### Abstract

The most general *t* *i* *m* *e*‐*d* *e* *p* *e* *n* *d* *e* *n* *t*, central force, classical particle dynamical systems (in *n*‐dimensional Euclidean space,*n*=2 or 3) of the form (a) **r**̈=*i*̂_{ r } *F*(*r*, *t*), (*r* ^{2}≡**r** ** **⋅** ** **r**, **r**=*i*̂_{ k } *x* ^{ k }, *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**̈=*i*̂_{ r }(*U*̈*U* ^{−} ^{1} *r*−μ_{0} *U* ^{−} ^{1} *r* ^{−} ^{2}), for which the concomitant vector constant of motion (b) takes the form (d) **I**=*U*(**L**×**v**)−*U*̇(**L**×**r**)+μ_{0} *r* ^{−} ^{1} **r**, 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 *v* *e* *l* *o* *c* *i* *t* *y*‐*i* *n* *d* *e* *p* *e* *n* *d* *e* *n* *t* 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.

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