The Hamilton-Jacobi problem in nonholonomic mechanics Manuel de Le - - PowerPoint PPT Presentation

The Hamilton-Jacobi problem in nonholonomic mechanics

Manuel de Le´

• n

Instituto de Ciencias Matem´ aticas (ICMAT)

Nonholonomic mechanics and optimal control Institut Henri Poincar´ e Institute, Paris, November 25th 28th, 2014

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A short introduction to Lagrangian mechanics

Let L = L(qi, ˙ qi) be a lagrangian function, where (qi) are coordinates in a configuration n-manifold Q, and (˙ qi) are the generalized velocities. The Hamilton ’s principle produces the Euler-Lagrange equations d dt ( ∂L ∂ ˙ qi ) − ∂L ∂qi = 0, 1 ≤ i ≤ n. (1) A geometric version of Eq. (1) can be obtained as follows. L : TQ − → R. Consider the (1,1)-tensor field S and the Liouville vector field ∆ defined on the tangent bundle TQ of Q: S = ∂ ∂ ˙ qi ⊗ dqi, ∆ = ˙ qi ∂ ∂ ˙ qi . We construct the Poincar´ e-Cartan 1 and 2-forms αL = S∗(dL), ωL = −dαL, where S∗ denotes the adjoint operator of S. The energy is given by EL = ∆(L) − L, so that we recover the classical expressions ωL = dqi ∧ dpi, EL = ˙ qipi − L,

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We say that L is regular if the 2-form ωL is symplectic, which in coordinates turns to be equivalent to the regularity of the Hessian matrix

• f L with respect to the velocities
• Wij =

∂2L ∂ ˙ qi∂ ˙ qj

• In this case, the equation

iX ωL = dEL (2) has a unique solution, X = ξL, called the Euler-Lagrange vector field; ξL is a second order differential equation (SODE) that means that its integral curves are tangent lifts of their projections on Q (these projections are called the solutions of ξL). The solutions of ξL are just the

• nes of Eqs (1).

If ♭L : TTQ − → T ∗TQ is the musical isomorphism, ♭L(v) = ivωL, then we have ♭L(ξL) = dEL.

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Legendre transformation The Legendre transformation FL : TQ − → T ∗Q is a fibred mapping (that is, πQ ◦ FL = τQ, where τQ : TQ − → Q and πQ : T ∗Q − → Q denote the canonical projections of the tangent and cotangent bundle of Q, respectively) defined by FL(qi, ˙ qi) = (qi, pi), L is regular if and only if FL is a local diffeomorphism. We will assume that FL is in fact a global diffeomorphism (in other words, L is hyperregular) which is the case when L is a lagrangian of mechanical type, say L = T − V where

• T is the kinetic energy defined by a Riemannian metric on Q,
• V : Q −

→ R is a potential energy.

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Hamiltonian description The hamiltonian counterpart is developed in the cotangent bundle T ∗Q

• f Q. Denote by ωQ = dqi ∧ dpi the canonical symplectic form, where

(qi, pi) are the canonical coordinates on T ∗Q. The Hamiltonian energy is just h = EL ◦ FL−1 and the Hamiltonian vector field is the solution of the symplectic equation iXh ωQ = dh. The integral curves (qi(t), pi(t)) of Xh satisfies the Hamilton equations. Since FL∗ωQ = ωL we deduce that ξL and Xh are FL-related, and consequently FL transforms the Euler-Lagrange equations into the Hamilton equations.

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Non-holonomic mechanics Consider a disc rolling without slipping in a rough plane. Let (x, y) be the coordinates of the point of contact of the disc with the ground, ψ the angle between a point fixed in the circle and the point of contact (the angle of rotation), φ the angle between the tangent to the disc at the point of contact and the axis x, and θ the angle of inclination

• f the disc.

The configuration space is Q = R

2 × S1 × S1 × S1. The lagrangian is

L = T − V where T = 1 2m(˙ x2 + ˙ y 2 + R2 ˙ θ2 + R2 ˙ φ2 sin2 θ) − mR( ˙ θ cos φ(˙ x sin φ − ˙ y cos φ) + ˙ φ sin θ(˙ x cos φ + ˙ y sin φ)) + 1 2I1( ˙ θ2 ˙ φ2 cos2 θ) + 1 2I2( ˙ ψ + ˙ φ sin θ)2 and V = mgR cos θ m is the mass of the disc, R is the radius, I1 and I2 are the principle moments of inertia.

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The non-slipping condition implies the following constraints. Φ1 = ˙ x − (R cos φ) ˙ ψ = 0, Φ2 = ˙ y − (R sin φ) ˙ ψ = 0. All the configurations are possible, but not all the velocities.

