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).Comment: v3: some misprints correcte
