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

Publisher Website
Google Scholar
Abstract
The most general time-dependent, central force, classical particle dynamical systems (in n-dimensional Euclidean space, n=2 or 3) of the form (a) r̈=îr F(r, t), (r2≡r ⋅ r, r=îkxk, 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≡ṙ) are obtained. It is found that the only class of such dynamical systems is (c) r̈=îr(ÜU−1r−μ0U−1r−2), for which the concomitant vector constant of motion (b) takes the form (d) I=U(L×v)−U̇(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 the Noether symmetries. By means of the vector constant of motion (d), the orbit equations of the dynamical system (c) are obtained. A one-dimensional procedure for obtaining constants of motion developed by Lewis and Leach is applied to the effective one-dimensional system concomitant to (c). Relations between constants of motion so obtained and those mentioned above are determined.