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Nonholonomic mechanical systems

A nonholonomic mechanical system is given by a lagrangian function L : TQ − → R subject to contraints determined by a linear distribution D

• n the configuration manifold Q. We will denote by D the total space of

the corresponding vector sub-bundle (τQ)|D : D − → Q defined by D, where (τQ)|D is the restriction of the canonical projection τQ : TQ − → Q. We will assume that the lagrangian L is defined by a Riemannian metric g on Q and a potential energy V ∈ C ∞(Q), so that L(vq) = 1 2 g(vq, vq) − V (q)

• r, in bundle coodinates (qi, ˙

qi) L(qi, ˙ qi) = 1 2 gij ˙ qi ˙ qj − V (qi)

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If {µa}, 1 ≤ a ≤ k is a local basis of the annihilator Do of D, then the constraints are locally given by µa

i (q) ˙

qi = 0, where µa = µa

i (q) dqi.

The nonholonomic equations can be written as d dt ∂L ∂ ˙ qi

• − ∂L

∂qi = λaµa

i (q)

µa(q) ˙ qi = 0, for some Lagrange multipliers λa to be determined. If we modify (11) as follows: iX ωL − dEL ∈ S∗((TD)o) (3) X ∈ TD (4) the unique solution Xnh is again a SODE whose solutions are just the

• nes of the nonholonomic equations.

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Let FL : TQ − → T ∗Q be the Legendre transformation given by FL(qi, ˙ qi) = (qi, pi = ∂L ∂ ˙ qi = gij ˙ qj) FL is a global diffeomorphism which permits to reinterpret the nonholonomical mechanical system in the hamiltonian side. Indeed, we denote by h = EL ◦ FL−1 the hamiltonian function and by M = FL(D) the constraint submanifold of T ∗Q. The nonholonomic equations are then given by dqi dt = ∂h ∂pi dpi dt = − ∂h ∂qi + ¯ λaµa

i ,

where ¯ λa are Lagrange multipliers to be determined.

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As above, the symplectic equation iXh ωQ = dh which gives the hamiltonian vector field Xh should be modified as follows: iX ωQ − dh ∈ F o (5) X ∈ TM (6) where F is a distribution along M whose annihilator F o is obtained from S∗((TD)o) throught FL. Equations (14) and (15) have a unique solution, the nonholonomic vector field Xnh.

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The Hamilton-Jacobi theory for Hamiltonian systems

The standard formulation of the Hamilton-Jacobi problem is to find a function S(t, qi) (called the principal function) such that ∂S ∂t + h(qi, ∂S ∂qi ) = 0, (7) where h = h(qi, pi) is the hamiltonian function of the system. If we put S(t, qi) = W (qi) − tE, where E is a constant, then W satisfies h(qi, ∂W ∂qi ) = E; (8) W is called the characteristic function. Equations (7) and (8) are indistinctly referred as the Hamilton-Jacobi equation.

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The Hamilton-Jacobi equation helps to solve the Hamilton equations for the hamiltonian h dqi dt = ∂h ∂pi , dpi dt = − ∂h ∂qi (9) Indeed, if we find a solution W of the Hamilton-Jacobi equation (8) then a solution (qi(t)) of the first set of equations (9) gives a solution of the Hamilton equations by taking pi(t) = ∂W

∂qi .

• R. Abraham, J.E. Marsden: Foundations of Mechanics (2nd edition).

Benjamin-Cumming, Reading, 1978.

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Let λ be a closed 1-form on Q, say dλ = 0; (then, locally λ = dW ) Hamilton-Jacobi Theorem The following conditions are equivalent: (i) If σ : I → Q satisfies the equation dqi dt = ∂h ∂pi then λ ◦ σ is a solution of the Hamilton equations; (ii) d(h ◦ λ) = 0

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Define a vector field on Q: X λ

h = TπQ ◦ Xh ◦ λ T ∗Q

πQ

• Xh

T(T ∗Q)

TπQ

• Q

λ

• Xλ

h

TQ The following conditions are equivalent: (i) If σ : I → Q satisfies the equation dqi dt = ∂h ∂pi then λ ◦ σ is a solution of the Hamilton equations; (i)’ If σ : I → Q is an integral curve of X λ

h , then λ ◦ σ is an integral

curve of Xh; (i)” Xh and X λ

h are λ-related, i.e.

Tλ(X λ

h ) = Xh ◦ λ

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Hamilton-Jacobi Theorem Let λ be a closed 1-form on Q. Then the following conditions are equivalent: (i) X λ

h and Xh are λ-related;

(ii) d(h ◦ λ) = 0 If λ = λi(q) dqi then the Hamilton-Jacobi equation becomes h(qi, λi(qj)) = const. and we recover the classical formulation when λi = ∂W ∂qi

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PROBLEM: How to extend the classical Hamilton-Jacobi theory for nonholonomic mechanical systems?

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V.V. Kozlov: On the integration theory of equations of nonholonomic mechanics, Regular and Chaotic Dynamics, 7 (2) (2002), 161-176.

• M. Pavon: Hamilton-Jacobi equations for nonholonomic dynamics, J. Math.
• Phys. 46, 032902.

