Abstract
The authors prove, for a particle moving in a plane under the influence of a conservative force, that when the motion is constrained by a 'second' invariant quadratic in the velocities, then the potential allows separability of the Hamilton-Jacobi equation in rectangular, polar, elliptical cylinder or parabolic cylinder coordinates. This link shows the intimate connection between quadratic invariants and the two-dimensional Hamilton-Jacobi equation. They give examples of the utility of parabolic cylinder coordinates in cases of recent study.
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