I’ve been going through Russ Tedrake’s notes on robotics, which got me thinking about their so-called monogram notation. The result of this deliberation was a new notation for spatial velocities that bridges the gap between abstract Lie groups and their applications to robotics and computer graphics.
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. In this post, I would like to collect my thoughts on their differential geometric treatment, which assumes a very simple and memorable form once we introduce the language of symplectic geometry.