G.E. Prince, J.E. Aldridge, G.B. Byrnes: A universal Hamilton-Jacobi equation for second-order ODEs, J. Phys. A: Math. Gen. 32 (1999), 827-844. V.V. Rumyantsev: Forms of Hamilton’s Principle for Nonholonomic Systems, Mechanics, Automatic Control and Robotics 2 (10) (2002), 1035-1048. A.S. Sumbatov: Nonholonomic Systems, Regular and Chaotic Dynamics, 7 (2) (2002), 221- 238.

• R. Van Dooren: The generalized Hamilton-Jacobi method for non-holonomic

dynamical systems of Chetaevs type, Zeitschrift Fur Angewandte Mathematik Und Mechanik 55, (1975) 407-411.

• R. Van Dooren: Motion of a rol ling disc by a new generalized Hamilton-Jacobi

method, Journal of Applied Mathematics and Physics 27 (1976) 501-505.

• R. Van Dooren: Second form of the generalized Hamilton-Jacobi method for

nonholonomic dynamical systems, Journal of Applied Mathematics and Physics 29 (1978) 828-834.

• R. Van Dooren: On the generalized Hamilton-Jacobi method for nonholonomic

dynamical systems, Dienst Analytische Mechanica, Tw, Vub (1979) 1-6.

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Hamilton-Jacobi theory for nonholonomic mechanical systems

Define a vector field on Q: X λ

nh = TπQ ◦ Xnh ◦ λ M

πQ

• Xnh

T(M)

TπQ

• Q

λ

• Xλ

nh

TQ Let λ be a 1-form on Q taking values into M and satisfying dλ ∈ I(Do). Then the following conditions are equivalent: (i) X λ

nh and Xnh are λ-related;

(ii) d(h ◦ λ) ∈ Do

• D. Iglesias, M. de Le´
• n, D. Mart´

ın de Diego: Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems, J. Phys. A: Math. Theor. 41 (2008),015205 (14 pp)

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• T. Oshawa, A.M. Bloch: Nonholonomic Hamilton-Jacobi equations and
• integrability. J. Geom. Mech. 1 4 (2009), 461–481.

When the system is completely nonholonomic, there is a nice result. Completely nonholonomic means that the iterared Lie brackets [D, D], [D, [D, D]], ... spans the tangent bundle. Then using the Chows theorem one can prove that there is no non-zero

• ne-form on Do.

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Using Lie algebroids

τD : D − → M a vector bundle, and τD∗ : D∗ − → M its dual vector bundle. D

τD

• D∗

τD∗

• M

A linear bivector1 ΛD∗ on D∗ (not Jacobi identity is required). We denote by { , }D∗ the corresponding almost-Poisson bracket. h : D∗ − → R a hamiltonian function.

1linear means that the bracket of two linear functions is a linear function

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The linear bivector ΛD∗ induces the following structure on D: an almost Lie bracket on the space Γ(τD) [ , ]D : Γ(τD) × Γ(τD) − → Γ(τD) (ξ1, ξ2) − → [ξ1, ξ2]D where

• [ξ1, ξ2]D = {

ξ1, ξ2}D∗ ([eα, eβ]D = C γ

αβeγ).

an anchor map ρD : Γ(τD) − → X(M) f ∈ C ∞(M), ξ ∈ Γ(D) ⇛ ρD(ξ)(f ) ◦ τD∗ = { ξ, f ◦ τD∗}D∗ (in coordinates, ρD(eα) = ρµ

α ∂ ∂xµ ).

Given local coordinates (xµ) in the base manifold M and a local basis of sections of D, {eα}, we induce local coordinates (xµ, yα) on D∗.

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Given Ω ∈ Γ(ΛkD∗) then dDΩ ∈ Γ(Λk+1D∗) and dDΩ(ξ0, ξ1, . . . , ξk) =

k

• i=0

(−1)iρD(ξi)(Ω(ξ0, . . . , ξi, . . . , ξk)) +

• i<j

Ω([ξi, ξj]D, ξ0, . . . , ξi, . . . , ξj, . . . , ξk) where ξ0, ξ1, . . . , ξk ∈ Γ(τD) From the definition, we deduce that (1) (dDf )(ξ) = ρD(ξ)(f ), f ∈ C ∞(M), ξ ∈ Γ(τD) (2) dDσ(ξ1, ξ2) = ρD(ξ1)(σ(ξ2)) − ρD(ξ2)(σ(ξ1)) − σ[ξ1, ξ2]D, σ ∈ Γ(τD∗), ξ1, ξ2 ∈ Γ(τD) (3) dD(Ω ∧ Ω′) = dDΩ ∧ Ω′ + (−1)kΩ ∧ dDΩ′, Ω ∈ Γ(ΛkD∗), Ω′ ∈ Γ(Λk′D∗) In general (dD)2 = 0 .

