When is an axisymmetric potential separable?

TL;DR

This paper derives conditions under which an axially symmetric potential in physics is separable, identifying specific ratios and constants that allow for additional integrals of motion beyond classical ones.

Contribution

It introduces new mathematical criteria involving ratios and constants that determine when an axially symmetric potential is separable and admits extra integrals of motion.

Findings

Potential is separable if s:k ratio is constant.

Existence of an additional integral I involving angular momentum and velocity components.

General condition for third integral involving constants a, b, c.

Abstract

An axially symmetric potential psi(R,z)=psi(r,theta) is completely separable if the ratio s:k is constant. Here r*s=d^2(r^2*psi)/dr/d(theta) and k=d^2(psi)/dR/dz. If beta=s/k, then the potential admits an integral of the form of I=(L^2+beta*v_z^2)/2+xi where xi is some function of positions determined by the potential psi. More generally, an axially symmetric potential respects the third axisymmetric integral of motion -- in addition to the classical integrals of the Hamiltonian and the axial component of the angular momentum -- if there exist three real constants a,b,c (not all simultaneously zero, a^2+b^2+c^2>0) such that a*s+b*h+c*k=0 where r*h=d^2(r*psi)/d(sigma)/d(tau) and (sigma,tau) is the parabolic coordinate in the meridional plane such that sigma^2=r+z and tau^2=r-z.