  As is widely recognized in Lyapunov analysis, linearized Hamilton's equations
of motion have two marginal directions for which the Lyapunov exponents vanish.
Those directions are the tangent one to a Hamiltonian flow and the gradient one
of the Hamiltonian function. To separate out these two directions and to apply
Lyapunov analysis effectively in directions for which Lyapunov exponents are
not trivial, a geometric method is proposed for natural Hamiltonian systems, in
particular. In this geometric method, Hamiltonian flows of a natural
Hamiltonian system are regarded as geodesic flows on the cotangent bundle of a
Riemannian manifold with a suitable metric. Stability/instability of the
geodesic flows is then analyzed by linearized equations of motion which are
related to the Jacobi equations on the Riemannian manifold. On some geometric
setting on the cotangent bundle, it is shown that along a geodesic flow in
question, there exist Lyapunov vectors such that two of them are in the two
marginal directions and the others orthogonal to the marginal directions. It is
also pointed out that Lyapunov vectors with such properties can not be obtained
in general by the usual method which uses linearized Hamilton's equations of
motion. Furthermore, it is observed from numerical calculation for a model
system that Lyapunov exponents calculated in both methods, geometric and usual,
coincide with each other, independently of the choice of the methods.
