Related First Integral Theorem: A Method for Obtaining Conservation Laws of Dynamical Systems with Geodesic Trajectories in Riemannian Spaces Admitting Symmetries

Publisher Website
Google Scholar
Abstract
In this paper we develop in detail a unified method, referred to as the Related First Integral Theorem, for obtaining ``derived'' first integrals (i.e., constants of the motion) of mass‐pole test particles with geodesic trajectories in a Riemannian spece. By this method, which is based upon a process of Lie differentiation, additional conservation laws in the form of mth order first integrals can be generated from a given mth order first integral (conservation law), provided the space admits symmetries in the form of continuous groups of projective collineations (which include affine collineations and motions as special cases). We give in tensor form a reformulation of the well‐known Poisson's theorem on constants of the motion for particles with geodesic trajectories. We then show for this class of trajectories that, as a method for generating mth order first integrals from a given mth order first integral, Poisson's theorem is a special case of the Related First Integral Theorem. It is also shown that dependency relations between generically related first integrals obtained by the Related First Integral Theorem are expressible in terms of the structure constants of the underlying continuous group of symmetries.