ:Π⊂Rm→M which satisfies the maximal rank condition everywhere, where the parameter space is Π and M = φ(Π) is the image of φ. If the maximal rank condition holds but the mapping is not one-to-one, then M is an immersion.

Note that a submanifold of the space Rn is a parametrically defined surface.

Pathologies

Regular Submanifold

p = t→∞lim φ(t)

N = R1 φ

immersion submanifold regular submanifold (imbedding)

M = R2

Definition: A regular submanifold N of Rn is a submanifold parameterized by a smooth mapping φ such that φ maps homeomorphically onto its image, i.e., for each x∈N there exists neighborhoods U of x in Rn such that φ-1[U∩N] is a connected open subset of the parameter space

Implicitly Defined Regular Manifolds

Proposition: Consider a smooth mapping F: Rm →Rn, n≤m. If F is of maximal rank on the set S = {x: F(x)=0}, then S is a regular, (m-n)-dimensional submanifold of Rm.

-1.5 -1 -0.5 0.5 1 1.5x

-1

-0.5

0.5

1

y

2 2

2 2

( , ) ( 1)

2 1 3

singular points: ( 1,0), (0, 1/ 3)

f x y x y y

Df xy x y

= + −

⎡ ⎤= − + +⎣ ⎦

± ±

Example

The Tangent SpaceDefinition: Let p: R→M be a Ck map so that p(t) is a curve in M. The tangent vector v to the curve p(t) at the point p0 = p(t0) is defined by

The set of tangent vectors to all curves in M passing through p0 is the tangent space to M at p0, denoted TM p0.

1≥k

⎭⎬⎫

⎩⎨⎧

−−

==→

0

00

)()(lim)(

0 tttptp

tpvtt

R

M

v

( )p t

t

0p

Tangent Space / Implicit Manifold

If M is an implicit submanifold of dimension m in Rm+k, i.e., F: Rm+k→Rk, M = {x∈Rm+k | F(x)=0} and DF satisfies the maximum rank condition on M, Then TMx is the (translated, of course to the point x). That is TMp is the tangent hyperplane to M at p.

TM Fxx =

∂∂LNMOQPKer

x

Im ∂∂FHGIKJ

LNMOQP

Fx

T

Μ

Rm k+

ker ( )xD F x

Tangent Vectors

( ) ( )( )( )

( ) ( )

1

The of the tangent vector to the curve in in local coordinates , are the numbers

, , where / .

Consider the map : . Let ,

denote the realizatio

m i

m

components vp t M U m

v v v d p t dt

F M R y f x x U R

ϕ

ϕ

ϕ

=

→ = ∈ ⊂

Definition :

…

( )( ) ( )

11

n of in the local coordinates , . Again,

denotes a curve in with its image in . Then therate of change of at a point on this curve is

m

mm

F U

p t M x t RF p

df f fv vdt x x

ϕ

∂ ∂= + +

∂ ∂

Tangent Vectors as Derivations

Rm

p t( )

x t( )

vϕ

The tangent vector is uniquely determined by theaction of the directional derivative operator (called a derivation)

mm x

vx

v∂

∂+

∂∂

=1

1v

[ ]1 mv v v=

Natural Basis0

1

0

th

i

v ix

⎡ ⎤⎢ ⎥⎢ ⎥ ∂⎢ ⎥= ⇔ =

∂⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

v

Definition: The set of partial derivative operators constitute a basis for the tangent space TMp for all points p∈U⊂M which is called the natural basis.The natural coordinate system on TMp induced by (U,ϕ) has basis vectors that are tangent vectors to the coordinate lines on M passing through p.

Tangent BundleDefinition: The union of all the tangent spaces to M is called the tangent bundle and is denoted TM,

pp MTM TM

∈=∪

Remark: The tangent bundle is a manifold with dim TM = 2 dim M. A point in TM is a pair (x,v) with x∈M, v∈TMx. If (x1,..,xm) are local coordinates on M and (v1,..,vm) components of the tangent vector in the natural coordinate system on TMx, then natural local coordinates on TM are

