Time-dependent dynamical symmetries and constants of motion. III. Time-dependent harmonic oscillator

Abstract

This paper is a continuation of previous Papers I and II [J. Math. Phys. 17, 1345 (1976); 18, 424 (1977)]. In the present paper we apply the theory (based upon Lagrangian dynamics) developed in I and II to obtain the dynamical symmetries and concomitant constants of motion admitted by the time-dependent n-dimensional oscillator (a) Ei≡χ̈i +2ω (t) xi=0. The dynamical symmetries are based upon infinitesimal transformations of the form (b) χ̄i=xi+δxi, δxi≡ξi(x,t) δa; t̄=t+δt, δt≡ξ0(x,t) δa which satisfy the condition (c) δEi=0, whenever Ei=0. It is shown that such symmetries of the oscillator (a) will be time-dependent projective collineations. For such symmetries which satisfy the R1 restriction (defined in I) it is shown there exist concomitant constants of motion C1 of the oscillator, which for n=1 are time-dependent cubic polynomials in the χ̇ variable, and for n⩾2 are time-dependent quadratic polynomials in the χ̇i variables. It is shown that those symmetries which satisfy the R2 restriction (Noether symmetry condition discussed in I) are time-dependent homothetic mappings consisting of time-dependent scale changes, time-dependent translations, and rotations. The concomitant Noether constants of motion C2 are time-dependent quadratic polynomials in the χ̇i variables for all n. The Noether constant of motion C2 [referred to as C2(B)] for which the associated underlying symmetry mapping is the time-dependent scale change is shown to include as a special case when n=1 a class of invariants formulated by Lewis [Phys. Rev. Lett. 18, 510 1967)] (by means of a phase space analysis which applies Kruskal’s theory in closed form). For the case of general n it is shown that the time-dependent symmetric tensor constant of motion Iij constructed by Günther and Leach [J. Math. Phys. 18, 572 (1977)] is included as a special case of a time-dependent symmetric tensor constant of motion Kij, where Kij is obtained by use of a time-dependent related integral theorem by means of the symmetry deformation of the constant of motion C2(B) with respect to the affine collineations; such collineations are a subset of the projective collineation symmetries mentioned above. The symmetries and their concomitant constants of motion of the oscillator (a) with ω (t) of the form ω (t) =a+bect are obtained.