Hamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics Manuel de León Institute of Mathematical Sciences CSIC, Spain joint work with J.C. Marrero (University of La Laguna) D. Martín de Diego (CSIC) Geometric Mechanics: Continuous and discrete, finite and infinite dimensional Banff, August 12 17, 2007
INDEX 1. Classical Hamilton-Jacobi theory (geometric version) 2. Geometric Hamilton-Jacobi Theory on almost Lie algebroids 3. An application to nonholonomic mechanical systems
The standard formulation of the Hamilton-Jacobi problem is to find a function S(t, q A ) (called the principal function) such that S t + H(qA, S qa) = 0. (1) If we put S(t, q A ) = W (q A ) te, where E is a constant, then W satisfies H(q A, W qa) = E; (2) W is called the characteristic function. Equations (1) and (2) are indistinctly referred as the Hamilton- Jacobi equation. R. Abraham, J.E. Marsden: Foundations of Mechanics (2nd edition). Benjamin-Cumming, Reading, 1978.
Classical Hamilton-Jacobi theory (geometric version) T M i Xh ω M = dh τ T M λ M Let λ be a closed 1-form (dλ = 0) T M X h T (T M) λ τ T M T τ T M M X λ h T M X λ h = T τ T M X h λ is a vector field on M
Hamilton-Jacobi Theorem Let λ be a closed 1-form (dλ = 0) (i)σ : I M integral curve of Xh λ λ σ integral curve of X h (ii) d(h λ) = 0 The condition (i) σ : I M integral curve of Xh λ λ σ integral curve of X h, is equivalent to (i) X h and Xh λ are λ-related (i.e. T λ(xλ h ) = X h).
Basic tools in Classical Hamilton-Jacobi theory T M τ T M M vector bundle over a manifold M The canonical symplectic 2-form ω M in T M The canonical Poisson 2-vector Λ T M on T M a linear bivector on the dual of the vector bundle. A hamiltonian function h : T M R A function h defined on the dual of the vector bundle A section λ : M T M such that dλ = 0 A section of the dual of the vector bundle which is closed with respect to the induced differential.
Geometric Hamilton-Jacobi Theory Ingredients: τ D : D M a vector bundle, and τ D vector bundle. D τ D M τ D D : D M its dual A linear bivector 1 Λ D on D (not Jacobi identity is required). We denote by {, } D the corresponding almost-poisson bracket. h : D R a hamiltonian function. 1 linear means that the bracket of two linear functions is a linear function
Proposition 1 We have that: Λ D is linear (a) ξ 1, ξ 2 Γ(τ D ) { ξ 1, ξ 2 } D on D, is a linear function (b) ξ Γ(τ D ), f C (M) { ξ, f τ D } D is a basic function with respect to τ D, (c) f, g C (M) {f τ D, g τ D } D = 0
Given local coordinates (x µ ) in the base manifold M and a local basis of sections of D, {e α }, we induce local coordinates (x µ, y α ) on D and the bivector Λ D is written as Λ E = ρ µ α y α x + 1 µ 2 Cγ αβ y γ y α y β The corresponding Hamiltonian vector field is or, in coordinates, ( X h = ρ µ h α y α x µ X h = ΛD (dh) ρ µ α Thus, the Hamilton equations are dx µ ( dt = h dy α ρµ α, y α dt = ) h x + µ Cγ αβ y h γ y β y α ρ µ α ) h x + µ Cγ αβ y h γ y β
Almost Lie algebroid structure on τ D : D M The linear bivector Λ D induces the following structure on D: an almost Lie bracket on the space Γ(τ D ) where [, ] D : Γ(τ D ) Γ(τ D ) Γ(τ D ) (ξ 1, ξ 2 ) [ξ 1, ξ 2 ] D [ξ 1, ξ 2 ] D = { ξ 1, ξ 2 } D ([e α, e β ] D = C γ αβ e γ). an anchor map ρ D : Γ(τ D ) X(M) f C (M), ξ Γ(D) ρ D (ξ)(f) τ D = { ξ, f τ D } D (in coordinates, ρ D (e α ) = ρ µ α x µ ).
Properties a) [, ] D is antisymmetric b) [ξ 1, fξ 2 ] D = f[ξ 1, ξ 2 ] D + ρ D (ξ 1 )(f)ξ 2 In general, [, ] D does not satisfy the Jacobi identity. In the case when it satisfies the Jacobi identity we say that (D, [, ] D, ρ D ) is a Lie algebroid.
