Poincaré’s 1901 paper introduces (in just a humble 3 pages) the Euler-Poincaré equations, which are the specialization of the Euler-Lagrange equations to the case where a Lie group acts on the configuration manifold. I work through Poincaré’s paper without making too many identifications.
Despite having encountered the Lagrangian and Hamiltonian formalisms of mechanics several times in a variety of engineering and physics settings, I had never been able to retain it in my memory. I had maintained a similar dissatisfaction with the many formulae of multivariable calculus, which only really ✨clicked✨ for me once I learned about the exterior derivative and the generalized Stokes’ theorem. In this post, I would like to collect my thoughts on the differential geometric treatment of Lagrangian and Hamiltonian mechanics, which assume a very simple and memorable form once we introduce the language of symplectic geometry . Of course, to have made our mathematical journey to this point, where we are able to say anything at all about symplectic geometry, was a not-so-simple task. ...