On the inverse problem of Lagrangian dynamics on Lie algebroids

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1 On the inverse problem of Lagrangian dynamics on Lie algebroids Liviu Popescu Abstract. In the present paper we start the study of the inverse problem of Lagrangian dynamics on Lie algebroids. Using the notion of J -regular section we prove the equivalence between the Helmholtz conditions and a Hamiltonian section on the prolongations of a Lie algebroid. M.S.C. 2010: 17B66, 45Q05. Key words: inverse problem; Lie algebroids. 1 Introduction The inverse problem of Lagrangian mechanics is a subject which has been studied intensively in the several decades and can be formulated as follows. Under what conditions a system of second order differential equations on a n-dimensional manifold M can be derived from a variational principle? One solution of this problem is known as the Helmholtz conditions. There are various approaches to derive the Helmholtz conditions and we refer to the survey [6] and the monographs [1], [5] for comments on the history of the problem. Generally, the framework of these studies is the tangent bundle T M of the manifold M. A generalization of the Helmholtz conditions can be found in [7] and the case when M is a Lie group is given in [2]. A Lie algebroid is a generalization of the tangent bundle. Using the geometry of Lie algebroids, Weinstain [14] developed a generalized theory of Lagrangian mechanics and obtained the equations of motions, using the Poisson structure on the dual of a Lie algebroid and Legendre transformation associated with a regular Lagrangian. Thus it is natural to extend the study of the inverse problem to the more general framework of Lie algebroids. In [12] the expressions of the Helmholtz conditions on Lie algebroids are given. In this paper we prove the equivalence between the Helmholtz conditions and a Hamiltonian section on the prolongations of a Lie algebroids over the vector bundle projections. BSG Proceedings 21. The International Conference Differential Geometry - Dynamical Systems DGDS-2013, October 10-13, 2013, Bucharest-Romania, pp c Balkan Society of Geometers, Geometry Balkan Press 2014.

2 140 Liviu Popescu 2 Preliminaries on Lie algebroids Let M be a differentiable, n-dimensional manifold and T M, π M, M its tangent bundle. A Lie algebroid [9] over the manifold M is the triple E, [, ] E, σ where π : E M is a vector bundle of rank m over M, whose C M-module of sections ΓE is equipped with a Lie algebra structure [, ] E and σ : E T M is a vector bundle homomorphism called the anchor which induces a Lie algebra homomorphism also denoted σ from ΓE to χm, satisfying the Leibniz rule [s 1, fs 2 ] E = f[s 1, s 2 ] E + σs 1 fs 2, for every f C M and s 1, s 2 ΓE. Therefore, we get σ[s 1, s 2 ] E = [σs 1, σs 2 ], [s 1, [s 2, s 3 ] E ] E + [s 2, [s 3, s 1 ] E ] E + [s 3, [s 1, s 2 ] E ] E = 0. If ω k E then the exterior derivative d E ω k+1 E is given by the formula d E ωs 1,..., s k+1 = k+1 1 i+1 σs i ωs 1,..., ŝi,..., s k+1 + i=1 + 1 i<j k+1 1 i+j ω[s i, s j ] E, s 1,..., ŝi,..., ˆ s j,...s k+1, where s i ΓE, i = 1, k + 1, and it follows that d E 2 = 0. Also, for ξ ΓE on can define the Lie derivative with respect to ξ by L ξ = i ξ d E +d E i ξ, where i ξ is the contraction with ξ. If we take the local coordinates x i on an open subset U M, a local basis {s α } of sections of the bundle π 1 U U generates local coordinates x i, y α on E. The local functions σαx, i L γ αβ x on M given by σs α = σα x i, [s α, s β ] E = L γ αβ s γ, i = 1, n, α, β, γ = 1, m, are called the structure functions of Lie algebroids. 2.1 The prolongation of a Lie algebroid over the vector bundle projection Let E, π, M be a vector bundle. For the projection π : E M we can construct the prolongation of E see [3], [10], [8]. The associated vector bundle is T E, π 2, E where T E = w E T w E with T w E = {u x, v w E x T w E σu x = T w πv w, πw = x M}, and the projection π 2 u x, v w = π E v w = w, where π E : T E E is the tangent projection. We have also the canonical projection π 1 : T E E given by π 1 u, v = u. The projection onto the second factor σ 1 : T E T E, σ 1 u, v = v will be the anchor of a new Lie algebroid over manifold E. An element of T E is said to be vertical if it is in the kernel of the projection π 1. We will denote V T E, π 2 V T E, E the vertical bundle of T E, π 2, E. The local basis of ΓT E is given by {X α, V α }, where [10] X α u = s α πu, σα x i, V α u = 0, u y α, u

