Manifolds, Vector Fields & Flowsb89089/link/Lecture2.pdfVector Fields Definition: A vector field v...

Nonlinear Control Theory ~ Lecture 2

Manifolds, Vector Fields & Flows

Harry G. KwatnyDepartment of Mechanical Engineering & MechanicsDrexel University


Outline• Manifolds• Maximum rank condition• Regular manifolds• Tangent space & tangent bundle• Differential map• Vector fields & flows• Lie bracket


Representations of Surfaces

• Explicit • Implicit• Parametric

( )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 m-dimensional manifold is a set M together with a countable

collection of subsets Ui⊂ M and one-to-one mappings ϕi:Ui→Vionto open subsets Vi of Rm, each pair (Ui,ϕi) called a coordinate chart, with the following properties:

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

if p∈ Ui and q∈ Uj are distinct points of M, then there are neighborhoods, W of ϕi(p) in Vi and U of ϕi(q) in Vj such that

ϕi-1(W) ∩ ϕj

-1(U) = ∅

)()(:1jijjiiij UUUUf ∩→∩= − ϕϕϕϕ


Definition: Manifold

1−= ijf ϕϕ

jϕ

iϕ

iU jU

iV

jV


Example: Circle• The 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 & Immersion

Definition: 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.


Pathologies


Regular SubmanifoldDefinition: A regular submanifold N of Rn is a submanifold parameterized by a smooth mapping φ such that φmaps 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

p = t→∞lim φ(t)

N = R1

φ

immersion submanifold regular submanifold (imbedding)

M = R2


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 1.5 x

-1

-0.5

0.5

1yExample

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

= + −

= − + +

± ±


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

M

Rm k+

ker ( )xD F x


Tangent VectorsDefinition: The components of the tangent vector v to the curve p(t) in M in the local coordinates (U,ϕ) are the m numbers v1,..,vm where vi=(dϕi/dt).

Consider the map F: M→R. Let y=f(x), x∈ϕ (U)⊂ Rm denote the realization of F in the local coordinates (U,ϕ). Again, suppose p(t) denotes a curve in M with x(t) its image in Rm. Then the rate of change of F at a point p on this curve is

mm x

fvxfv

dtdf

∂∂

++∂∂

=1

1


Tangent Vectors as Derivations

Rm

UV

M

p t( )

x t( )ϕ

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

[ ]1 mv v v=

v

mm x

vx

v∂∂

+∂∂

=1

1v


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


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→


Differential Map ~ local coordinatesIn 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 CurvesIn local coordinates (U,ϕ), the image of an integral curve

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

dt=

Rm

UV

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.

Properties of flows:

• satisfies ode

• semigroup property

( ) ( )( ) ( ), , , 0,d t p v t p p pdtΨ = Ψ Ψ =

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


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

( )1 2 1 2

0

( )

t t

t t t t

e p pd e p v e pdte p e e p

⋅

+

=

=

=

v

v v

v v v


Series Representation of Exp Map

( ) ( )( )0 !

kt k

k

tf e x v f xk

∞

=

=∑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 v x xk

∞

=

=∑v


Examples

( )( )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 tt x e x v x x x e xk

x e x t t x x tx

k

x

∂ ∞ ∞∂

= =

∂∂

= ⇒ = ⇒Ψ = +

∂ ∂Ψ = =

=

+ + + = + ∂ ∂

⇒ = ⇒Ψ

=

Ψ = = = =∑ ∑


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 kvv F x L F x=


Lie BracketDefinition: 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

∂∂

−∂∂

=],[Property:

( )( ) [ ]0

,,

xt

dw t xv w

dt=

Ψ=


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=

xeeeex vv εεεεεψ ww −−=),(


Lie Bracket Interpretation Continued

y

v

w

-v-w

[v,w]

x

zu

x),(εψ

( ) [ ]0 , ,x

d x v wdε

+Ψ =

Manifolds, Vector Fields & Flowsb89089/link/Lecture2.pdfVector Fields Definition: A vector field v...

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