The almost differential d D : Γ(Λ k D ) Γ(Λ k+1 D ) Given Ω Γ(Λ k D ) then d D Ω Γ(Λ k+1 D ) and d D Ω(ξ 0, ξ 1,..., ξ k ) = k ( 1) i ρ D (ξ i )(Ω(ξ 0,..., ξ i,..., ξ k )) i=0 + Ω([ξ i, ξ j ] D, ξ 0,..., ξ i,..., ξ j,..., ξ k ) i<j where ξ 0, ξ 1,..., ξ k Γ(τ D ) From the definition, we deduce that (1) (d D f)(ξ) = ρ D (ξ)(f), f C (M), ξ Γ(τ D ) (2) d D σ(ξ 1, ξ 2 ) = ρ D (ξ 1 )(σ(ξ 2 )) ρ D (ξ 2 )(σ(ξ 1 )) σ[ξ 1, ξ 2 ] D, σ Γ(τ D ), ξ 1, ξ 2 Γ(τ D ) (3) d D (Ω Ω ) = d D Ω Ω + ( 1) k Ω d D Ω, Ω Γ(Λ k D ), Ω Γ(Λ k D ) In general (d D ) 2 0.
A linear bivector Λ D on D An almost Lie algebroid structure ([, ] D, ρ D ) on D An almost differential d D : Γ(Λ k D ) Γ(Λ k+1 D ) satisfying (1) and (2)
The inverse process also works An almost differential d D :Γ(Λ k D ) Γ(Λ k+1 D ) satisfying(1)and (2) An almost Lie algebroid structure ([, ] D, ρ D ) on D ρ D (ξ)(f) = d D (f)(ξ), ω([ξ, ξ ] D ) = (d D ω)(ξ, ξ ) + d D (ω(ξ ))(ξ) d D (ω(ξ))(ξ ) ξ, ξ Γ(τ D ), f C (D), ω Γ(τ D )
An almost Lie algebroid structure ([, ] D, ρ D ) on D A linear bivector Λ D on D with almost Poisson bracket {, } D { ξ, ξ } D = [ξ, ξ ] D, { ξ, f τ D } D = ρ D (ξ)(f) τ D, {f τ D, f τ D } D = 0 f, f C (M), ξ, ξ Γ(τ D )
In conclusion A linear bivector Λ D on D An almost Lie algebroid structure ([, ] D, ρ D )on D An almost differential d D :Γ(Λ k D ) Γ(Λ k+1 D )satisfying(1)and (2)
Hamilton-Jacobi Theorem Let Λ D be a linear bivector on D and λ : M D be a section of τ D : D M D X h T D λ τ D T τ D M We define X λ h = T τ D X h λ X λ h T M It is easy to show that X λ h (x) ρ D(D x ), x M Indeed, look the local expressions Xh λ = ρ µ h h α = ρ( e y α x µ α ) y α
Hamilton-Jacobi Theorem Assume that d D λ = 0. (i) σ : I M integral curve of X λ h λ σ integral curve of X h (ii) d D (h λ) = 0
For the proof, we will need the following preliminary results (Propositions 2 and 3). Proposition 2 Let λ : M D be a section of τ D. Then, λ is a 1-cocycle with respect to d D (i.e. d D λ = 0) for all x M the subspace L λ,d (x) = (T x λ)(ρ D (D x )) T λ(x) D is Lagrangian with respect to Λ D, that is, ΛD (L λ,d ) o = L λ,d Remark: Proposition 2 is the generalization of the well-known result for the particular case D = T M and Λ D = Λ T M: Let λ be a 1-form on M; then, λ is closed if and only if λ(m) is a Lagrangian submanifold of T M.
Proposition 3 Let λ : M D be a section of τ D d D λ = 0. Then such that (ker ΛD ) λ(x) (L λ,d ) o, for all x M Remark: In the particular case when D = T M this Proposition is trivial since ker ΛT M = {0} ((T M, ω M ) is a symplectic manifold). Remember that ΛD : T λ(x) D T λ(x) D
Proof of the Theorem Let λ : M D be a section such that d D λ = 0. (i) (ii) We assume that the integral curves of X λ h and X h are λ-related, that is, X λ h and X h are λ-related. Moreover, we know that X λ h (x) ρ D(D x ), x M. Therefore, X h (λ(x)) (T x λ)(ρ D (D x )) = L λ,d (x), for all x M. From Proposition 1 (L λ,d is lagrangian) we deduce that X h (λ(x)) = ΛD (η λ(x) ), for some η λ(x) (L λ,d ) o Moreover from the definition of hamiltonian vector field Thus, X h (λ(x)) = ΛD (dh(λ(x)) η λ(x) dh(λ(x)) ker ΛD (λ(x)) L λ,d (x) o (by Proposition 2) Then, dh(λ(x)) L λ,d (x) o, x M. Finally, if a x D x, then d D (h λ)(x)(a x ) = ρ D (a x )(h λ) = (T x λ)(ρ D (a x ))(h) = dh(λ(x))(t x λ)(ρ D (a x )) = 0
(ii) (i) The condition d D (h λ) = 0 implies that Then dh(λ(x)) L λ,d (x) o, x M X H (λ(x)) = ΛD (dh)(λ(x)) ΛD (L λ,d (x) o ) = L λ,d (Proposition 1) = T x λ(ρ D (D x )) Therefore X λ h and X h are λ-related and we conclude (i).
