Quadratic Integrals of Motion and Stellar Orbits in the Absence of Axial Symmetry of the Potential

Abstract

As is well known, in the case of an axially symmetric and time-invariant gravitational potential, if the potential satisfies one particular additional constraint, there exist three isolating integrals of motion: the energy integral, the area integral, and the third integral which is quadratic in the velocities. This work discusses the case in which there exist quadratic integrals in the absence of axial symmetry of the potential. Such a case has already been examined by Eddington [1], but in their explicit form, the integrals were introduced by Clark [2].