Dirac-Weinstein Anchored Vector Bundle Reduction for ... · one for symmetry reduction and that one can do reduction by stages; that is, repeated reduction without leaving the context

Dirac-Weinstein Anchored VectorBundle Reduction

for Mechanical Systems with Symmetry

by Hernan endra, UNS and CONICET, ARGENTINA-(based on joint work with Tudor Ratiu, EPFL

and Hiroaki Yoshimura, Waseda Univ).

Manuel de Leon Fest,ICMAT, Madrid, December 18, 2013

1/36

Basic Ideas.

Symmetry, Reduction

Unification.

Categorical languaje

2/36

Basic Ideas.

3/36

Contents.

Linear Dirac Structures.

Dirac Anchored Vector Bundles

Equations of Motion

Morphisms of Dirac Anchored Vector Bundles

Reduction of Dirac Anchored Vector Bundles orDirac-Weinstein Reduction

Reduction of Equations of Motion

4/36

Linear Dirac Structures.

A linear Dirac structure D on a vector space V is defined bythe following conditions:D ⊆ V ⊕ V ∗ subspacepV : V ⊕ V ∗ → V projectionED = Im ρVωD : ED × ED → R skew symmetricD = (X,α) ∈ V ⊕ V ∗ : ω(X,Y ) = α(Y ), for all Y ∈ ED.

Examples include:

ED = V presymplectic form,ED = Imπ]D, πD : V ∗ × V ∗ → R skew symmetric, which inducesnaturally a presymplectic ωD(X,Y ) := πD(α, β), whereX = π(α), Y = π(β).

5/36

Dirac Anchored Vector Bundles.

An anchored vector bundle is a pair (π(V,M), ρV ) whereπ(V,M) : V →M vector bundle over MρV : V → TM vector bundle map over the identity(ρV (u) ∈ Tπ(V,M)(u)M for u ∈ V ) called the anchor.

An anchored morphismΦA : (π(V,P ), ρV )→ (π(W,P ), ρW )is a vector bundle map Φ : V →W , covering a map φ : M → P ,such thatρW Φ = Tφ ρV .

6/36

Dirac Anchored Vector Bundles.

If ΨA : (π(W,P ), ρW )→ (π(U,N), ρU ) is another anchored

morphism of anchored vector bundles, the composition ΨA ΦA

is defined by ΨA ΦA = (Ψ Φ)A, with an obviousinterpretation of the notation. This defines the category ofanchored vector bundles AV.

The most fundamental example is the anchored vector bundle(TM, 1TM ).

Let π(V,M), ρV ) an anchored vector bundle andπ(V ∗,M) : V ∗ →M dual vector bundle.

A Dirac structure on the bundle π(V,M) is a vector subbundleDV ⊂ V ⊕ V ∗ such that for each m ∈M , (DV )m ⊂ Vm ⊕ V ∗m isa Dirac structure on the vector space Vm. We call a triple(π(V,M), ρV , DV )) a Dirac anchored vector bundle over M .

7/36

Equations of Motion.

Let (π(V,M), ρV , DV ) be a Dirac anchored vector bundle over Mand let a section ϕ of the bundle V ∗ called energy form.Equations of motion are defined to be the following twoconditions on a curve u(t) ∈ V :

1. The pointwise condition

u⊕ ϕ(π(V,M)(u)) ∈ (DV )π(V,M)(u) (1)

2. The curve u(t) is admissible; that is,

ρV (u(t)) =d

dtπ(V,M)(u(t)), (2)

for all t for which u(t) is defined.

8/36


We show how several fundamental systems fit into this unifiedcontext. Our main goal is to show that this context is a naturalone for symmetry reduction and that one can do reduction bystages; that is, repeated reduction without leaving the contextof Dirac anchored vector bundles. This goal is inspired byLagrangian Reduction by Stages, and HamiltonianReduction by Stages. In the course of doing this, we give anumber of concrete examples.

