1. The problem statement, all variables and given/known data The problem is that i have heard about things like congruences of curves and geodesically equidistant hyper-surfaces, but have no idea how to use Hamilton-Jacobi theory to solve problems( especially in mechanics), or why it works. Why do we use this method to solve problems, and how do we use this method to solve problems ? Any insight at all is much appreciated. 2. Relevant equations ??? The hamiltonian function, hamilton's equations ? 3. The attempt at a solution Not sure what to solve, are we trying to get a curve that displays information about a dynamical system? if so what part do the hyper surfaces play?