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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
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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
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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 .
Li (WUSTL) Mech on fibered mfld GoneFishing 16 3 / 14
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)
Li (WUSTL) Mech on fibered mfld GoneFishing 16 3 / 14
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.
Li (WUSTL) Mech on fibered mfld GoneFishing 16 4 / 14
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.
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)
Li (WUSTL) Mech on fibered mfld GoneFishing 16 4 / 14
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.
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 ∇.
Li (WUSTL) Mech on fibered mfld GoneFishing 16 4 / 14
Lagrangian mechanics Lagrangian mechanics on A
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 .
Li (WUSTL) Mech on fibered mfld GoneFishing 16 5 / 14
Lagrangian mechanics Lagrangian mechanics on A
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)
Li (WUSTL) Mech on fibered mfld GoneFishing 16 5 / 14
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.
Li (WUSTL) Mech on fibered mfld GoneFishing 16 6 / 14
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).
Li (WUSTL) Mech on fibered mfld GoneFishing 16 6 / 14
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 `.
Li (WUSTL) Mech on fibered mfld GoneFishing 16 6 / 14
Lagrangian mechanics Reduction by groupoid action
Example
I G is a Lie group.I L ∈ C∞(TQ)G is (left) invariant.
Li (WUSTL) Mech on fibered mfld GoneFishing 16 7 / 14
Lagrangian mechanics Reduction by groupoid action
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 .
Li (WUSTL) Mech on fibered mfld GoneFishing 16 7 / 14
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)
Li (WUSTL) Mech on fibered mfld GoneFishing 16 8 / 14
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)
Li (WUSTL) Mech on fibered mfld GoneFishing 16 8 / 14
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.
Li (WUSTL) Mech on fibered mfld GoneFishing 16 9 / 14
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∗).
Li (WUSTL) Mech on fibered mfld GoneFishing 16 9 / 14
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∗).
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)
Li (WUSTL) Mech on fibered mfld GoneFishing 16 9 / 14
Hamilton-Pontryagin mechanics H-P mechanics on (A, A∗)
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;
Li (WUSTL) Mech on fibered mfld GoneFishing 16 10 / 14
Hamilton-Pontryagin mechanics H-P mechanics on (A, A∗)
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)
Li (WUSTL) Mech on fibered mfld GoneFishing 16 10 / 14
Hamilton-Pontryagin mechanics H-P mechanics on (A, A∗)
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)
Li (WUSTL) Mech on fibered mfld GoneFishing 16 11 / 14
Hamilton-Pontryagin 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.
Li (WUSTL) Mech on fibered mfld GoneFishing 16 12 / 14
Hamilton-Pontryagin 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).
Li (WUSTL) Mech on fibered mfld GoneFishing 16 12 / 14
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 `.
Li (WUSTL) Mech on fibered mfld GoneFishing 16 13 / 14
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!
Li (WUSTL) Mech on fibered mfld GoneFishing 16 14 / 14
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