Abstract
The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles, Lie algebras, principal bundles, reduced systems, and constrained are included in such description. In this paper, we investigate how to derive the dynamics associated with a Lagrangian system defined on the set of admissible elements of a given Lie algebroid using Tulczyjew’s triple on Lie algebroids and constructing a Lagrangian Lie subalgebroid of a symplectic Lie algebroid, by building on the geometric formalism for mechanics on Lie algebroids developed by M. de Leon, J.C. Marrero and E. Martinez on “Lagrangian submanifolds and dynamics on Lie algebroids”.
Highlights
Lie algebroids have deserved a lot of interest in the last decades in the field of Geometric Mechanics since these spaces generalize the traditional framework of tangent bundles to more general situations (Lie algebras, principal bundles, semi-direct products), including, for instance, systems with symmetries and constrained [15], [16], [22], [26]
If a Lagrangian specifying the dynamics of a mechanical systems is given, and it is invariant under a Lie group of symmetries, the description of the system relies on the geometry of a quotient space under the action of the Lie group and the dynamics is governed by Lagrange-Poincare equations
The geometrical description of higher-order mechanical systems on Lie algebroids have been recently studied by the interest in the applications mentioned before and its subjacent geometry [2], [3], [18], [19], [29]
Summary
Lie algebroids have deserved a lot of interest in the last decades in the field of Geometric Mechanics since these spaces generalize the traditional framework of tangent bundles to more general situations (Lie algebras, principal bundles, semi-direct products), including, for instance, systems with symmetries and constrained [15], [16], [22], [26]. In this work we study the unification of both approaches of higher-order mechanics on Lie algebroids for second-order Lagrangian systems into a symplectic framework by understanding the derivation of the dynamics using a Lagrangian Lie subalgebroid of a symplectic Lie algebroid based, and building, on the work done by de Leon, Marrero and Martınez in [21] In such a work the authors propose an interpretation of the equations of motions based in the construction of a Tulczyjew triple in prolongations of Lie algebroids which consists of three prolongations of Lie algebroids retaining the dynamics into a symplectic framework that does not depends on the choice of the Lagrangian and its regularity, and it connect all the symplectic structures of the symplectic Lie algebroids that appear in the triple by symplectomorphisms.
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