9/36


Example:Lagrangian, Hamiltonian and Nonholonomic MechanicsLet ∆Q ⊆ TQ distribution representing nonholonomicconstraint on the configuration space Q and let M = TQ⊕ T ∗Qbe the Pontryagin bundle.Define an induced Dirac structure D∆M

⊂ TM ⊕ T ∗M on M ,in local coordinates (q, v, p) ∈M , by

D∆M(q, v, p) = ((q, v, p), (α, γ, β)) (3)

such that

q ∈ ∆(q), α+ p ∈ ∆(q), β = q, γ = 0 (4)

One can prove that this definition is independent of thecoordinates and gives a Dirac structure.

10/36


Nonholonomic Mechanics We will show that nonholonomicmechanical systems are Dirac dynamical systems.

Choose the vector bundle V on M to be V = TM , with theanchor ρV being the identity map Id on TM . The Diracstructure will be DV = D∆M

given equivalently by either of thetwo preceding displayed equations, and the energy form ischosen to be ϕ = dE, where E : M → R is given byE(q, v, p) = 〈p, v〉 − L(q, v). LetD∆M

= (π(V,M) = τM , ρV = Id, D∆M) be a Dirac anchored

vector bundle over M . Then, it can be checked directly that theDirac dynamical system (ϕ = dE,D∆M

) that satisfies

(q, v, p, q, v, p)⊕ dE(q, v, p) ∈ D∆M(q, v, p),

leads to the system of equations

11/36


p− ∂L

∂q∈ ∆(q), q = v, p− ∂L

∂v= 0, q ∈ ∆(q), (5)

which is equivalent to the ones obtained by the d’AlembertPrinciple.

For the case in which ∆Q = TQ, namely,∆M = TM = T (TQ⊕ T ∗Q) one obtains the Euler-Lagrangeequations. Also LC circuits can be described as anonholonomic system where Q is the charge space, TQ is thecurrent space, ∆Q represents the KCL and the Lagrangian iswritten in terms of Inductances and capacitors

1

2

n∑k=0

Lk(vk)2 − 1

2

n∑k=0

q2k

Ck.

Hamilton’s equations: choosing E = π∗(M,T ∗Q)H, where H isa given Hamiltonian on T ∗Q one obtains an equivalent form ofHamilton’s equations.

12/36

Morphisms of Dirac Anchored Vector Bundles.

We will define the push forward and pull back of Diracstructures. Let V and W be vector spaces and let Dir(V ) andDir(W ) be the set of Dirac structures on V and W respectively.Let f : V →W be a linear map and f∗ : W ∗ → V ∗ its dual.Define the forward Dirac map Ff : Dir(V )→ Dir(W ) andthe backward Dirac map Bf : Dir(W )→ Dir(V ) as follows:

Ff(DV ) = (fv, w∗) ∈W ⊕W ∗ | v ∈ V,w∗ ∈W ∗, (v, f∗w∗) ∈ DV (6)

Bf(DW ) = (v, f∗w∗) ∈ V ⊕ V ∗ | v ∈ V,w∗ ∈W ∗, (fv, w∗) ∈ DW .(7)

In general, FL and BL are not inverses to each other. However

13/36


Proposition. If L : V →W is surjective, thenFL BL = IDir(W ). If L : V →W is injective, thenBL FL = IDir(V ).

Proposition. Let K : U → V and L : V →W be linear maps.Then F(L K) = FL FK and B(L K) = BK BL.

Next we shall define the Forward-Dirac Category and theBackward-Dirac Category, which are equivalent categories.

14/36


The Forward-Dirac Category FD Objects of FD are pairs(V,DV ) where V is a vector space and DV is a Dirac structureon V. A morphism φF : (V,DV )→ (W,DW ) is given by a linearmap φ : V →W such that Fφ(DV ) = DW . The compositionrule of a morphism ψF : (U,DU )→ (V,DV ) and a morphismφF : (V,DV )→ (W,DW ) is given by φF ψF = (φ ψ)F , whichwe can prove easily that is well defined, using the fact thatFφ (Fψ(DU )) = F(φ ψ)DU .

