Distributions, Frobenious Theorem & Transformations

Transcription

1 Distributions Frobenious Theorem & Transformations Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University

2 Outline Distributions & integral surfaces Involutivity Frobenious Theorem Codistributions & computing Invariant distributions Transformation of vector fields Involutive closure

3 Distributions v v is a set of vector fields on M r { } Δ = span v( p) v ( p) is a subspace of TM p r p Definition: A smooth distribution Δ on M is a map which assigns to each point p M a subspace of the tangent space to M at p Δ p TMp such that Δ p is the span of a set of smooth vector fields v..v r evaluated at p. We write Δ=span{v..v r }. Definition: An integral submanifold of a set of vector fields v..v r is a submanifold N M whose tangent space TNp is spanned by {v (p)..v r (p)} for each p N. The set of vector fields is (completely) integrable if through every point p M there passes an integral submanifold.

4 Involutive Distributions Definition: A system of smooth vector fields {v..v r } on M is in involution if there exist smooth real valued functions c ij k (p) p M and ijk =..r such that for each ij r ij i j ckvk k = v v = Proposition: (Froebenius) Let {v..v r } be an involutive system of vector fields with dim [span{v..v r }]=k on M. Then the system is integrable with all integral manifolds of dimension k. Proposition: (Hermann) Let {v..v r } be a system of smooth vector fields on M. Then the system is integrable if and only if it is in involution.

5 Example 3 M = R Δ = span{ v w} v = y x w = 2 2x 2y + x 2 y 2 [ vw ] so the distribution Δ is completely integrable. The distribution is singular because dim Δ = 2 everywhere except on the -axis x = y = 2 2 and on the circle x + y = = where dim Δ = The -axis and the circle are one-dimensional integral manifolds. All others are the tori: T c = {( ) ( ) ( ) 2} / x y R x + y x + y + + = c >

6 Example

7 Maximal Integral Submanifolds Definition: A maximal integral submanifold N of a distribution Δ on a manifold M is a connected integral manifold of Δ which is not a proper subset of any other connected integral manifold of Δ. A distribution Δ on M is said to have the maximal integral manifolds property if through every point p M passes a maximal integral submanifold of Δ. Proposition: Let Δ be a d-dimensional involutive smooth distribution on M. Let p M. Then through p there passes a unique maximal connected integral manifold of Δ.

8 Covector Fields A covector field w on M assigns to each point p M an element w(p) T*Mp. A codistribution Ω on M is a mapping which assigns a subspace Ω(p) of T*Mp to each point p M. As with distributions we write Ω = span{w..w r }. Distributions are sometimes associated with special codistributions. As an example for each p M the annihilator of the distribution Δ(p) is the set of all covectors which annihilate vectors in Δ(p) Δ (p) := {w T*Mp : w(v)= for all v Δ(p)} Conversely given a codistribution Ω we define its annihilator a distribution Ω Ω (p):= {v TMp : v(w)= for all w Ω(p)}.

9 Representing Integral Surfaces Δ= span{ v v } a k-dim involutive distribution k on m-dim M. Consider x=ψ( x ) where ( ) Ψ x = x 2 ψ ψ ψ k ( ) k 2 m ( ) ( ) t t2 notation : ψ ψ ( ) = ϕ ϕ x t t x 2 2 x 3 2 x =Ψ( 2 x) x 2 integral surface passing through x x

10 Local Coordinate Transformation Δ= span{ v v } a k-dim distribution on m-dim M define v v such that span{ v v } = R k+ m m consider x=ψ( x ) where k 2 ( ) Ψ x =ψ ψ ψ m ( x ) m 2 m m x 3 2 x =Ψ( 2 c x) 3 x x x 2 First k coordinates map onto integral surface. c identifies the surface. Mapping is defined on a neighborhood of.

11 Invariant Distributions Definition: A distribution Δ = { v v r } on M is invariant with respect to a vector field f on M if the Lie bracket [fv i ] for each i = r is a vector field of Δ. Notation: [ f Δ] = span{[ f vi ] i = r} that Δ is invariant with respect to f may be stated [fδ] Δ. In general Δ+[fΔ] = Δ+span{[fv i ] i=..r } = span{v..v r [fv ]..[fv r ]}

12 Example: Invariant Linear Subspace V n = span{ v v r } vi R that is invariant under the linear mapping A i.e. AV V. Define a distribution on R n and a vector field Δ V ( x ) = span{ v vr} f A ( x) = Ax We prove that Δ V is invariant under the vector field f A. We need to show that [ f v ] Δ A i V Compute vi f A [ f A vi ] = f A vi = Avi By assumption x x Av i V

13 Transformation of Vector Fields: Lemma Lemma: Let Δ be a be an involutive distribution of constant dimension d on an open subset U of and suppose that Δ is invariant under a vector field f. Then at each point there exists a neighborhood of in U and a coordinate transformation defined on it in which the vector field f is of the form n R x U x Φ(x) = U = ) ( ) ( ) ( ) ( ) ( n d n n d d n d d d n d d f f f f f

14 Transformation of Vector Fields sketch of proof ~ (a) Δhas dimension d integral surfaces through every x U d+ { } =Φ( x) x U withspan dφ dφ =Δ i τ i i k i define the basis for TM : τ ( ) = τk = i k = i τ n f i f f τi = τ = i n

15 Transformation of Vector Fields sketch of proof ~ (b) v { d note : Δ= span τ τ } v Δ v=v d in coords Δinvariant w.r.t. f f Δ Δ f k i = d + k n i d

16 Transformation of Vector Fields = Φ(x) 2 x 2 τ Δ τ 2 Δ x integral manifolds x x = f ( ) = f( x) 2 = f ( ) 2 2 2

17 Slice e c { ( ) ( εf ) 2 2 } S = x U x = e c εf = f ( ) 2 = f ( ) c { 2( ) 2( )} S = x U x = c

18 Example span { v v } =span x2 x Δ= 2 Δ is involutive and invariant w.r.t. f. f x x 3 = xx 3 4 xxx sin x3 + x2 + xx3 2 v3 = v4 = span { v v2 v3 v4} = span [ ] [ ] Ψ ( ) = + Ψ ( x) = x x x x x + x f 2 3 = 34 sin 3

19 Involutive Closure ~ Problem : find the smallest distribution with the following properties It is nonsingular It contains a given distribution Δ It is involutive It is invariant w.r.t. a given set of vector fields τ τ q τ τ Δ q

20 Involutive Closure ~ 2a Problem 2: find the largest distribution with the following properties It is nonsingular It is contained in a given distribution Δ It is involutive It is invariant w.r.t. a given set of vector fields τ τ q

21 Involutive Closure ~ 2b Problem 2 is equivalent to: find the smallest codistribution with the following properties It is nonsingular It contains a given codistribution Δ It is spanned by a set of exact covector fields It is invariant w.r.t. a given set of vector fields τ τ q τ τq Ω

22 Algorithms Algorithm for Problem : Δ =Δ [ τ ] Δ =Δ + Δ k k i= i k stop when * k q Δ =Δ k Algorithm for Problem 2: Ω =Ω=Δ q k k L i= τ i k Ω =Ω + Ω stop when Ω k Ω =Ω k * * k