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New tangent structures for Lie algebroids and Lie groupoids

Banff International Research Station for Mathematical Innovation and Discovery

The tangent bundle on a smooth manifold is, in a sense, sufficient structure for 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, using recent advances in involution algebroids. We will use the fact that, in smooth manifolds, Lie algebroids are the same thing as involution algebroids. This means 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. We propose that tangent categories provide the proper algebraic framework to describe a theory of Lagrangian mechanics that extends to Weinstein's program when using these tangent bundles.

This talk draws on joint work with Matthew Burke and Richard Garner.

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

[2] Marle CM. On Henri Poincaréâ s note â Sur une forme nouvelle des équations de la Mécaniqueâ . 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).

Mathematics; Category Theory; Homological Algebra; Algebraic Topology; Algebraic Structures

video/mp4

Author affiliation: University of Calgary

2023-10-28

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86378

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/