Recall the natural ‘unit vectors’ on TMx are ,…,

),,,,,(),,,,,( 1111 mmmm xxxxvvxx ………… =

11 x∂∂=v mm x∂∂=v

Summary

Regular ManifoldParametrically definedImplicitly defined

Tangent Space, Tangent Vector, Tangent Bundle

1x

2x

3x

1z

2z ( )1 2,x z z= Ψ

1 1, continous (homeomorphic)Ψ ⇔

( ){ }3 1 11 1 2 3

1 2

, , 0 rank =1 on f fM x R f x x x Mx x

⎡ ⎤∂ ∂= ∈ = ⎢ ⎥∂ ∂⎣ ⎦

Mechanical System State SpaceA mechanical system is a collection of mass particles which interact through physical constraints or forces. A configuration is a specification of the position for each of its constituent particles. The configuration space is a set M of elements such that any configuration of the system corresponds to a unique point in the set M and each point in M corresponds to a unique configuration of the system. The configuration space of a mechanical system is a differentiable manifold called the configuration manifold. Any system of local coordinates qon the configuration manifold are called generalized coordinates. The generalized velocities are elements of the tangent spaces to M, TMq. The state space is the tangent bundle TM which has local coordinates .

q),( qq

Example: Pendulum

θΤΜθθ

θ

TM

TM

θ

Μ

M=S1

Differential Map

R

p

F(p)

φ( )t φ φ( ) ( ( ))t F t=

Fddtφ

ddtφ

Given the map , the differential map is the induced mapping

that takes tangent vectors into tangent vectors.

:F M N→

* ( ): p F pF TM TN→

( )( ) ( )

an arbitray curve on passing through point

maps into on passing through point

t M p

t N F p

φ

φ

Differential Map ~ local coordinates

In local coordinates, the chain rule yields

d F d Fv vdt x dt xφ φ∂ ∂

= ⇒ =∂ ∂

• The map F* is also denoted dF• The Jacobian is the representation of the differential map in local coordinates

Vector FieldsDefinition: A vector field v on M is a map which assigns to each point p∈M, a tangent vector v(p)∈TMp. It is a Ck-vector field if for each p∈M there exist local coordinates (U,ϕ) such that each component vi(x), i=1,..,m is a Ck function for each x∈ϕ(U).

Definition: An integral curve of a vector field v on M is a parameterized curve p=φ(t), t∈(t1,t2)⊂R whose tangent vector at any point coincides with v at that point.

Integral Curves

In local coordinates (U,ϕ), the image of an integral curve

satisfies the ode( ) ( )x t tϕ φ=( )dx v x

dt=

Rm

ΥV

M

p t( )

x t( )

vϕ

FlowDefinition: Let v be a smooth vector field on M and denote the parameterized maximal integral curve through p∈M by Ψ(t,p) and Ψ(0,p)=p. Ψ(t,p) is called the flow generated by v.

( ) ( )( ) ( ), , , 0,d t p v t p p pdt

Ψ = Ψ Ψ =

( )( ) ( )2 1 1 2, , ,t t p t t pΨ Ψ = Ψ +

Properties of flows:

• satisfies ode

• semigroup property

Exponential MapWe will adopt the notation

( ): ,te p t p= Ψv

The motivation for this is that the flow satisfies the three basic properties ordinarily associated with exponentiation –from properties of Ψ(t,p).

( )1 2 1 2

0

( )

boundary condition

differential equation

semi-group property

t t

t t t t

e p pd e p v e pdt

e p e e p

⋅

+

=

=

=

v

v v

v v v

Series Expansion Along Trajectory

( ) ( ) ( )

( )( ) ( ) ( )( ) ( )( )

( )( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( ) ( )

0

22

0 20 0

11

22

2

2 20 0 0

Suppose satisfies , 0 . Let : .

12

12

m p

t t

mm

x t x v x x x f R R

d df x t f x f x t t f x t tdt dt

d f f ff x t v x v x v x f xdt x x x

d ff x t v x v x f xdt x x

f x t f x f x t f x t

= =

= = →

⎡ ⎤⎡ ⎤= + + +⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦∂ ∂ ∂

= = + + =∂ ∂ ∂

∂ ∂⎛ ⎞= =⎜ ⎟∂ ∂⎝ ⎠

= + + +

v

v

v v

Series Representation of Exp Map

( ) ( ) ( )0 !

kt k

k

tf e x f xk

∞

=

= ∑v v

For f a scalar or vector, we can derive the Taylor expansion of f(x(t)) about t=0

Choose f(x)=x, to obtain

( )( )0 !

kt k

k

te x x xk

∞

=

= ∑v v

Example: scalar linear fields

( ) ( )0 0

22

2

( , )

( , )

1 1 ( , )

1( ,

! !

