Nonseparable Solutions to the Hamilton-Jacobi Equation

Abstract

The Hamilton-Jacobi partial differential equation is solved for potential energy functionals of constant, linear, and quadratic form using a class of nonseparable solutions; these solutions give a geometric property to the generating solution, embedding it into the class of conics. These solutions have two basic components, that designated as a kernel component which belongs to the system regardless of the specific dynamics of the system and the primary and secondary system functions that are dependent on the specific initial conditions. Solutions are obtained for the linear oscillator, a rheonomic oscillator and a two-degree-of-freedom system, the latter suggesting an approach for general multidegree-of-freedom systems.