Lie algebroid generalization of geometric mechanics (Q2782572)

This paper is an excellent report presenting recent progress in the geometric formulation of mechanics based on Lie algebroid approach. In the second section, the authors give a concise but complete review of geometric mechanics, and in the third section the fundamentals of Lie algebroids are recalled, including the notion of Nijenhuis structure. The exterior algebra of Lie algebroid and the corresponding differential operator are reviewed in section 4. In section 5, the proposals by \textit{P. Libermann} [Arch. Math., Brno 32, No. 3, 147-162 (1996; Zbl 0912.70009)] and \textit{A. Weinstein} [in Shadwick, William F.(ed.) et al., Mechanics days. Proceedings of a workshop, June 12, 1992. Providence, RI: American Mathematical Society. Fields Inst. Commun. 7, 207-231 (1996; Zbl 0844.22007)] are reviewed. A different approach is given by the authors in the present paper, based in a convenient prolongation of Lie algebroid. This approach permits to extend the notions of canonical endomorphism of the tangent bundle of a manifold, in such a way that it is possible to construct a convenient Poincaré-Cartan two-form and to derive the equations of motion. Finally, a variational principle in this framework is presented in the last section. More details can be obtained in the paper by \textit{E. Martínez} [Acta Appl. Math. 67, No. 3, 295-320 (2001; Zbl 1002.70013)].NEWLINENEWLINEFor the entire collection see [Zbl 0980.00030].