Local expression of the Hamilton-Jacobi equations Take local coordinates (x µ ) in the base manifold M, a local basis of sections of D, {e α }, and induced coordinates (x µ, y α ) on D. Then if λ : (x µ ) (x µ, λ α (x µ )) (x, λ(x)) we have is locally written as d D (h λ) = 0 0 = d D (h λ)(e α ) x = ρ D (x)(e α (x))(h λ) = ρ µ α(x) x µ(h λ) x [ h = ρ µ h α(x) x µ(x, λ(x)) + (x, λ(x)) λ ] β y β x µ(x) The Hamilton-Jacobi Equations [ h ρ µ h α(x) x µ(x, λ(x)) + (x, λ(x)) λ ] β y β x µ(x) = 0, α
M. de León, J.C. Marrero, E. Martínez: Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen. 38 (2005), R241 308. (Topical review). J. Cortés, M. de León, J.C. Marrero, E. Martínez, D. Martín de Diego: A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Meth. Mod. Phys. 3 (3) (2006), 509 558. J. Cortés, M. de León, J.C. Marrero, E. Martínez: Nonholonomic Lagrangian systems on on Lie algebroids, Preprint (2006). D. Iglesias, M. de León, D. Martín de Diego: Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems, Preprint (2007).
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: with V a function on M L(a) = 1 G(a, a) V (τ(a)), a E, 2 The nonholonomic constraints determined by a subbundle D of E
Consider the orthogonal decomposition E = D D, and the associated orthogonal projectors P : E D Q : E D Take local coordinates (x µ ) in the base manifold M and a local basis of sections of E (moving basis), {e α }, adapted to the nonholonomic problem (L, D), in the sense that (i) {e α } is an orthonormal basis with respect to G (that is G(e α, e β ) = δ αβ ) (ii) {e α } = {e a, e A } where D = span{e a }, D = span{e A }. D D
Denoting by (x µ, y α ) = (x µ, y a, y A ) the induced coordinates on E, the constraint equations determining D just read y A = 0. Therefore we choose (x µ, y a ) as a set of coordinates on D D i D τ D τ M In these coordinates we have the inclusion and the dual map i D : E D E (x µ, y a ) (x µ, y a, 0) i D : E D (x µ, y a, y A ) (x µ, y a ) where (x µ, y α ) are the induced coordinates on E by the dual basis of {e α }.
Moreover, from the orthogonal decomposition we have that and its dual map P : E D (x µ, y a, y α ) (x µ, y a ) P : D E (x µ, y a ) (x µ, y a, 0)
In these coordinates, the nonholonomic system is given by i) The Lagrangian L(x µ, y α ) = 1 2 α (yα ) 2 V (x µ ), ii) The nonholonomic constraints y A = 0.
In this case, the Legendre transformation associated with L is the isomorphism F L : E E induced by the metric G. Therefore, locally, the Legendre transformation is F L : E E (x µ, y α ) (x µ, y α = y α ) and we can define the nonholonomic Legendre transformation F L nh = i D F L i D : D D F L nh : D D (x µ, y a ) (x µ, y a = y a ) Summarizing, we have the following diagram E F L E P i D i D P D F L nh D
The nonholonomic bracket A.J. Van der Schaft, B.M. Maschke: On the Hamiltonian formulation of nonholonomic mechanical systems, Reports on Mathematical Physics, 34 (2) (1994), 225-233. Ch.M. Marle: Various approaches to conservative and nonconservative nonholonomic systems, Reports on Math Phys., 42, (1998) 211-229. W.S. Koon, J.E. Marsden: Poisson reduction of nonholonomic mechanical systems with symmetry, Reports on Math Phys., 42, (1998) 101-134. L. Bates: Examples of singular nonholonomic reduction. Rep. Math. Phys. 42 (1998), no. 1-2, 231 247. A. Ibort, M. de León, J.C. Marrero, D. Martín de Diego: Dirac brackets in constrained dynamics. Forschritte der Physik. 47 (1999) 5, 459-492. F. Cantrijn, M. de León, D. Martín de Diego: On almost Poisson structures in nonholonomic mechanics, Nonlinearity 12 (1999), 721-737. F. Cantrijn, M. de León, J.C. Marrero, D. Martín de Diego: On almost Poisson structures in nonholonomic mechanics II: The time-dependent framework, Nonlinearity 13 (2000), no. 4, 1379 1409. Juan-Pablo Ortega, V. Planas-Bielsa: Dynamics on Leibniz manifolds. J. Geom. Phys. 52 (2004), no. 1, 1 27. H. Cendra, S. Grillo: Generalized nonholonomic mechanics, servomechanisms and related brackets. J. Math. Phys. 47 (2006), no. 2, 022902.
