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- W2089764512 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) Ei(χ̈,χ̇,x,t) ≡ Ei(χ̈1,...,χ̈n; χ̇1,...,χ̇n; x1,...,xn;t) = 0. Such dynamical symmetries are based upon infinitesimal transformations of the form (b) χ̄i=xi +δxi, δxi≡ξi(x,t) δa, (c) t̄=t +δt, δt≡ξ0(x,t) δa, which satisfy the condition (d) δEi=0 whenever Ej=0. It is shown that if (ξiA, ξ0A), 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) Ei(χ̈,χ̇,x) =0, it is shown that if (ξi, ξ0) defines a symmetry mapping then in general (∂Kξi/∂tK, ∂Kξo/∂tK), 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 (ξi, ξ0) lead to further contants of motion." @default.
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- W2089764512 date "1977-03-01" @default.
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- W2089764512 title "Time‐dependent dynamical symmetry mappings and associated constants of motion for classical particle systems. II" @default.
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- W2089764512 doi "https://doi.org/10.1063/1.523286" @default.
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