Time−dependent dynamical symmetries, associated constants of motion, and symmetry deformations of the Hamiltonian in classical particle systems

Abstract

A study is made of time−dependent dynamical symmetry mappings of Hamilton’s equations for classical particle systems. The conditions that an infinitesimal mapping (δxA, δt) ≡ (ξA(x,t) δa, ξ0(x,t) δa), A = 1, ..., 2n, be a symmetry mapping are expressed in terms of a ’’symmetry vector’’ ZA(x,t) ≡ ξA − ξ0 ηAB∂BH (x,t) (where ηAB defines the symplectic matrix of phase space). These conditions imply that ξ0 is arbitrary. It is shown that the symmetry deformation of a constant of motion M (x,t) will also be a (’’derived’’) constant of motion (time−dependent related integral theorem). It follows for the case H = H (x) that every time−dependent symmetry deformation of H (x) is a constant of motion, and it is shown conversely that every constant of motion M (x,t) can be expressed as a symmetry deformation of the Hamiltonian, that is, there exists a symmetry vector ZA(x,t) such that M = ZA∂AH. It is found that if ZA(≠0) is a symmetry vector, then M (x,t) ZA will be a (scaled) symmetry vector if and only if M is a constant of motion. The existence of groups of symmetry vectors is considered, and it is shown that a complete set of r symmetry vectors ZAα, α = 1, ..., r, determines an r−parameter continuous group of symmetries. A special class of symmetry vectors ZA(P) (x,t) ≡ ηAB (∂BM − N∂BH), (’’extended Poisson vectors’’), where M (x,t), N (x,t) are constants of motion is defined and conditions that such vectors determine a symmetry group are obtained. Poisson vectors are also used to show that the related integral theorem mentioned above may be considered as a generalization of Poisson’s theorem on constants of motion. Dependency relations between derived constants of motion with respect to vectors of a symmetry group are obtained.