(E, [, ], ρ) is a Lie algebroid Λ E is a linear Poisson structure on E If f 1 and f 2 are functions on M, and ξ 1 and ξ 2 are sections of E, then: {f 1 τ E, g 1 τ E } E = 0, { ξ 1, f 1 τ E, } E = (ρ(ξ 1 )) f 1 τ E, { ξ 1, ξ 2 } E = [ξ 1, ξ 2 ] In the induced coordinates (x µ, y α ), the Poisson bracket relations on E are {x µ, x η } E = 0, {y α, x µ, } E = ρ µ α, {y α, y β } E = C γ αβ y γ In other words Λ E = ρ µ α y α x + 1 µ 2 Cγ αβ y γ y α y β
The nonholonomic bracket on D, {, } nh,d, is defined by for all F, G C (D ) The induced bivector Λ nh,d is {F, G} nh,d = {F i D, G i D } E P That is, Λ nh,d = ρ µ a y a x µ + 1 2 Cc ab y c y a y b {x µ, x η } nh,d = 0, {y a, x µ } nh,d = ρ µ a, {y a, y b } nh,d = C c ab y c Λ nh,d is a linear bivector on D, but in general, does not satisfy Jacobi identity. So, we are in the case considered in the very beginning.
Particular cases 1. E = T M. 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 {y a, y b } nh,d ± = ±C c aby c
3. E = the Atiyah algebroid associated with a principal G-bundle π : Q Q/G E = T Q/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 the Hamilton-Poincare bracket on T Q/G (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 = T Q/G Thus, we obtain a reduced non-holonomic bracket {, } nh,d /G (the non-holonomic Hamilton-Poincaré bracket on D /G)
We return to the general case Taking the hamiltonian function H : E R defined by H(x µ, y α ) = 1 (y α ) 2 + V (x µ ) 2 α then we induce a hamiltonian function h : D R by taking h = H P. In coordinates, h(x µ, y a ) = 1 (y a ) 2 + V (x µ ) 2 a
The nonholonomic dynamics is determined on D by the linear bivector Λ nh,d and the hamiltonian function h : D R, that is F = {F, h} nh,d or, in coordinates, by the equations ẋ µ ẏ a = ρ µ a h y a = ρ µ ay a = C c aby c h y b ρ µ a = C c aby c y b ρ µ a h x µ V x µ we can apply Hamilton-Jacobi theory to nonholonomic mechanics!
An example: The mobile robot with fixed orientation The robot has three wheels with radius R, which turn simultaneously about independent axes, and perform a rolling without sliding over a horizontal floor. Let (x, y) denotes the position of the centre of mass, θ the steering angle of the wheel, ψ the rotation angle of the wheels in their rolling motion over the floor. So, the configuration manifold is M = S 1 S 1 R 2
The lagrangian L is L = 1 2 mẋ2 + 1 2 mẏ2 + 1 2 J θ 2 + 3 2 J ωψ 2 where m is the mass, J is the moment of inertia and J ω is the axial moment of inertia of the robot.
The constraints giving the distribution D are induced by the conditions that the wheels roll without sliding, in the direction in which they point, and that the instantaneous contact point of the wheels with the floor have no velocity component orthogonal to that direction: ẋ sin θ ẏ cos θ = 0, ẋ cos θ + ẏ sin θ R ψ = 0. The vector fields e 1 = 1 J θ [ 1 e 2 = R cos θ mr2 + 3J ω x + R sin θ y + ] ψ are an orthonormal basis generating D.
Moreover, Therefore [e 1, e 2 ] = The linear bivector is ( 1 R sin θ J(mR2 + 3J ω ) x + R cos θ ) y Λ nh,d = 1 J y 1 θ + [e 1, e 2 ] D = 0 C 1 12 = C 2 12 = 0 R sin θ + mr2 + 3J ω R cos θ mr2 + 3J ω y 2 x y 2 y + 1 mr2 + 3J ω y 2 ψ
For any section λ : M D : the condition (x, y, θ, ψ) (x, y, θ, ψ, λ 1 (x, y, θ, ψ), λ 2 (x, y, θ, ψ)) d D λ = 0 e 1 (λ 2 ) e 2 (λ 1 ) = 0 Now it is trivial to show that taking λ 1 = k 1 and λ 2 = k 2 where k 1, k 2 are constants then since d D (h λ) = 0 h = 1 2 (y2 1 + y 2 2) Then, to integrate the nonholonomic problem is equivalent to integrate the vector fields on the configuration space: [ Xh λ = k 1 θ + k 2 R cos θ x + R sin θ y + ] ψ where (k 1, k 2) R 2.