3 On the inverse problem of Lagrangian dynamics on Lie algebroids 141 and / x i, / y α is the local basis on T E. The structure functions of T E are given by the following formulas σ 1 X α = σα x i, σ1 V α = y α, [X α, X β ] T E = L γ αβ X γ, [X α, V β ] T E = 0, [V α, V β ] T E = 0. The differential of sections of T E is determined by d E X α = 1 2 Lα βγx β X γ, d E V α = 0. Other canonical geometric objects see [8] are Euler section C = y α V α and the vertical endomorphism or tangent structure J = X α V α. A section S of T E is called semispray or second order differential equation -SODE if JS = C. In local coordinates a semispray has the expression 2.1 Sx, y = y α X α + S α x, yv α A nonlinear connection N on T E is an m dimensional distribution called horizontal distribution N : u E HT u E T E that is supplementary to the vertical distribution. This means that we have the following decomposition T u E = HT u E V T u E, for u E. A connection N on T E induces two projectors h, v : T E T E such that hρ = ρ h and vρ = ρ v for every ρ ΓT E, where h = 1 2 id + N, v = 1 2 id N. Locally, a connection can be expressed as NX α = X α 2Nα β V β, NV β = V β, where Nα β = Nα β x, y are the local coefficients of N. The sections δ α = X α h = X α N β α V β, generate a basis of HT E. The frame {δ α, V α } is a local basis of T E called the Berwald basis. The dual basis is {X α, δv α } where δv α = V α Nβ αx β. A semispray S with local coefficients S α determines an associated nonlinear connection N = L S J with local coefficients 2.2 Nα β = 1 Sβ 2 y α + yε L β αε. The inverse problem of Lagrangian dynamics on Lie algebroids is to give necessary and sufficient conditions for a system of second order differential equation to be the Euler-Lagrange equations of some regular Lagrangian function. One solution is known as the Helmholtz conditions. If S is a semispray on T E given by 2.1 and Nβ α are the coefficients of the associated nonlinear connection 2.2 then the Helmholtz conditions are concerned with the local existence of the functions g αβ such that [12] i det g αβ 0, g αβ = g βα, g αβ y ε = g αε y β, 2.3 ii Sg αβ g γβ Nα γ g γα N γ β = yε g γβ L γ εα + g γα L γ εβ, iii g αγ σβ S γ x i + SN γ β + N βn ε ε γ L δ εβn γ δ + Lγ δε N βy δ ε = g βγ σα S γ x i + SN α γ + NαN ε ε γ L δ εαn γ δ + Lγ δε N αy δ ε.

4 142 Liviu Popescu 2.2 The prolongation of a Lie algebroid to its dual bundle Let τ : E M be the dual bundle of π : E M and E, [, ] E, σ a Lie algebroid structure over M. One can construct a Lie algebroid structure over E, by taking the prolongation over τ : E M see [3], [8], [4]. The associated vector bundle is T E, τ 1, E where T E = u E T u E and T u E = {u x, v u E x T u E σu x = T u τv u, τ u = x M}, and the projection τ 1 : T E E, τ 1 u x, v u = u. The anchor is the projection σ 1 : T E T E, σ 1 u, v = v. Notice that if T τ : T E E, T τu, v = u then V T E, τ 1 V T E, E with V T E = KerT τ is a subbundle of T E, τ 1, E, called the vertical subbundle. If q i, µ α are local coordinates on E at u and {s α } is a local basis of sections of π : E M then a local basis of ΓT E is {Q α, P α } where [8] Q α u = s α τu, σα q i u, P α u = 0, u The structure functions on T E are given by the following formulas and therefore σ 1 Q α = σα q i, σ1 P α =,. [Q α, Q β ] T E = L γ αβ Q γ, [Q α, P α ] T E = 0, [P α, P β ] T E = 0, d E Q γ = 1 2 Lγ αβ Qα Q β, d E P α = 0, d E q i = σ i αq α, d E µ α = P α, where {Q α, P α } is the dual basis of {Q α, P α }. In local coordinates the Liouville section is given by θ E = µ α Q α. The canonical symplectic structure ω E is defined by ω E = d E θ E. It follows that ω E is a non degenerate 2-section, d E ω E = 0 and 2.4 ω E = Q α P α µ αl α βγq β Q γ. 3 Regular sections and the inverse problem of Lagrangian dynamics on Lie algebroids An almost tangent structure J on T E is a bundle morphism J : T E T E of τ 1 : T E E of rank m, such that J 2 = 0. An almost tangent structure J on T E is called adapted if imj = ker J = V T E. Locally, an adapted almost tangent structure is given by J = t αβ Q α P β, where the matrix t αβ x, µ is nondegenerate. structure if and only if [4] It follows that J is an integrable 3.1 where t αγ t γβ = δ α β. t αγ µ β = tβγ,

