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Lagrangian mechanics and groupoids. (English) Zbl 0844.22007
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).
From the author’s introduction: “A recent paper of J. Moser and A. P. Veselov [Comm. Math. Phys. 139, No. 2, 217-243 (1991; Zbl 0754.58017)] on the complete integrability of certain discrete dynamical systems uses the Lagrangian and Hamiltonian formalisms for discrete mechanics in two different settings. In the first, one begins with a suitably nondegenerate Lagrangian function \(L\) on the Cartesian square \(M \times M\) of a configuration space \(M\) and obtains a second-order recursion relation on \(M\); i.e. a map which assigns to each pair \((x,y)\) a pair \((y,z)\). The corresponding Hamiltonian system is the canonical transformation of \(T^*M\) for which \(L\) is the generating function in a very classical sense. In the second setting, the Lagrangian \(L\) is defined on a Lie group \(G\), and the dynamical system is given by a diffeomorphism from \(G\) to itself. The corresponding Hamiltonian system is the mapping from the dual Lie algebra \({\mathfrak g}^*\) to itself for which \(L\) is the generating function in the sense of Z. Ge and J. E. Marsden [Phys. Lett., A 133, 134-139 (1988)].”
Tangent bundles \(TM\) and Lie algebras \({\mathfrak g}\) are the most basic examples of Lie algebroids – corresponding to the Lie groupoids \(M\times M\) and \(G\) – and in this paper the author shows how the Lagrangian formalism extends to arbitrary Lie groupoids and Lie algebroids in a way which subsumes and clarifies the two examples of Moser and Veselov, as well as, for example, a Lagrangian formalism on Lie algebras due to PoincarĂ©. – This is a very clearly written paper, whose ideas are likely to lead to many further developments. The first sections also provide an admirably concise and accessible account of the use of Lie algebroids in Poisson geometry.
For the entire collection see [Zbl 0833.00041].

22A22 Topological groupoids (including differentiable and Lie groupoids)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58H05 Pseudogroups and differentiable groupoids
70H99 Hamiltonian and Lagrangian mechanics
Zbl 0754.58017