Nonholonomic and discrete Hamilton-Jacobi theory

The first part of the thesis discusses an extension of Hamilton-Jacobi theory to nonholonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. The major advantage of our result is that it provides us with a method of integrating the equations of motion just as the unconstrained Hamilton---Jacobi theory does. We develop nonholonomic Hamilton-Jacobi theory from two different perspectives; one is a direct approach based on the standard formulation of nonholonomic systems, and the other uses the technique of the Chaplygin Hamiltonization. We also establish a link between these two approaches by providing an explicit formula that relates the solutions of the Hamilton-Jacobi equations resulting from both approaches.;The second part of the thesis develops a discrete analogue of Hamilton-Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton-Jacobi equation is discrete only in time, and is shown to recover the Hamilton-Jacobi equation in the continuous-time limit. The correspondence between discrete and continuous Hamiltonian mechanics naturally gives rise to a discrete analogue of Jacobi's solution to the Hamilton-Jacobi equation. We also prove a discrete analogue of the geometric Hamilton-Jacobi theorem. These results are readily applied to the discrete optimal control setting, and some well-known results in discrete optimal control theory, such as the Gellman equation (discrete-time Hamilton-Jacobi-Bellman equation) of dynamic programming, follow immediately. We also apply the theory to discrete linear Hamiltonian systems, and show that the discrete Riccati equation follows as a special case of the discrete Hamilton-Jacobi equation.