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A linear bivector ΛD∗ on D∗

• An almost Lie algebroid structure ([ , ]D, ρD) on D
• An almost differential dD : Γ(ΛkD∗) → Γ(Λk+1D∗) satisfying (1) and (2)

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Let ΛD∗ be a linear bivector on D and λ : M − → D∗ be a section of τD∗ : D∗ − → M D∗

τD∗

• Xh

TD∗

TτD∗

• M

λ

• X λ

h

TM We define X λ

h = TτD∗ ◦ Xh ◦ λ

It is easy to show that X λ

h (x) ∈ ρD(Dx), ∀x ∈ M

Hamilton-Jacobi Theorem Assume that dDλ = 0. Then (i) σ : I → M integral curve of X λ

h ⇒ λ ◦ σ integral curve of Xh

• (ii)

dD(h ◦ λ) = 0

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Application: Mechanical systems with nonholonomic constraints

Let G : E ×M E → R be a bundle metric on a Lie algebroid (E, [· , ·], ρ) The class of systems that were considered is that of mechanical systems with nonholonomic constraints determined by The Lagrangian function L: L(a) = 1 2G(a, a) − V (τ(a)), a ∈ E, with V a function on M The nonholonomic constraints determined by a subbundle D of E

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Consider the orthogonal decomposition E = D ⊕ D⊥, and the associated

• rthogonal projectors

P : E − → D Q : E − → D⊥ Take local coordinates (xµ) in the base manifold M and a local basis of sections of E (moving basis), {eα}, adapted to the nonholonomic problem (L, D), in the sense that (i) {eα} is an orthonormal basis with respect to G (that is G(eα, eβ) = δαβ) (ii) {eα} = {ea, eA} where D = span{ea}, D⊥ = span{eA}. ✲ ✑✑✑ ✑ ✸ ✻ D D⊥ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜

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Denoting by (xµ, y α) = (xµ, y a, y A) the induced coordinates on E, the constraint equations determining D just read y A = 0. Therefore we choose (xµ, y a) as a set of coordinates on D D

iD

• τD
• E

τ

• M

In these coordinates we have the inclusion iD : D − → E (xµ, y a) − → (xµ, y a, 0) and the dual map i∗

D :

E ∗ − → D∗ (xµ, ya, yA) − → (xµ, ya) where (xµ, yα) are the induced coordinates on E ∗ by the dual basis of {eα}.

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Moreover, from the orthogonal decomposition we have that P : E − → D (xµ, y a, y α) − → (xµ, y a) and its dual map P∗ : D∗ − → E ∗ (xµ, ya) − → (xµ, ya, 0) In these coordinates, the nonholonomic system is given by i) The Lagrangian L(xµ, y α) = 1

2

• α(y α)2 − V (xµ),

ii) The nonholonomic constraints y A = 0.

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The Legendre transformation associated with L is the isomorphism FL : E − → E ∗ induced by the metric G. Therefore, locally, the Legendre transformation is FL : E − → E ∗ (xµ, y α) − → (xµ, yα = y α) and we can define the nonholonomic Legendre transformation FLnh = i∗

D ◦ FL ◦ iD : D −

→ D∗ FLnh : D − → D∗ (xµ, y a) − → (xµ, ya = y a) Summarizing, we have the following diagram E

P

• FL

E ∗

i∗

D

• D
• iD
• FLnh

D∗

P∗

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The nonholonomic bracket (E, [ , ], ρ) is a Lie algebroid ⇓ ΛE ∗ is a linear Poisson structure on E ∗ If f1 and f2 are functions on M, and ξ1 and ξ2 are sections of E, then: {f1 ◦ τE ∗, g1 ◦ τE ∗}E ∗ = 0, { ξ1, f1 ◦ τE ∗, }E ∗ = (ρ(ξ1)) f1 ◦ τE ∗, { ξ1, ξ2}E ∗ = [ξ1, ξ2] In the induced coordinates (xµ, yα), the Poisson bracket relations on E ∗ are {xµ, xη}E ∗ = 0, {yα, xµ, }E ∗ = ρµ

α,

{yα, yβ}E ∗ = C γ

αβyγ

In other words ΛE ∗ = ρµ

α

∂ ∂yα ∧ ∂ ∂xµ + 1 2C γ

αβyγ

∂ ∂yα ∧ ∂ ∂yβ

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The nonholonomic bracket on D∗, { , }nh,D∗, is defined by {F, G}nh,D∗ = {F ◦ i∗

D, G ◦ i∗ D}E ∗ ◦ P∗

for all F, G ∈ C ∞(D∗) The induced bivector Λnh,D∗ is Λnh,D∗ = ρµ

a ∂ ∂ya ∧ ∂ ∂xµ + 1 2C c abyc ∂ ∂ya ∧ ∂ ∂yb

That is, {xµ, xη}nh,D∗ = 0, {ya, xµ}nh,D∗ = ρµ

a ,

{ya, yb}nh,D∗ = C c

abyc

Λnh,D∗ is a linear bivector on D∗, but in general, does not satisfy Jacobi identity.

