Time‐dependent vector constants of motion, symmetries, and orbit equations for the dynamical system r̈=îr{[Ü(t)/U(t)]r −[μ0/U(t)]r−2}

The most general time‐dependent, central force, classical particle dynamical systems (in n‐dimensional Euclidean space, n=2 or 3) of the form (a) r=ir F(r, t), (r2≡r ⋅ r, r=ikxk, k=1,...,n), which admit vector constants of motion of the form (b) I=U(r, t)(L×v)+Z(r, t)(L×r) +W(r, t)r (L≡r×v, v≡r) are obtained. It is found that the only class of such dynamical systems is (c) r=ir(UU−1r−μ0U−1r−2), for which the concomitant vector constant of motion (b) takes the form (d) I=U(L×v)−U(L×r)+μ0r−1r, where in (c) and (d) U=U(t) is arbitrary (≠0). The dynamical system (c) includes both the time‐dependent harmonic oscillator and a time‐dependent Kepler system. Based upon infinitesimal velocity‐independent mappings the complete symmetry group for the dynamical system (c) is obtained. This complete group of [2+n(n−1)/2] parameters contains a complete Noether symmetry subgroup of [1+n(n−1)/2] parameters. In addition to the n(n−1)/2 angular momenta, there is an energy‐like constant of motion also associated with...