The Backward-Dirac Category BD.Objects of BD are pairs (V,DV ) where V is a vector space andDV is a Dirac structure on V. A morphismφB : (V,DV )→ (W,DW ) is given by a linear map φ : V →Wsuch that Bφ(DW ) = DV . The composition rule of a morphismψB : (U,DU )→ (V,DV ) and a morphismφB : (V,DV )→ (W,DW ) is given by φB ψB = (φ ψ)B, whichwe can prove easily that is well defined, using the fact that,Bψ (Bφ(DW )) = B(φ ψ)DW .

15/36


A forward-Dirac anchored morphism from(π(V,M), ρV , DV ) to (π(W,P ), ρW , DW ), generically denoted

ΦFDA : (π(V,M), ρV , DV )→ (π(W,P ), ρW , DW )

means that the vector bundle map Φ : V →W is both, amorphism of anchored vector bundles and and a forward Diracmorphism.Let ΨFDA : (π(W,P ), ρW , DW )→ (π(U,Q), ρU , DU ) be a givenforward-Dirac anchored morphism from (π(W,P ), ρW , DW ) to(π(U,Q), ρU , DU ), then

(Ψ Φ)FDA : (π(V,M), ρV , DV )→ (π(U,P ), ρU , DU ) is aforward-Dirac anchored morphism from (π(V,M), ρV , DV ) to

(π(U,P ), ρU , DU ), denoted ΨFDA ΦFDA.

16/36


A backward-Dirac anchored morphism from(π(V,M), ρV , DV ) to (π(W,P ), ρW , DW ), generically denoted

ΦBDA : (π(V,M), ρV , DV )→ (π(W,P ), ρW , DW )

means that the vector bundle map Φ : V →W is both, amorphism of anchored vector bundles and and a backwardDirac morphism.Let ΨBDA : (π(W,P ), ρW , DW )→ (π(U,Q), ρU , DU ) be a givenbackward-Dirac anchored morphism from (π(W,P ), ρW , DW ) to(π(U,Q), ρU , DU ), then

(Ψ Φ)BDA : (π(V,M), ρV , DV )→ (π(U,P ), ρU , DU ) is abackard-Dirac anchored morphism from (π(V,M), ρV , DV ) to

(π(U,P ), ρU , DU ), denoted ΨBDA ΦFDA.

17/36


Existence of Morphisms of Dirac Anchored VectorBundles(a) Let V →M and W → N be vector bundles and letΦ : V →W be a vector bundle map covering a map φ : M → N,and assume, in addition, that ker Φ has locally constantdimension. Let DW be a Dirac structure on W . ThenDV := BΦ(DW ) is a Dirac structure on V.(b) Let V →M and W → N be vector bundles and letΦ : V →W be a vector bundle map covering a submersionφ : M → N. Assume that for any given y ∈ N andx1, x2 ∈ φ−1(y), the condition Φ (Vx1) = Φ (Vx2) is satisfied. LetDV be a Dirac structure on V such that, for given y ∈ N andx1, x2 ∈ φ−1(y), the condition FΦx1 (DV x1) = FΦx2 (DV x2) issatisfied. Then DW := FΦ(DV ) is a Dirac structure on W .

18/36


Under the additional assumption that FΦ is a morphism ofanchored vector bundles gives existence of correspondingmorphism of Dirac anchored vector bundles


19/36

Reduction of Dirac Anchored Vector Bundles or DW Reduction

We shall describe the notions of forward reduction,backward reduction and reduction, within the category ofDirac anchored vector bundles.DefinitionA forward reduction from the Dirac anchored vector bundle(π(V,M), ρV , DV ) to another Dirac anchored vector bundle(π(W,P ), ρW , DW ) is a fiberwise surjective forward morphism

ΦFDA : (π(V,M), ρV , DV )→ (π(W,P ), ρW , DW ) of anchored vectorbundles, that is, Φx is onto Wφ(x) and, moreover, φ : M → P isa submersion and TxφρV (Vx) = ρW (Wφ(x)), for each x ∈M .

20/36

Description of the CA for Dirac Manifolds.