) 12

t

t

k ktx k tx

x

k k

dxv

dxv t x x tdt

t

x x t x e xdt

t t

x e x t t x x tx x

t x e x x x x e xk k

∂ ∞ ∞∂

= =

∂∂

= ⇒ = ⇒ Ψ = +

⎛ ⎞∂ ∂Ψ = =

= ⇒ = ⇒ Ψ =

Ψ = = = =

+ + + = +⎜ ⎟∂ ∂⎝ ⎠

∑ ∑v

Example: general linear field

( )

( )

( ) ( ) ( )( ) ( ) ( )

( )( )

1

21

1

1

2 2

1 1

0 0

1 00

00 1

! !

nn

n

n

k k k k

k kt k k At

k k

aa

v x Ax x a x a xx x

a

x a x a x Ax

x Ax A x A x

x A x A x A x

t te x x x A x e xk k

− −

∞ ∞

= =

⎡ ⎤⎢ ⎥ ∂ ∂⎢ ⎥= = ⇒ = + +⎢ ⎥ ∂ ∂⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= + + =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= = =

= = =

⎛ ⎞= = =⎜ ⎟

⎝ ⎠∑ ∑v

v

v

v v v

v v v

v

Example: Affine Field

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )( )

1 1

2 21 1

1

1 1

1 2 1 1

1

0

1 00

00 1

! ! !

n nn

n n

n n

k k k k k k

k k kt k k k

k k

a ba b

v x Ax b x a x b a x bx x

a b

x a x b a x b Ax b

x A x A b A x A x A b

t t te x x x A x A bk k k

− − − −

∞−

= =

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ∂ ∂⎢ ⎥ ⎢ ⎥= + = + ⇒ = + + + +⎢ ⎥ ⎢ ⎥ ∂ ∂⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= + + + + = +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= + = = +

= = +∑v

v

v

v v v

v 1

0 0

At At

ke x e A b

∞ ∞−

=

⎛ ⎞= +⎜ ⎟

⎝ ⎠∑ ∑

Examples, Cont’d

( )

( ) ( )

( ) ( )3

3 3

2 4 2 6 3 8 4 10 50 20 35 63120 0 0 0 0 08 8 8 82

0

2 4 2 6 3 8 4 10 520 35 63120 0 0 0 0 0 08 8 8 8

Exact solution:

11 2

via exponential map:

1x

x

v x x x x

xx t x t x t x t x t x t xx t

x t e x x t x t x t x t x t x∂

−∂

= − ⇔ = −

= = − + − + − ++

= = − + − + − +

Lie DerivativeDefinition: Let v(x) denote a vector field on M and F(x) a mapping from M to Rn, both in local coordinates. Then the Lie derivative of order 0,…,k is

( )1

0 , ( )k

k vv v

LL F F L F vx

−∂= =

∂

With this notation we can write

( ) ( )( ) ( )k kvF x L F x=v

Example: Exponential Map of a Nonlinear Field

( )

( )( )

( )

( ) ( )

( ) ( ) ( )( ) ( )( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )

1

21

1

2 2

1 1

0 0! !

nn

n

v

v

k k k kv

k kt k k

vk k

v xv x

v x v x v xx x

v x

x L x v x

x v x L x

x A x A x L x

t te x x x L xk k

− −

∞ ∞

= =

⎡ ⎤⎢ ⎥

∂ ∂⎢ ⎥= ⇒ = + +⎢ ⎥ ∂ ∂⎢ ⎥⎢ ⎥⎣ ⎦

= =

= =

= = =

⎛ ⎞= = ⎜ ⎟

⎝ ⎠∑ ∑v

v

v

v v

v v v

v

Lie Bracket

Definition: If v,w are vector fields on M, then their Lie bracket[v,w] is the unique vector field defined in local coordinates by the formula

wxvv

xwwv

∂∂

−∂∂

=],[

( )( ) [ ]0

,,

xt

dw t xv w

dt=

Ψ=

Property:

The rate of change of w along the flow of v

Lie Bracket Interpretation

Let us consider the Lie bracket as a commutator of flows. Beginning at point xin M follow the flow generated by v for an infinitesimal time which we take as for convenience. This takes us to point

Then follow w for the same length of time, then -v, then -w. This brings us to a point ψ given by

ε

xy )exp( vε=

( , )x e e e e xε ε ε εψ ε − −= w v w v

Lie Bracket Interpretation Continued

y

v

w

-v-w

[v,w]

x

zu

x),(εψ

( ) [ ]0 , ,x

d x v wdε

+Ψ =

Summary

Definition of regular manifoldImplicitly defined & parametrically defined

Local coordinatesTangent space, vector field, integral curveDifferential map, exponential mapLie derivativeLie bracket