mechanics on fibered manifolds - math.wustl.edumath.wustl.edu/~sli/gonefishing16.pdf · gone...

Mechanics on fibered manifolds

Songhao Li

Washington University in St. Louis

Gone Fishing Poisson geometry meeting 2016March, 2016

Based on joint work with Ari Stern and Xiang Tang:arXiv:1511.00061

Li (WUSTL) Mech on fibered mfld GoneFishing 16 1 / 14

http://arxiv.org/abs/1511.00061

Outline

Lagrangian mechanicsLagrangian mechanics on TQLagrangian mechanics on AReduction by groupoid action

Hamilton-Pontryagin mechanicsH-P mechanics on TQ ⊕ T ∗QH-P mechanics on (A,A∗)Reduction by groupoid action


Lagrangian mechanics Lagrangian mechanics on TQ

I Q is a manifold.I L ∈ C∞(TQ) is a smooth function.I P(Q) = {q : I → Q | q is C2}.I δq ∈ TqP(Q) with δq(0) = δq(1) = 0.

I S : P(Q)→ R, S(q) =∫ 1

0 L(q(t), q(t))dt .


Lagrangian mechanics Lagrangian mechanics on TQ

I Q is a manifold.I L ∈ C∞(TQ) is a smooth function.I P(Q) = {q : I → Q | q is C2}.I δq ∈ TqP(Q) with δq(0) = δq(1) = 0.

I S : P(Q)→ R, S(q) =∫ 1

0 L(q(t), q(t))dt .

Lagrangian mechanics on TQA path q ∈ P(Q) satisfies Hamilton’s variational principle if

δS = dS(δq) = 0, ∀ δq. (1)

In coordinates, we have the Euler-Lagrange equation:

∂L∂qi =

ddt

∂L∂qi . (2)


Lagrangian mechanics Lagrangian mechanics on A

I Q is a manifoldI ρ : A→ TQ is a Lie algebroid.I Pρ(A) = {a : I → A | ρ(a) = q} is the space of A-paths, where q is

the base path of a.




the base path of a.

Choosing TQ-connection on A, ∇ : Γ(TQ)× Γ(A)→ Γ(A), we have twoinduced A-connections on A, ∇ and ∇:

∇X Y = ∇ρ(X)Y , ∇X Y = ∇ρ(Y )X + [X ,Y ]. (3)




the base path of a.

Choosing TQ-connection on A, ∇ : Γ(TQ)× Γ(A)→ Γ(A), we have twoinduced A-connections on A, ∇ and ∇:

∇X Y = ∇ρ(X)Y , ∇X Y = ∇ρ(Y )X + [X ,Y ]. (3)

An A-variation of a is δa = Xb,a ∈ TaPρ(A) where b ∈ P(A) is a path inA such that b(0) = 0 and b(1) = 0.Reletive to a chosen TM-connection ∇ on A, the horizontal componentof Xb,a is ρ(b), and the vertical component is ∇ab, which areindependent of the choice of ∇.



I Q is a manifoldI ρ : A→ TQ is a Lie algebroid.I the space of A-paths:Pρ(A) = {a : I → A | ρ(a) = q, q is the base path of a.}

I δa = Xb,a ∈ TaPρ(A) with b(0) = 0 and b(1) = 0.

I S : Pρ(A)→ R, S(a) =∫ 1

0 L(a(t))dt .



I Q is a manifoldI ρ : A→ TQ is a Lie algebroid.I the space of A-paths:Pρ(A) = {a : I → A | ρ(a) = q, q is the base path of a.}

I δa = Xb,a ∈ TaPρ(A) with b(0) = 0 and b(1) = 0.

I S : Pρ(A)→ R, S(a) =∫ 1

0 L(a(t))dt .

Lagrangian mechanics on A (Weinstein ’96, Martinez ’01, L-S-T)A A-path a ∈ Pρ(A) satisfies Hamilton’s variational principle if

δS = dS(Xb,a) = 0, for all A-variations Xb,a. (4)

Using ∇, we have the Euler-Lagrange-Poincare equation:

ρ∗dLhor(a) +∇∗adLver(a) = 0. (5)


Lagrangian mechanics Reduction by groupoid action

I G⇒ M is a Lie groupoid.I µ : Q → M is a fibered manifold equipped with a free and proper

action by G⇒ M.I VQ is the vertical tangent bundle.I L ∈ C∞(VQ)G is G-invariant.





1. VQ/G is a Lie algebroid over Q/G.2. L ∈ C∞(VQ)G reduces to ` ∈ C∞(VQ/G).






The mechanics for L on VQ is equivalent to the mechanics for ` onVQ/G. That is,

a ∈ PV (Q) satisfies the E-L-P equation for L⇐⇒ a ∈ Pρ(VQ/G) satisfies the E-L-P equation for `.



Example

I G is a Lie group.I L ∈ C∞(TQ)G is (left) invariant.



Example

I G is a Lie group.I L ∈ C∞(TG)G is (left) invariant.

It follows that L reduces to ` ∈ C∞(g). The Euler-Lagrange equationfor L reduces to the Euler-Poincare equation:

ad∗ξµ = µ, (6)

where ξ(t) = a(t) and µ = δ`δξ .


