A geometrical approach to the Hamilton-Jacobi form of dynamics and its generalizations

The paper under review is devoted to the task of expressing the principles of classical dynamics in a coordinate-free manner and aims to show that the Hamilton-Jacobi formalism is as complete as the Hamiltonian or Lagrangian treatment of dynamical systems. This concerns mainly the incorporation of symmetry group actions in Hamilton-Jacobi theory. Stated in geometrical form, the unknown in this theory becomes a Lagrangian submanifold in a phase space, instead of a function over the configuration manifold in the ordinary formulation. The time-dependent Hamilton-Jacobi theory is also cast in this form after promoting time and its conjugate variable to the level of dynamical variables. Once this is done the analysis copies that of the time-independent case. The paper ends with interesting comments on the relationship between the Hamilton-Jacobi theory and Schrödinger's quantum mechanics. Plenty of other comments, examples and remarks are spread throughout the text. The exposition is quite informative and lucid