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Particular cases

1

E = TM. Then the linear Poisson structure on E ∗ = T ∗M is the canonical symplectic

• structure. Thus, D is a distribution on M and { , }nh,D∗ is the nonholonomic bracket

studied by A.J. Van der Schaft, B.M. Maschke, and others.

2

E = g, where g is a Lie algebra. E is a Lie algebroid over a single point (the anchor map is the zero map). In this case, the linear Poisson structure on E ∗ = g∗ is the ± Lie-Poisson

• structure. Thus, if D = h is a vector subspace of g, we obtain that the nonholonomic

bracket (nonholonomic Lie-Poisson bracket) is given by {F, G}nh,D∗±(µ) = ±

• µ, P

δF δµ , δG δµ

• for µ ∈ h∗, and F, G ∈ C ∞(h∗). In adapted coordinates

{ya, yb}nh,D∗± = ±C c

abyc 3

E = the Atiyah algebroid associated with a principal G-bundle π : Q − → Q/G E = TQ/G The linear Poisson structure on E ∗ = T ∗Q/G is characterized by the following condition: the canonical projection T ∗Q − → T ∗Q/G is a Poisson epimorphism (See J.P. Ortega and T. S. Ratiu: Momentum maps and Hamiltonian reduction, Progress in Math., 222 Birkhauser, Boston 2004) D a G-invariant distribution on Q, D/G is a vector subbundle of E = TQ/G Thus, we obtain a reduced non-holonomic bracket { , }nh,D∗/G (the non-holonomic Hamilton-Poincare bracket on D∗/G)

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We return to the general case Taking the hamiltonian function H : E ∗ − → R defined by H(xµ, yα) = 1 2

• α

(yα)2 + V (xµ) then we induce a hamiltonian function h : D∗ − → R by taking h = H ◦ P∗. In coordinates, h(xµ, ya) = 1 2

• a

(ya)2 + V (xµ) The nonholonomic dynamics is determined on D∗ by the linear bivector Λnh,D∗ and the hamiltonian function h : D∗ − → R, that is ˙ F = {F, h}nh,D∗ So we can apply Hamilton-Jacobi theory to nonholonomic mechanics!

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A pure Poisson approach

Let (E, Λ) be an almost Poisson manifold, that is, Λ is a (2, 0) tensor field on a manifold E. We denote by ♯ : T ∗E − → TE the mapping defined by ♯(α), β = Λ(α, β) We denote by C the characteristic distribution defined by Λ, that is C(p) = ♯(T ∗

p E)

for all p ∈ E. Notice that C is a generalized distribution that is not (in general) integrable since Λ is not Poisson in principle. Definition A submanifold N of E is said to be a lagrangian submanifold if the following equality holds ♯(TN◦) = TN ∩ C

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Assume that π : E − → M is a fibration of E over a manifold M. So we have the diagram (E, Λ)

π

• M

To have dynamics we need to consider a hamiltonian function h : E − → R, and thus we obtain the corresponding hamiltonian field Xh = ♯(dh)

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Assume that γ is a section of π : E − → M, i.e. π ◦ γ = idM. We can use γ to project Xh on M just defining a vector field X γ

h on M by

X γ

h = Tπ ◦ Xh ◦ γ

The following diagram summarizes the above construction: E

π

• Xh

TE

Tπ

• M

γ

• X γ

h

TM

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Assume that Im(γ) is a lagrangian submanifold. Then we have the following theorem. Theorem [Hamilton-Jacobi] Under the previous assumptions, the following conditions are equivalent

1

dh ∈ (TIm(γ) ∩ C)◦.

2

Xh and X γ

h are γ-related.

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Assume that (E, Λ) is a transitive Poisson manifold, that is, C = TE. Then, we have Proposition A submanifold N of E is a lagrangian submanifold if and only if ♯(TN◦) = TN Note: Symplectic and cosymplectic manifolds are transitive Poisson manifolds. Therefore, the above theorem takes this form. Theorem Assume that γ(M) is a lagrangian submanifold of (E, Λ). Then the following assertions are equivalent:

1

Xh and X γ

h are γ-related;

2

d(h ◦ γ) = 0.

