Remarks on Geometric Mechanics

This paper gives a few new developments in mechanics, as well as some remarks of a historical nature. To keep the discussion focussed, most of the paper is confined to equations of “rigid body”, or “hydrodynamic” type on Lie algebras or their duals. In particular, we will develop the variational structure of these equations and will relate it to the standard variational principle of Hamilton. Even this small area of mechanics is fascinating from the historical point of view. In fact, it is quite surprising how long it can sometimes take for fundamental results of the masters to be tied together and to filter into the main literature and to become “well-known”. In particular, part of our story follows a few fragments of a thread through the works of Euler, Lagrange, Lie, Poincaré, Clebsch, Ehrenfest, Hamel, Arnold, and many others. Although Newton’s discoveries were directly motivated by planetary motion, the realm of mechanics expanded well beyond particle mechanics with the work of Euler, Lagrange, and others to include fluid and solid mechanics. Today we see its methods permeating large areas of physical phenomena besides these, including electromagnetism, plasma physics, classical field theories, general relativity, and quantum mechanics. Part of what makes this unified point of view possible is the abstraction, often in a geometric way, of the underlying structures in mechanics. Two general points of view emerged early on concerning the basic structures in mechanics. One, which is commonly referred to as “Lagrangian mechanics” can be based in variational principles, and the other, “Hamiltonian mechanics”, rests on symplectic and Poisson geometry. As we shall see shortly, the history of this development is actually quite complex. How rigid body mechanics, fluid mechanics and their generalizations fit into this story is quite interesting because of the way their equations fit into the schemes of Lagrange and Hamilton. For example, the way the equations are normally presented (in body representation for the rigid body, and in spatial representation for