On Lagrangian and Hamiltonian systems with homogeneous trajectories

TL;DR

This paper investigates homogeneous trajectories in Lagrangian and Hamiltonian systems, generalizing the geodesic lemma, and explores conditions for their existence and properties in the context of symmetric spaces.

Contribution

It extends the geodesic lemma to Lagrangian and Hamiltonian systems and analyzes homogeneous trajectories and g.o. spaces in these frameworks.

Findings

Criteria for orbits to solve Euler-Lagrange or Hamiltonian equations

Existence results for homogeneous trajectories in Lagrangian systems

Relation between Hamiltonian g.o. spaces and coadjoint invariants

Abstract

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solution of the Euler-Lagrange or Hamiltonian equations. In particular, we generalize the `geodesic lemma' known in Riemannian geometry to Lagrangian and Hamiltonian systems. We present results on the existence of homogeneous trajectories of Lagrangian systems. We study Hamiltonian and Lagrangian g.o. spaces, i.e. homogeneous spaces G/H with G-invariant Lagrangian or Hamiltonian functions on which every solution of the equations of motion is homogeneous. We show that the Hamiltonian g.o. spaces are related to the functions that are invariant under the coadjoint action of G. Riemannian g.o. spaces thus correspond to special Ad*(G)-invariant functions. An Ad*(G)-invariant function that is related to a g.o. space also serves as a potential for the mapping called `geodesic graph'. As illustration we discuss the Riemannian g.o. metrics on SU(3)/SU(2).