On symmetries and first integrals of dynamical systems

Abstract

A previous result according to which, given a symmetry vector of a dynamical system, under certain co-ordinate-dependent conditions, a first integral could be expressed as the divergence of the symmetry vector has been generalized. By introducing convenient connections it is found that not only one but a set of first integrals can, in general, be interpreted as divergences of a single symmetry vector. It is also shown that any symmetry vector is a linear combination of a maximal independent set of symmetry vectors of the same dynamical system with first integrals as coefficients. This leads to other relationships between first integrals andsets of symmetry vectors. Finally it is seen how a type-(r, s) tensor invariant of the one-parameter group generated by a dynamical system yieldsr generally different type-(r−1),s) tensor invariants. This is, in a sense, a further generalization of the former result. The type-(r, s) tensor invariant also leads to a type-(r, s+1) tensor invariant of the same dynamical system.