Lagrangian and Hamiltonian formalism for constrained variational problems

Abstract

ABSTRACT. We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ T M on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set Ω P Q (M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M . Under suitable assumptions, the set Ω P Q (M, D) has the structure of a smooth Banach manifold and we can thus study the critical points of L. If the Lagrangian L satisfies an appropriate hyper-regularity condition, we associate to it a degenerate Hamiltonian H on T M * using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L. In the particular case where L is given by the quadratic form corresponding to a positive definite metric on D, we obtain the well-known characterization of the normal geodesics in sub-Riemannian geometry (see [10]); by adding a potential energy term to L, we reobtain the equations of motion for the Vakonomic mechanics with non holonomic constraints (see [8]).

Variational calculus on Lie algebroids

“…It is well known that C(J, F ) is a Banach manifold (see [1,29]). In particular we will consider the case of curves on the tangent bundle, F = T M, and the case of curves on our Lie algebroid, F = E. The set of E-paths…”

Lie Algebroids in Classical Mechanics and Optimal Control

“…We remark that P(J, E) m 1 m 0 is a Banach submanifold of P(J, E), since it is a disjoint union of Banach submanifolds (the E-homotopy classes of curves with base path connecting such points). On the contrary, there is no guaranty that the analog set A(J, E) m 1 m 0 is a manifold (see [22]). Theorem 2 ( [20]).…”

Variational integrators in discrete vakonomic mechanics

“…This definition is no more than the translation to the discrete case of the stationarity notion of vakonomic mechanics (see [1,9,11,13,21]), arising here again the question whether the stationarity of an admissible section s = (q 0 , . .…”

“…Remark As can be seen, the essential difference with the smooth case is that, in the discrete formalism that we are considering, all the manifolds of sections involved in the subject are finite-dimensional, eliminating in this way any functional analysis difficulties present when we have infinite dimensional differentiable manifolds (see, for example [21]). …”

“…A doctrine of renewed interest in the last years is, certainly, the nonholonomic mechanics, either in its properly speaking nonholonomic version or in its vakonomic formulation [3,4,8,11,13,17,20,21]. With roots in the famous d'Alambert principle and in the well-known Lagrange's variational problem, respectively, its actual interest has been addressed very specially to the field theory (see, e.g.…”