**Title**: New tangent structure on Lie algebroids and Lie groupoids

**Abstract**: The tangent bundle on a smooth manifold is, in a sense, sufficient structure to develop Lagrangian mechanics. In a famous note from 1901, Poincare reformulated Lagrangian mechanics by replacing the tangent bundle with a Lie algebra acting on a smooth manifold [1, 2]. Poincare’s formalism leads to the Euler-Poincare equations, which capture the usual Euler-Lagrange equations as a specific example. In 1996, Weinstein sketched out a general program building on Poincare’s ideas to formulate mechanics on Lie groupoids using Lie algebroids [3], which motivates the work of Martinez et al. [4,5], Libermann [6], and the recent thesis by Fusca [7].

In this talk, we will look at Weinstein’s program through the lens of tangent categories, which are a categorical abstract of the Weil functor formalism. We show that Lie algebroids can be reformulated as a certain category of tangent functors from Weil algebras into smooth manifolds. This tangent structure on the category of Lie algebroids agrees with Martinez’s presentation of Lie algebroids as generalized tangent bundles.

[1] Poincaré H. Sur une forme nouvelle des équations de la mécanique. CR Acad. Sci. 1901;132:369-71.

[2] Marle CM. On Henri Poincaré’s note “Sur une forme nouvelle des équations de la Mécanique”. Journal of geometry and symmetry in physics. 2013;29:1-38.

[3] Weinstein A. Lagrangian mechanics and groupoids. Fields Institute Proc. AMS. 1996;7:207-31.

[4] Martínez E. Lagrangian mechanics on Lie algebroids. Acta Applicandae Mathematica. 2001 Jul;67(3):295-320.

[5] de León M, Marrero JC, Martínez E. Lagrangian submanifolds and dynamics on Lie algebroids. Journal of Physics A: Mathematical and General. 2005 Jun 1;38(24):R241.

[6] Libermann P. Lie algebroids and mechanics. Archivum mathematicum. 1996;32(3):147-62.

[7] Fusca D. A groupoid approach to geometric mechanics (Doctoral dissertation, University of Toronto).