5 On the inverse problem of Lagrangian dynamics on Lie algebroids 143 Definition 3.1. An adapted almost tangent structure J on T E is called symmetric if 3.2 ω E J ρ 1, ρ 2 = ω E J ρ 2, ρ 1, ρ 1, ρ 2 ΓT E. Considering ρ 1 = ξ α 1 Q α + ρ 1β P β and ρ 2 = ξ α 2 Q α + ρ 2β P β we obtain ω E ξ α 1 t αβ P β, ξ α 2 Q α + ρ 2β P β = ω E ξ α 2 t αβ P β, ξ α 1 Q α + ρ 1β P β, which lead to the symmetry of t αβ. If g is a pseudo-riemannian metric on the vertical subbundle V T E i.e. a 0, 2-type symmetric E-tensor g = g αβ q, µp α P β of rank m on T E then there exists a unique symmetric adapted almost tangent structure on T E such that 3.3 gj ρ, J υ = ω E J ρ, υ, ρ 1, ρ 2 ΓT E, and we say that J is induced by the metric g. Locally, the relation 3.3 implies t αβ = g αβ. Definition 3.2. Let J be an adapted almost tangent structure on T E. A section ρ of T E is called J regular if 3.4 J [ρ, J ν] T E = J ν, ν ΓT E. Locally, the section ρ = ξ α Q α + ρ β P β is J regular if and only if [4] t αβ = ξβ, where t αβ t αγ = δ β γ. We have to remark that if the equation 3.4 is satisfied for any section ν ΓT E with rank[t αβ ] = m, then J is an integrable structure. Definition 3.3. A section ψ on T E is called the Hamilton section if it is J regular and L ψ ω E = 0, where ω E is the canonical symplectic section. If ψ = ξ α Q α + ρ α P α then by direct computation we obtain σα ξ β q i + ρ α ξ γ L β γα µ ε L ε γα µ β ξ γ Q α P β + µ β L ψ ω E = ξβ P α P β + σα ρ β q i ρ γl γ αβ + µ εξ γ σα L ε γβ q i + µ ε L ε γβσα ξ γ q i 1 2 µ εξ θ L ε θγl γ αβ and L ψ ω E = 0 leads to the equations [13] Q α Q β, 3.5 ξ β = ξα µ β, 3.6 σα ξ β q i + ρ α = ξ γ L β γα + µ ε L ε ξ γ γα, µ β µ β

