### Abstract

This paper is a continuation of a previous paper with a similar title [J. Math. Phys. 17, 1345 (1976)]. In this paper we develop further properties of time‐dependent symmetries of dynamical systems expressible in the form (a) *E* ^{ i }(χ̈,χ̇,*x*,*t*) ≡ *E* ^{ i }(χ̈^{1},...,χ̈^{ n }; χ̇^{1},...,χ̇^{ n }; *x* ^{1},...,*x* ^{ n };*t*) = 0. Such dynamical symmetries are based upon infinitesimal transformations of the form (b) χ̄^{ i }=*x* ^{ i } +δ*x* ^{ i }, δ*x* ^{ i }≡ξ^{ i }(*x*,*t*) δ*a*, (c) =*t* +δ*t*, δ*t*≡ξ^{0}(*x*,*t*) δ*a*, which satisfy the condition (d) δ*E* ^{ i }=0 whenever *E* ^{ j }=0. It is shown that if (ξ^{ i } _{ A }, ξ^{0} _{ A }), *A*=1,...,ρ, is a complete set of solutions of the symmetry equations as determined by (d), then these solutions generate a ρ‐parameter complete group of symmetry mappings, and the group structure implies linear dependency relations between first and second derived time‐dependent constants of motion as obtained by a related integral theorem. The complete groups of time‐dependent symmetry mappings are obtained for all conservative systems (*n*≳1) with spherically symmetric potentials. These groups are classified into six types according to the associated form of the potential. A similar analysis leads to three types of Noether symmetries. In the case where (a) takes the form (e) *E* ^{ i }(χ̈,χ̇,*x*) =0, it is shown that if (ξ^{ i }, ξ^{0}) defines a symmetry mapping then in general (∂^{ K }ξ^{ i }/∂*t* ^{ K }, ∂^{ K }ξ^{ o }/∂*t* ^{ K }), *K*=1,2,..., will also define symmetry mappings; similar properties are shown for Noether symmetries. These results when applied to a large class of time‐dependent constant of motion defined in terms of (ξ^{ i }, ξ^{0}) lead to further contants of motion.

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