Geodesic Compatibility: Goldfish Systems

Abstract

To capture a multidimensional consistency feature of integrable systems in terms of geometry, we give a condition called geodesic compatibility implying the existence of integrals in involution of the geodesic flow. The geodesic compatibility condition is constructed from a concrete example namely the integrable Calogero's goldfish system through the Poisson structure and the variational principle. The geometrical view of the geodesic compatibility gives compatible parallel transports between two different Hamiltonian vector fields