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Computations in local coordinates Assume that (xi, y a) are local coordinates adapted to the fibration π : E − → M, that is, π(xi, y a) = (xi), where (xi) are local coordinates in M. Proposition γ(M) is a lagrangian submanifold of (E, Λ) if and only if Λab + Λbj ∂γa ∂xj − Λaj ∂γb ∂xj + Λij ∂γa ∂xi ∂γb ∂xj = 0 (10)

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Classical hamiltonian systems A classical hamiltonian system is given by a hamiltonian function h defined on the cotangent bundle T ∗Q of the configuration manifold Q. In this case, E = T ∗Q and Λ is the canonical Poisson structure ΛQ on T ∗Q provided by the canonical symplectic form ωQ on T ∗Q. Recall that now we can take bundle coordinates (qi, pi) where πQ(qi, pi) = (qi), and πQ : T ∗Q − → Q is the canonical projection. Since in bundle coordinates ωQ = dqi ∧ dpi we deduce that ΛQ = ∂ ∂qi ∧ ∂ ∂pj

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Therefore, Xh = ∂h ∂pi ∂ ∂qi − ∂h ∂qi ∂ ∂pi and if a section γ : Q − → T ∗Q (that is, a 1-form on Q) is locally expressed by γ(qi) = (qi, γi(qi)) we obtain X γ

h = ∂h

∂pi ∂ ∂qi

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The notion of lagrangian submanifold defined in the almost-Poisson setting reduces to the well-known in the symplectic setting, that is, it is isotropic and coisotropic with respect to the symplectic form ωQ. If we compute the condition (10) in this case we obtain ∂γi ∂qj = ∂γj ∂qi which just means that γ is a closed form, i.e., dγ = 0. So we recover the classical result. Proposition Given a 1-form γ on Q, we have that γ(Q) is a lagrangian submanifold

• f (T ∗Q, ΛQ) if and only if γ is closed.

As a consequence, we deduce the classical result. Theorem Let γ be a closed 1-form on Q. Then the following assertions are equivalent:

1

Xh and X γ

h are γ-related;

2

d(h ◦ γ) = 0.

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Nonholonomic mechanical systems A nonholonomic mechanical system is given by a lagrangian function L : TQ − → R subject to contraints determined by a linear distribution D

• n the configuration manifold Q. We will denote by D the total space of

the corresponding vector sub-bundle (τQ)|D : D − → Q defined by D, where (τQ)|D is the restriction of the canonical projection τQ : TQ − → Q. We will assume that the lagrangian L is defined by a Riemannian metric g on Q and a potential energy V ∈ C ∞(Q), so that L(vq) = 1 2 g(vq, vq) − V (q)

• r, in bundle coodinates (qi, ˙

qi) L(qi, ˙ qi) = 1 2 gij ˙ qi ˙ qj − V (qi)

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If {µa}, 1 ≤ a ≤ k is a local basis of the annihilator Do of D, then the constraints are locally given by µa

i (q) ˙

qi = 0, where µa = µa

i (q) dqi.

The nonholonomic equations can be written as d dt ∂L ∂ ˙ qi

• − ∂L

∂qi = λaµa

i (q)

µa(q) ˙ qi = 0, for some Lagrange multipliers λi to be determined.

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Let S (respectively, ∆) the canonical vertical endomorphism (respectively the Liouville vector field) on TQ. In local coordinates, we have S = dqi ⊗ ∂ ∂ ˙ qi , ∆ = ˙ qi ∂ ∂ ˙ qi Therefore, we can construct the Poincar´ e-Cartan 2-form ωL = −dS∗(dL) and the energy function function EL = ∆(L) − L, such that the equation iξL ωL = dEL (11) has a unique solution, ξL, which is a SODE on TQ (that is, S(ξL) = ∆). Furthermore, its solutions coincide with the solutions of the Euler-Lagrange equations for L: d dt ∂L ∂ ˙ qi

• − ∂L

∂qi = 0 If we modify (11) as follows: iX ωL − dEL ∈ S∗((TD)o) (12) X ∈ TD (13) the unique solution Xnh is again a SODE whose solutions are just the

• nes of the nonholonomic equations.

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Let FL : TQ − → T ∗Q be the Legendre transformation given by FL(qi, ˙ qi) = (qi, pi = ∂L ∂ ˙ qi = gij ˙ qj) FL is a global diffeomorphism which permits to reinterpret the nonholonomical mechanical system in the hamiltonian side. Indeed, we denote by h = EL ◦ FL−1 the hamiltonian function and by M = FL(D) the constraint submanifold of T ∗Q. The nonholonomic equations are then given by dqi dt = ∂h ∂pi dpi dt = − ∂h ∂qi + ¯ λaµa

i ,

where ¯ λi are Lagrange multipliers to be determined.

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As above, the symplectic equation iXh ωQ = dh which gives the hamiltonian vector field Xh should be modified as follows: iX ωQ − dh ∈ F o (14) X ∈ TM (15) where F is a distribution along M whose annihilator F o is obtained from S∗((TD)o) throught FL. Equations (14) and (15) have a unique solution, the nonholonomic vector field Xnh.

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Another way to obtain Xnh is to consider the Withney sum decomposition T(T ∗Q)|M = TM ⊕ F ⊥ where the complement is taken with respect to ωQ. If P : T(T ∗Q)|M − → TM is the canonical projection onto the first factor, one easily proves that Xnh = P(Xh) Moreover, one can introduce an almost-Poisson tensorΛnh on M by Λnh(α, β) = ΛQ(P∗α, P∗β) which is called the nonholonomic bracket. Obviously, we have Xnh = ♯(dh)

• F. Cantrijn, M. de Le´
• n, D. Mart´

ın de Diego: On almost-Poisson structures in nonholonomic mechanics. Nonlinearity 12 (1999), 721–737.

