Mathematics

Ever since Lagrange proved the invariance of the Euler-Lagrange equation under arbitrary nonlinear coordinate changes, and recognized the enormous importance of this observation, the search for coordinate-invariant conditions for optimality of curves, and for coordinate-free formulations of these conditions, has been one of the driving forces leading to new and better results. In this talk we will first show how this search logically leads from the Euler-Lagrange conditions to two different possible Hamiltonian formulations. One of them represents the path that was actually followed, while the other one, the path not taken (and dramatically missed by giants such as Weierstrass and Caratheodory), could have led rather quickly to the discovery of more general conditions, and to the control point of view, in which the momentum is no longer defined as the gradient of the Lagrangian with respect to the velocity. We will then pursue the coordinate-invariant path, showing how it leads naturally to manifestly coordinate-free conditions involving Lie brackets and, in our most recent work, to conditions expressed in terms of flows rather than vector fields, and generalized rather than ordinary derivatives. These conditions apply to systems that are non-smooth in the sense of \"non-smooth analysis\" (i.e., involving locally Lipschitz vector fields) and also to many systems that are even less smooth than that, such as the reflected brachistochrone problem, which is Holder-1/2 but not Lipschitz.