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libermann p symplectic geometry analytical mechanics 1987

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Textbook in PDF format Preface Symplectic vector spaces and symplectic vector bundles Symplectic vector spaces Properties of exterior forms of arbitrary degree Properties of exterior 2-forms Symplectic forms and their automorphism groups The contravariant approach Orthogonality in a symplectic vector space Forms induced on a vector subspace of a symplectic vector space Additional properties of Lagrangian subspaces Reduction of a symplectic vector space. Generalizations Decomposition of a symplectic form Complex structures adapted to a symplectic structure Additional properties of the symplectic group Symplectic vector bundles Properties of symplectic vector bundles Orthogonality and the reduction of a symplectic vector bundle Complex structures on symplectic vector bundles Remarks concerning the operator Λ and Lepage's decomposition theorem The decomposition theorem in a symplectic vector space Decomposition theorem for exterior differential forms A first approach to Darboux's theorem Semi-basic and vertical differential forms in mechanics Definitions and notations Vector bundles associated with a surjective submersion Semi-basic and vertical differentia! forms The Liouville form on the cotangent bundle Symplectic structure on the cotangent bundle Semi-basic differential forms of arbitrary degree Vector fields and second-order differential equations The Legendre transformation on a vector bundle The Legendre transformation on the tangent and cotan gent bundles Applications to mechanics: Lagrange and Hamilton equations Lagrange equations and the calculus of variations The Poincaré-Cartan integral invariant Mechanical systems with time dependent Hamiltonian or Lagrangian functions Symplectic manifolds and Poisson manifolds Symplectic manifolds; definition and examples Special submanifolds of a symplectic manifold Symplectomorphisms Hamiltonian vector fields The Poisson bracket Hamiltonian systems Presymplectic manifolds Poisson manifolds Poisson morphisms Infinitesimal automorphisms of a Poisson structure The local structure of Poisson matlifolds The symplectic foliation of a Poisson manifold The local structure of symplectic manifolds Reduction of a symplectic manifold The Darboux-Weinstein theorems Completely integrable Hamiltonian systems Exercises Action of a Lie group on a symplectic manifold Symplectic and Hamiltonian actions Elementary properties of the momentum map The equivariance of the momentum map Actions of a Lie group on its cotangent bundle Momentum maps and Poisson morphisms Reduction of a symplectic manifold by the action of a Lie group Mutually orthogonal actions and reduction Stationary motions of a Hamiltonian system The motion of a rigid body about a fixed point Euler's equations Special formulae for the group S0(3) The Euler-Poinsot problem The Euler-Lagrange and Kowalevska problems Additional remarks and comments Exercises Contact manifolds Background and notations Pfaffian equations Principal bundles and projective bundles The class of Pfaffian equations and forms Darboux's theorem for Pfaffian forms and equations Strictly contact structures and Pfaffian structures Projectable Pfaffian equations Homogeneous Pfaffian equations Liouville structures Fibered Liouville structures The automorphisms of Liouville structures The infinitesimal automorphisms of Liouville structures The automorphisms of strictly contact structures Some contact geometry formulae in local coordinates Homogeneous Hamiltonian systems Time-dependent Hamiltonian systems The Legendre involution in contact geometry The contravariant point of view Basic notions of differential geometry Differentiable maps, immersions, submersions The flow of a vector field Lie derivatives Infinitesimal automorphisms and conformal infinitesimal transformations Time-dependent vector fields and forms Tubular neighborhoods Generalizations of Poincaré's lemma Infinitesimal jets Generalities Velocity spaces Second-order differential equations Sprays and the exponential mapping Covelocity spaces Liouville forms on jet spaces Distributions, Pfaffian systems and foliations Distributions and Pfaffian systems Completely integrable distributions Generalized foliations defined by families of vector fields Differentiable distributions of constant rank Integral invariants Integral invariants of a vector field Integral invariants of a foliation The characteristic distribution of a differential form Lie groups and Lie algebras Lie groups and Lie algebras; generalities The exponential map Action of a Lie group on a manifold The adjoint and coadjoint representations Semi-direct products Notions regarding the cohomology of Lie groups and Lie algebras Affine actions of Lie groups and Lie algebras The Lagrange-Grassmann manifold The structure of the Lagrange-Grassmann manifold The signature of a Lagrangian triplet The fundamental groups of the symplectic group and of the Lagrange-Grassmann manifold Morse families and Lagrangian submanifolds Lagrangian submanifolds of a cotangent bundle Hamiltonian systems and first-order partial differential equations Contact manifolds and first-order partial differential equations Jacobi's theorem The Hamilton-Jacobi equation for autonomous systems The Hamilton-Jacobi equation for nonautonomous systems Bibliography Index

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libermann p symplectic geometry analytical mechanics 1987