A backward reduction from the Dirac anchored vectorbundle (π(W,P ), ρW , DW ) to another Dirac anchored vectorbundle (π(V,M), ρV , DV ) is a fiberwise injective backwards

morphism ΦBDA : (π(V,M), ρV , DV )→ (π(W,P ), ρW , DW ), that isΦx is injective and, moreover, Φ(V ) is a subbundle of Wφ : M → P is an embedding and the condition ρW (w) ∈ Tφ(M)implies that w = Φ(v) for some v ∈ V .

A reduction is a chain of a finite number of forward andbackward reductions of the type (R1, ..., Rk), where eachRi : (π(Vi,Mi), ρVi , DVi)→ (π(Vi+1,Mi+1), ρVi+1 , DVi+1) is a forwardor a backward reduction. Reductions can be composed byconcatenation.

21/36

Reduction of Dirac Anchored Vector Bundles or DW Reduction

MW reductionLet (M,ΩM ) be a connected symplectic manifold. Let a freeand proper action of a Lie group G on M . Let J : M → g∗ bean equivariant momentum map and for µ ∈ g∗, let Gµ be thecorresponding coadjoint isotropy group given by

Gµ := g ∈ G | g · µ = µ, for µ ∈ g∗.

Assume that µ ∈ g∗ is a regular value point of J and that theaction of Gµ on J−1(µ) is free and proper. Then, we can formthe symplectic reduced space defined by the quotient space

Mµ = J−1(µ)/Gµ,

which has the reduced symplectic form Ωµ that is uniquelydefined by

π∗µΩµ = ι∗µΩM , (8)

where πµ : J−1(µ)→Mµ is the projection and ιµ : J−1(µ) →Mis the inclusion.

22/36

Examples: Nonholonomic Systems and L-C circuits.

Symplectic reduction can be systematically explained in thecontext of Dirac anchored vector bundle reduction by simplyobserving that it is a chain of a backward reduction from(τM , DΩM , 1TM ) to (τJ−1(µ), Dι∗µΩM , 1TJ−1(µ)) via the embeddingιµ followed by a forward reduction from(τJ−1(µ), Dι∗µΩM , 1TJ−1(µ)) to (τMµ, DΩµ , 1TMµ), via theprojection πµ.Of course, we are not giving here a an entirely new proof ofMW-reduction but simply showing that it can be written as aDW-reduction. In fact we are using that the kernel of ι∗µΩM isthe space tangent to the orbit of Gµ.

23/36

Reduction of equations of Motion

Reduction of Equations of Motion Let a forward reduction


and let ϕV and ϕW be energy forms such that ϕV = Φ∗ϕW .Using the fact that ρWΦv(t) = TφρV v(t) we can see that anadmissible curve v(t), that is, a curve satisfying

ρV v(t) =d

dtπ(V,M)v(t)

is transformed into a curve w(t) = Φv(t) which is alsoadmissible, that is

ρWw(t) =d

dtπ(W,M)w(t).

If v(t) satisfies the condition v(t)⊕ ϕV (t) ∈ DV (φ(V,M)v(t) thenw(t) satisfies the condition w(t)⊕ ϕW (t) ∈ DW (φ(W,M)w(t). Wecall w(t) forward reduced solution, it satisfies the reduced Diracdynamical system.

24/36

Reduction of equations of Motion

A similar situation occurs with backward reduction. In fact, inthis case we can assume without loss of generality that M is asubmanifold of P , that V is a subbundle of W |M thatρV = ρW |V . Assume that the energy form ϕW is such that forany wx, x ∈M satisfying wx ⊕ ϕw(x) ∈ DWx the conditionρW (wx) ∈ TxM is satisfied. Then an admissible curve w(t)satisfying also w⊕ϕw ∈ DW and the initial condition w(t0) ∈ Vwill also satisfy that w(t) ∈ V , for all t and it is an admissiblecurve of (π(V,M), ρV ).

25/36

LARGA VIDA Y FELICIDADES MANUEL!

26/36

Documents

Dirac-Weinstein Anchored Vector Bundle Reduction for ... · one for symmetry reduction and that one can do reduction by stages; that is, repeated reduction without leaving the context