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Calculus on Manifolds
Hamiltonian Systems with Symmetry
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action assume atlas bifurcation bijection bundle chart called canonical classical closed orbit compact support complete components computation connected constant coordinates corresponding cotangent critical point defined Definition denote derivative diffeomorphism differential dimension domain eigenvalue elliptic energy surface equations equivalent ergodic example Exercise fiber finite finite-dimensional formula function geodesic given Hamiltonian system Hamiltonian vector field Hausdorff Hence identity induced inner product integral curve invariant isomorphism Lagrange left-invariant Lemma Let G Lie algebra Lie group linear map locally matrix mechanics momentum mapping motion neighborhood nondegenerate norm Note one-form open set orbit cylinder orientation Poincare Poisson bracket Proof Proposition prove real vector space regular value result Riemannian metric Sect smooth stability subspace Suppose symmetric symplectic diffeomorphism symplectic form symplectic manifold symplectic structure tangent bundle tensor theory three-body problem topological space topology transversal unique vector bundle vector bundle mapping zero