Rayleigh dissipation from the general recurrence of metrics in path spaces

Abstract

Given a pair (metric g, symmetric 2-covariant tensor field H though as a Rayleigh dissipation) on a path space (manifold M, semispray S), the family of nonlinear connections N such that H equals the dynamical derivative of g with respect to (S,N) is determined by using the Obata tensors. In this way, we generalize the case of metric nonlinear connections as well as that of recurrent metrics. As applications, we treat firstly the case of Finslerian (α,β)-metrics finding all nonlinear connections for which the associated Finsler–Sasaki metric is exactly the dynamical derivative of the Riemannian–Sasaki metric. Secondly, we apply our results for the case of Beil metrics used in Relativity and field theories.