The related integral theorem in phase space: A generalization of Poisson’ theorem on constants of the motion

Abstract

The related integral theorem 0 -3) is a unified method which shows that , if a dynamical system admits a symmetry, then in general a new constant of the motion will result from the deformation of a given one under this symmetry mapping. As originally formulated 0) the theorem provided a method for generating quadrat ic first integrals of dynamical systems with geodesic trajectories based upon the deformation of the metrical quadrat ic integral under projective collineations. A later generalization (3) extended the domain of the theorem to include conservative dynamical systems governed by the equation D2x~/dt2+ gV.~= 0(~). Dynamical symmetries for such systems were called t ra jectory collineations. Based upon the deformations of the energy integral under the t ra jec tory collineation mappings addi t ional constants of the motion were obtained in a systematic manner. Application of the theory to the Kepler problem and three-dimensional isotropic harmonic oscillator showed that the well-known RungeLenz vector constant of the motion and symmetr ic tensor constant of the motion could be obtained in a simple manner by this unified approach. In the cases thus far considered the related integral theorem has been formulated in terms of point mappings of the generalized co-ordinates of configuration space. We now show how the related integral theorem may be formulated in phase space for dynamical systems governed by Hamil ton 's equations. As a result we find tha t the wellknown Poisson's theorem on constants of the inotion (b) is a special caae of the related integral theorem. Consider a dynamical system of n degrees of freedom with Hamiltonian H(~). I Iamil ton 's equations may be expressed in the form (s e)