Time-dependent quadratic constants of motion, symmetries, and orbit equations for classical particle dynamical systems with time-dependent Kepler potentials

Abstract

It is shown there are only two classes of time-dependent Kepler potentials [V2≡λ0(at2+bt+c)−1/2/r, (b2−4ac≠0), and V3≡λ0(αt+β)−1/r] for which the associated classical dynamical equations will admit quadratic first integrals more general than quadratic functions of the angular momentum. In addition to the angular momentum the system defined by V2 admits only a ’’generalized time-dependent energy integral,’’ while the system defined by V3 admits in addition to these a time-dependent vector first integral that is a generalization of the Laplace–Runge–Lenz vector constant of motion (associated with the time-independent Kepler system). For the V3 system the time-dependent vector first integral is employed to obtain in a simple manner the orbit equations in completely integrated form. The complete group of (velocity-independent) symmetry mappings is obtained for each of these two classes of dynamical systems and used to show that the generalized energy integral is expressible as a Noether constant of motion.