  In this article we study the Hofer geometry of a compact Lie group $K$ which
acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized
Hofer norms on the Lie algebra of $K$ are introduced and analyzed with tools
from group invariant convex geometry, functional and matrix analysis. Several
global results on the existence of geodesics and their characterization in
finite dimensional Lie groups $K$ endowed with bi-invariant Finsler metrics are
proved. We relate the conditions for being a geodesic in the group $K$ and in
the group of Hamiltonian diffeomorphisms. These results are applied to obtain
necessary and sufficient conditions on the moment polytope of the momentum map,
for the commutativity of the Hamiltonians of geodesics. Particular cases are
studied, where a generalized non-crossing of eigenvalues property of the
Hamiltonians hold.
