On projective symmetries of dynamical systems

Abstract

The present paper is concerned with symmetry transformations of a dynamical system defined on the tangent bundle of a Riemannian manifold. Of present interest are infinitesimal symmetry transformations of the vector field which defines the dynamical system on the tangent bundle. It is known that a class of such transformations entails infinitesimal projective transformations leaving the vector field invariant. Symmetry algebras formed by such projective transformations are studied. It is shown which dynamical systems admit large symmetry algebras. As a result, two kinds of dynamical systems are determined, which have the base Riemannian manifolds of constant curvature with dimensions n?4. The systems are generalizations of the classical harmonic oscillator and Kepler problem usually considered in Euclidean spaces. First integrals quadratic in the velocities are obtained, which are also generalizations of the well-known quadratic integrals for the above classical systems.