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An alternative way to define the nonholonomic bracket is as follows.

• L. Bates, J. Sniatycki: Nonholonomic reduction. Rep. Math. Phys. 32

(1993), no. 1, 99-115. Consider the distribution TM ∩ F along M. A direct computation shows that the subspace TpM ∩ Fp is symplectic whithin the symplectic vector space (Tp(T ∗Q), ωQ(p)), for all p ∈ M. Thus we have a second Withney sum decomposition T(T ∗Q)|M = (TM ∩ F) ⊕ (TM ∩ F)⊥ where the complement is taken with respect to ωQ.

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If ˜ P : T(T ∗Q)|M − → TM ∩ F is the canonical projection onto the first factor, one easily proves that Xnh = ˜ P(Xh) Moreover, the nonholonomic almost-Poisson tensorΛnh on M is given by Λnh(α, β) = ΛQ(˜ P∗α, ˜ P∗β)

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Consider now the fibration (M, Λnh)

πQ |M

• Q

and the hamiltonian h|M (also denoted by h in the following). We can easily prove that Cp = TpM ∩ Fp Indeed,we have ♯nh(α), β = ωQ(˜ PXα, Xβ) = −ωQ(β, ˜ PXα) = (iXβ ωQ)(˜ PXα) = −β, ˜ PXα which implies ♯nh(α) = −˜ P(Xα) Furthermore, the symplectic structure Ω on Cp at a point p ∈ M is given by the restriction of the canonical symplectic structure on T ∗Q.

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Proposition Let γ : Q → M be a section of πQ|M : M − → Q, then Im(γ) is a lagrangian submanifold if and only if dγ(X, Y ) = 0 for all X, Y ∈ D. Proof: We notice that F = {v ∈ T(T ∗Q) such that TπQ(v) ∈ D} and an easy computation in local coordinates shows that dim(F ∩ TM) = 2 dim(D). Thus, we have TIm(γ) ∩ C = Tγ(D) On the other hand, it is clear that our definition of lagrangian submanifold is equivalent to TIm(γ) ∩ C be lagrangian with respect to the simplectic structure Ω on the vector space C. Since Ω is the restriction of ΩQ, given X, Y ∈ D we have Ω(Tγ(X), Tγ(Y )) = ΩQ(Tγ(X), Tγ(Y )) = dγ(X, Y ) So, after a careful counting of dimensions, we deduce that Im(γ) is lagrangian with respect to Λnh if and only if dγ(X, Y ) = 0 for all X, Y ∈ D.

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Using this proposition we can recover the Nonholonomic Hamilton-Jacobi Theorem as a consequence of the general Theorem. Theorem [Nonholonomic Hamilton-Jacobi] Given a hamiltonian h : M → R, and γ a 1-from on Q taking values in M, such that dγ(X, Y ) = 0 for all X, Y ∈ D. The following conditions are equivalent

1

Xh and X γ

h are γ-related.

2

dh ∈ (Tγ(D))◦ (which is in turns equivalent to d(h ◦ γ) ∈ D◦).

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• M. de Le´
• n, D. Iglesias-Ponte, D. Mart´

ın de Diego: Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems. Journal of Physics A: Math. Gen. (2008), no. 1, 015205, 14 pp.

• M. de Le´
• n, J.C. Marrero, D. Mart´

ın de Diego: Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic

• mechanics. J. Geom. Mech. 2 2 (2010), 159–198.
• T. Oshawa, A.M. Bloch: Nonholonomic Hamilton-Jacobi equations and
• integrability. J. Geom. Mech. 1 4 (2009), 461–481.

J.F. Cari˜ nena, X. Gracia, G. Marmo, E. Mart´ ınez, M. Mu˜ noz-Lecanda, N. Rom´ an-Roy: Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems. Int. J. Geom. Meth. Mod. Phys. 7 3 (2010), 431–454.

• M. Leok, T. Ohsawa, D. Sosa: Hamilton-Jacobi Theory for Degenerate

Lagrangian Systems with Holonomic and Nonholonomic Constraints. J.

• Math. Phys. 53 (7) (2012), DOI: 10.1063/1.4736733 .
• M. de Le´
• n, Manuel, D. Mart´

ın de Diego, David, M. Vaquero: A Hamilton-Jacobi theory on Poisson manifolds. J. Geom. Mech. 6 (2014), 1, 121-140.

