Lyapunov Analysis on a Geometric Method
山口義幸
3月9日(金)13:30
3F北演習室
As is well know 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 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.
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