Variational calculus on Lie algebroids

Abstract

Abstract. It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.Mathematics Subject Classification. 49S05, 49K15, 58D15, 70H25, 17B66, 22A22.

Variational calculus with constraints on general algebroids

“…Note that we use here 'infinite-dimensional manifold' structures in a very intuitive sense. However, we could have put rigorously a Banach manifold structure on M, N , etc, similarly as it has been done in [33]. On the other hand, because the Implicit Function Theorem will be not used, a less formal language is completely satisfactory for our purposes, so we will skip technical complications associated with the Banach manifold setting.…”

“…In [33] admissible variations are constructed out of homotopies of admissible paths as defined in [8]. For general algebroids we need different way of constructing admissible variations, since we have to accept the fact that they are not tangent to the submanifold of admissible paths in general.…”

“…For a general algebroid the admissible variations are constructed from the vertical ones by means of the double vector bundle relation κ = κ ε : TE−−✄TE which is dual to the morphism ε. Of course, for Lie algebroids our admissible paths coincide with the infinitesimal homotopies of admissible paths associated with the lifts of time-dependent sections, as they appear in [33,8]. We prefer a more fundamental approach which uses κ ε to produce admissible variations out of the vertical ones, instead of lifting whole sections extending paths in E and showing that the result does not depend on the extension.…”

Lie Algebroids in Classical Mechanics and Optimal Control

“…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]). Let L ∈ C ∞ (E) be a Lagrangian function on the Lie algebroid E and fix two points m 0 , m 1 ∈ M .…”

Nonholonomic constraints: A new viewpoint

“…In the simplest situation, for D being a linear nonholonomic constraint, i.e. just a vector subbundle of TM (or, of E in the algebroid context), this procedure describes the nonholonomic EulerLagrange equations by means of the d'Alembert principle, having analogs also in the algebroid case [7,18,27,8]. We should stress that our nonholonomic constraints are linear in the broader sense, i.e.…”