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Calculus on Manifolds
Hamiltonian Systems with Symmetry
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action algebra assume Banach space basis bifurcation called canonical chart clearly closed orbit compact complete computation condition connected consider constant continuous converges coordinates corresponding critical point defined Definition denote derivative determined diffeomorphism differential dimension domain elements energy equations equivalent example Exercise exists fact fiber Figure flow formula function given gives Hamiltonian Hence identity implies induced integral curve invariant isomorphism Lagrangian Lemma linear locally manifold means measure mechanics metric momentum motion natural neighborhood Note obtain operator orbit orientation periodic Poincare preserving problem Proof Proposition prove regular relation relative representative respect restricted result Riemannian satisfying Sect smooth stability structure submanifold subset Suppose symplectic symplectic manifold tangent tensor theorem theory topology transformation unique vector bundle vector field vector space volume write zero