Hamilton-Pontryagin mechanics H-P mechanics on TQ ⊕ T∗Q

I Q is a manifold.I L ∈ C∞(TQ) is a smooth function.I P(TQ ⊕ T ∗Q) = {(q, v ,p) : I → TQ ⊕ T ∗Q}.I (δq, δv , δp) ∈ Tq,v ,pP(TQ ⊕ T ∗Q) with δq(0) = δq(1) = 0.I S : P(TQ ⊕ T ∗Q)→ R is defined by

S(q, v ,p) =

∫ 1

0L (q(t), v(t)) + 〈p(t), q(t)− v(t)〉 dt . (7)


Hamilton-Pontryagin mechanics H-P mechanics on TQ ⊕ T∗Q

I Q is a manifold.I L ∈ C∞(TQ) is a smooth function.I P(TQ ⊕ T ∗Q) = {(q, v ,p) : I → TQ ⊕ T ∗Q}.I (δq, δv , δp) ∈ Tq,v ,pP(TQ ⊕ T ∗Q) with δq(0) = δq(1) = 0.I S : P(TQ ⊕ T ∗Q)→ R is defined by

S(q, v ,p) =

∫ 1

0L (q(t), v(t)) + 〈p(t), q(t)− v(t)〉 dt . (7)

H-P mechanics on TQ ⊕ T ∗QA path (q, v ,p) ∈ P(TQ ⊕ T ∗Q) satisfies H-P principle if

δS = dS(δq, δv , δp) = 0, ∀ (δq, δv , δp). (8)

In coordinates, we have the implicit E-L equations:

pi =∂L∂qi , pi =

∂L∂v i , vi = qi . (9)


Hamilton-Pontryagin mechanics H-P mechanics on (A, A∗)

I ρ : A→ TQ is a Lie algebroid.I L ∈ C∞(A) is a smooth function.




(A,A∗)-pathsAn (A,A∗)-path consists of

1. an A-path a ∈ Pρ(A) over the base path q ∈ P(Q);2. a path on A, v ∈ P(A), over q;3. a path on A∗, p ∈ P(A∗), over q;

The space of (A,A∗)-paths: P(A,A∗).




(A,A∗)-pathsAn (A,A∗)-path consists of

1. an A-path a ∈ Pρ(A) over the base path q ∈ P(Q);2. a path on A, v ∈ P(A), over q;3. a path on A∗, p ∈ P(A∗), over q;

The space of (A,A∗)-paths: P(A,A∗).

The action SS : P(A,A∗)→ R is defined by

S(a, v ,p) =

∫ 1

0L (v(t)) + 〈p(t),a(t)− v(t)〉 dt . (10)



Variations of (A,A∗)-pathsFor a (A,A∗)-path (a, v ,p),

1. we vary the A-path a by an A-variation δa = Xb,a ∈ TaPρ(A) withb(0) = 0 and b(1) = 0, whose base variation is δq ∈ TqP(Q);

2. we vary the v ∈ P(A) by a free variation δv ∈ TvP(A) over δq;3. we vary the p ∈ P(A∗) by a free variation δp ∈ TpP(A∗) over δq;



Variations of (A,A∗)-pathsFor a (A,A∗)-path (a, v ,p),

1. we vary the A-path a by an A-variation δa = Xb,a ∈ TaPρ(A) withb(0) = 0 and b(1) = 0, whose base variation is δq ∈ TqP(Q);

2. we vary the v ∈ P(A) by a free variation δv ∈ TvP(A) over δq;3. we vary the p ∈ P(A∗) by a free variation δp ∈ TpP(A∗) over δq;

A (A,A∗)-path (a, v ,p) satisfies H-P principle if

δS = dS(δa, δv , δp) = 0, for all variations (δa = Xb,a, δv , δp). (11)



Theorem (Li-Stern-Tang)Choosing a TM-connection ∇ on A, an (A,A∗)-path (a, v ,p) satisfiesH-P principle if and only if (a, v ,p) satisfies the implicit E-L-Pequations:

ρ∗dLhor(v) +∇∗ap = 0,dLver(v)− p = 0,a = v .

(12)


Hamilton-Pontryagin mechanics Reduction by groupoid action





Theorem (Li-Stern-Tang)The H-P mechanics for L on VQ ⊕ V ∗Q is equivalent to the H-Pmechanics for ` on (A,A∗) where A = VQ/G. That is,

(q, v , p) ∈ PV (VQ ⊕ V ∗Q) satisfies the implicit E-L-P equation for L.⇐⇒ (a, v ,p) ∈ P(A,A∗) satisfies the implicit E-L-P equation for `.


References

A. Weinstein, Lagrangian mechanics and groupoids, vol. 7 of FieldsInst. Commun. 1996, pp. 207–231.

E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl.Math., 67 (2001), pp. 295–320.

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. ofMath. (2), 157 (2003), pp. 575–620.

H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangianmechanics. I. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006),pp. 133–156.

S. Li, A. Stern and X. Tang, Mechanics on fibered manifolds,arXiv:1511.00061

Thank you!


http://arxiv.org/abs/1511.00061

mechanics on fibered manifolds - math.wustl.edumath.wustl.edu/~sli/gonefishing16.pdf · gone...

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