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We will get a suitable expression for the nonholonomic bracket Λnh defined on the constraint submanifold M of T ∗Q. Let us recall that the constraints were defined through a distribution D

• n Q. Let D′ a complementary distribution of D in TQ and assume that

{Xα}, 1 ≤ α ≤ n − k is a local basis of D and that {Ya}, 1 ≤ a ≤ k is a local basis of D′. Notice that µa(Xα) = 0. Next we introduce new coordinates in T ∗Q as follows: ˜ pα = X i

αpi , ˜

pn−k+a = Y i

api

where Xα = X i

α

∂ ∂qi , Ya = Y i

a

∂ ∂qi In these new coordinates we deduce that the constraints become ˜ pn−k+a = 0 Therefore, we can take local coordinates (qi, ˜ pα) on M.

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A direct computation shows now that the nonholonomic bracket Λnh on M is given by Λnh(dqi, dqj) = 0 , Λnh(dqi, d˜ pα) = X i

α

Λnh(d˜ pα, d˜ pβ) = X i

βpj ∂X j

β

∂qi − X i αpj ∂X j

β

∂qi

In the sequel, we will apply then general theory developed in Section 2 to the almost-Poisson structure (M, Λnh). Assume that γ : Q − → M is a section of π : M − → Q. Then, we have γ(qi) = (qi, ˜ γα(qi)) Since γ can also be considered as a 1-form on Q taking values on M we have γ(qi) = (qi), γi(qi)) and since it takes values in M we get ˜ γα = X i

α γi

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A direct computation from equation (10) gives Λαβ

nh + Λβj nh

∂˜ γα ∂qj − Λαj

nh

∂˜ γβ ∂qj = X i

βγj

∂X j

α

∂qi − X i

αγj

∂X j

β

∂qi − X j

β

∂ ∂qj

• X i

αγi

• − X j

α

∂ ∂qj

• X i

βγi

• = X i

αX j β

∂γj ∂qi − ∂γi ∂qj

• = 0.

which can be equivalently written as dγ(Xα, Xβ) = 0 (16) Therefore, γ(Q) is a lagrangian submanifold of (M, Λnh) if and if dγ ∈ I(Do), where I(Do) denotes the ideal of forms generated by Do.

• Indeed. notice that (16) holds if and only if dγ =

a ξa ∧ µa, for some

1-forms ξa.

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Jacobi systems Let (E, Λ, Z) a Jacobi manifold. This means that Λ is a bivector field and Z a vector field such that [Λ, Λ] = 2 Z ∧ Λ; LZ Λ = 0. Jacobi manifolds include: Poisson manifolds Locally conformal symplectic manifolds contact manifolds We have the map ♯ : T ∗E − → TM defined by ♯(α), β = Λ(α, β).

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The hamiltonian vector field associated to a hamiltonian function h : E − → R is Xh = ♯(dh) + h Z The Jacobi bracket is defined by {f , g} = Λ(df , dg) + fZ(g) − gZ(f ) Notice that we do not have the Leibniz rule supp {f , g} ⊂ supp f ∩ supp g However [Xf , Xg] = X{f ,g} The characteristic distribution is now C = Im ♯ + Z

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An (2n + 1)-dimensional manifold E equipped with a 1-form η such that η ∧ (dη)n = 0 is called a contact manifold. A contact manifold is a Jacobi manifold equipped with the bivector given by ♭(X) = iX dη + η(X) η and the Reeb vector field iR dη = 0, iR η = 1 Definition A submanifold N of a contact manifold E is said to be a legendrian submanifold if it is an integral manifold of maximal dimension (n) of the distribution η = 0

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We have extended this definition for arbitray Jacobi manifolds as follows. Definition A submanifold N of a Jacobi manifold E is said to be a lagrangian-legendrian submanifold if the following equality holds ♯(TN◦) = TN ∩ C

• R. Ibanez, M. de Le´
• n, J.C. Marrero, D. Martin de Diego: Co-isotropic

and Legendre-Lagrangian submanifolds and conformal Jacobi morphisms. J- Phys. A: Math. Gen. 30 (1997), 5427–5444. Everything works if we delete the integrability condition.

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Assume that π : E − → M is a fibration of E over a manifold M. So we have the diagram (E, Λ)

π

• M

To have dynamics we need to consider a hamiltonian function h : E − → R, and thus we obtain the corresponding hamiltonian field Xh = ♯(dh) + h Z

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Assume that γ is a section of π : E − → M, i.e. π ◦ γ = idM. We can use γ to project Xh on M just defining a vector field X γ

h on M by

X γ

h = Tπ ◦ Xh ◦ γ

Also, we can project Z: Z γ = Tπ ◦ Z ◦ γ The following diagram summarizes the above constructions: E

π

• Xh

TE

Tπ

• M

γ

• X γ

h

TM

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Assume that Im(γ) is a lagrangian-legendrian submanifold of (E, Λ, Z) such that Z and Z γ are γ-related. Then we have the following theorem. Theorem [Hamilton-Jacobi] Under the previous assumptions, the following conditions are equivalent

1

dh ∈ (TIm(γ) ∩ C)◦.

2

Xh and X γ

h are γ-related.

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