6 144 Liviu Popescu 3.7 σβ ρ α q i ρ β σi α q i = µ ε ξ γ σβ L ε γα q i σα L ε γβ q i + L ε νγl ν αβ + ξ γ +µ ε σ i q i β L ε γα σαl i ε γβ ργ L γ αβ. Next, we consider a local diffeomorphism Φ to E to E given locally by 3.8 x i = q i, y α = ξ α q, µ, and its inverse Φ 1 has the following local coordinates expression 3.9 q i = x i, µ α = ζ α x, y, There always exists a local diffeomorphism Φ to E to E given, for instance, by Legendre transformation associated with a regular Hamiltonian on E and the Lagrangian is Lx, y = ζ α y α Hx, µ where the components ζ α x, y define a 1-section on E. From the condition for Φ 1 to be the inverse of Φ we get the following formulas: 3.10 V β ζ α Φ = g αβ, X β ζ α Φ = g αγ Q β ξ γ, 3.11 Φ P α = g αβ Φ 1 V β, Φ Q α = X α + Q α ξ β Φ 1 V β, 3.12 Φ 1 V α = g αβ P β, Φ 1 X α = Q α g γε Q α ξ ε P γ, where Φ is the tangent map of Φ and g αβ = ξ α / µ β, g αβ g βγ = δ α γ. We denote g αβ = g αβ Φ 1 by abuse. Let us consider S = y α X α + S α x, yv α a semispray on T E and Φ 1 : E E the diffeomorphism given by 3.9. Then we set: Theorem 3.1. The section ρ = Φ 1 S is a Hamiltonian section on T E if and only if S and the function g αβ = ζ α / y β satisfy the Helmholtz conditions and the equation X α ζ θ X θ ζ α = N ε θ g εα N ε αg εθ + ζ ε L ε αθ. Proof. Since Φ is a local diffeomorphism it results detg αβ 0. obtain Using 3.12 we Φ 1 S = Φ 1 y α X α + Φ 1 S α V α = ξ α Q α g γε Q α ξ ε P γ + S α g αβ P β = ξ α Q α + g γε ξ α Q α ξ ε + S α g αγ P γ. From the condition 3.5 it results ξβ = ξα µ β which leads to the first Helmholtz condition. The condition 3.6 lead to σγ ξ α q i + and using 3.10 it results ξ β g γε Q β ξ ε + S β g βγ = ξ β L α βγ + µ ε L ε ξ β βγ, Q γ ξ α +g βα X β ζ γ +ξ β X β g εγ g αε +P α S β g βγ +S β P α g βγ +ξ β L α γβ+µ ε L ε γβg βα = 0

7 On the inverse problem of Lagrangian dynamics on Lie algebroids 145 which is equivalent with g θβ X θ ζ α X α ζ θ + V θ S ε g εα + ξ ε X ε g αθ + S ε V ε g αθ + ζ ε L ε αθ + y ε L β αε = 0. From 2.2 we obtain 3.13 X α ζ θ X θ ζ α = Sg αθ 2N ε θ g εα + y β L ε θβg εα + y β L ε αβg εθ + ζ ε L ε αθ. The antisymmetric part of 3.13 leads to the equation Sg αθ N ε θ g εα N ε αg εθ + y β L ε θβg εα + L ε αβg εθ = 0 which is the second Helmholtz condition. The symmetric part of 3.13 leads also to the equation X α ζ θ X θ ζ α = N ε θ g εα N ε αg εθ + ζ ε L ε αθ. Finally, the section ρ = Φ 1 S is a Hamilton section if the condition 3.7 is satisfied. In the same way, by straightforward computation the third Helmholdz condition is obtained, which ends the proof. Acknowledgements. The author wishes to express his thanks to the referee for useful remarks concerning this paper and for the references [5], [7]. References [1] I. M. Anderson, G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Mem. Amer. Math. Soc., [2] M. Crampin, T. Mestdag, The inverse problem for invariant Lagrangians on a Lie group, J. Lie Theory, 18, no [3] P. J. Higgins, K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, [4] D. Hrimiuc, L. Popescu, Nonlinear connections on dual Lie algebroids, Balkan J. Geom. Appl., 11, [5] O. Krupkova, The geometry of ordinary variational equations, New York: Springer Verlag, [6] O. Krupkova, G. E. Prince, Second-order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations, In the Handbook of Global Analysis ed. D. Krupka and D. Saunders, Elsevier [7] O. Krupkova., M. Radka, Helmholtz conditions and their generalizations, Balkan J. Geom. Appl., 15, [8] M. de Leon, J. C. Marrero, E. Martinez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., [9] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, Cambridge, 124, [10] E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67, [11] L. Popescu, Geometrical structures on Lie algebroids, Publ. Math. Debrecen, 72,

8 146 Liviu Popescu [12] L. Popescu, Metric nonlinear connections on Lie algebroids, Balkan J. Geom. Appl., 16, [13] L. Popescu, Dual structures on the prolongations of a Lie algebroid, An. Stiinţ. Univ. Al. I. Cuza Iaşi. Mat. N.S., 59, [14] A. Weinstain, Lagrangian mechanics and groupoids, Fields Inst. Comm., Author s address: Liviu Popescu Department of Applied Mathematics, University of Craiova 13, Al. I. Cuza, st., Craiova , Romania. liviupopescu@central.ucv.ro, liviunew@yahoo.com