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Symplectic Geometry by V. I. Arnol’d, A. B. Givental
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About this book :-
Symplectic Geometry written by V. I. Arnol’d, A. B. Givental
Symplectic geometry is the mathematical apparatus of such areas of physics as classical mechanics, geometrical optics and thermodynamics. Whenever the equations of a theory can be gotten out of a variational principle, symplectic geometry clears up and systematizes the relations between the quantities entering into the theory. Symplectic geometry simplifies and makes perceptible the frightening formal apparatus of Hamiltonian dynamics and the calculus of variations in the same way that the ordinary geometry of linear spaces reduces cumbersome coordinate computations to a small number of simple basic principles.
Book Detail :-
Title: Symplectic Geometry
Author(s): V. I. Arnol’d, A. B. Givental
Publisher: Springer-Verlag Berlin Heidelberg
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Book Contents :-
Symplectic Geometry written by V. I. Arnol’d, A. B. Givental cover the following topics.
1. Linear Symplectic Geometry
Symplectic Space, The Skew-Scalar Product, Subspaces, The Lagrangian Grassmann Manifold, Linear Hamiltonian Systems, The Symplectic Group and its Lie Algebra, The Complex Classification of Hamiltonians, Linear Variational Problems, Normal Forms of Real Quadratic Hamiltonians, Sign-Definite Hamiltonians and the Minimax Principle, Families of Quadratic Hamiltonians, The Concept of the Miniversal Deformation, Miniversal Deformations of Quadratic Hamiltonians, Generic Families, Bifurcation Diagrams, The Symplectic Group, The Spectrum of a Symplectic Transformation, The Exponential Mapping and the Cayley Parametrization, Subgroups of the Symplectic Group, The Topology of the Symplectic Group, Linear Hamiltonian Systems
2. Symplectic Manifolds
Local Symplectic Geometry, The Darboux Theorem, Example. The Degeneracies of Closed 2-Forms on [w4 , Germs of Submanifolds of Symplectic Space, The Classification of Submanifold Germs, The Exterior Geometry of Submanifolds, The Complex Case, Examples of Symplectic Manifolds, Cotangent Bundles, Complex Projective Manifolds, Symplectic and KHhler Manifolds, The Orbits of the Coadjoint Action of a Lie Group, The Poisson Bracket, The Lie Algebra of Hamiltonian Functions, Poisson Manifolds, Linear Poisson Structures, The Linearization Problem., Lagrangian Submanifolds and Fibrations, Examples of Lagrangian Manifolds, Lagrangian Fibrations, Intersections of Lagrangian Manifolds and Fixed Points of Symplectomorphisms
3. Symplectic Geometry and Mechanics
Variational Principles, Lagrangian Mechanics, Hamiltonian Mechanics, The Principle of Least Action, Variational Problems with Higher Derivatives, The Manifold of Characteristics, The Shortest Way Around an Obstacle, Completely Integrable Systems, Integrability According to Liouville, The “Action-Angle” Variables, Elliptical Coordinates and Geodesics on an Ellipsoid, Poisson Pairs, Functions in Involution on the Orbits of a Lie Coalgebra, The Lax Representation, Hamiltonian Systems with Symmetries, Poisson Actions and Momentum Mappings, The Reduced Phase Space and Reduced Hamiltonians, Hidden Symmetries, Poisson Groups, Geodesics of Left-Invariant Metrics and the Euler Equation, Relative Equilibria, Noncommutative Integrability of Hamiltonian Systems, Poisson Actions of Tori
4. Contact Geometry
Contact Manifolds, Contact Structure, Examples, The Geometry of the Submanifolds of a Contact Space, Degeneracies of Differential l-Forms on iw”, Symplectification and Contact Hamiltonians, Symplectification, The Lie Algebra of Infinitesimal Contactomorphisms, Contactification, Lagrangian Embeddings in iw’“, The Method of Characteristics, Characteristics on a Hypersurface in a Contact Space, The First-Order Partial Differential Equation, Geometrical Optics, The Hamilton-Jacobi Equation
5. Lagrangian and Legendre Singularities
Lagrangian and Legendre Mappings, Fronts and Legendre Mappings, Generating Families of Hypersurfaces, Caustics and Lagrangian Mappings, Generating Families of Functions, Summary, The Classification of Critical Points of Functions, Versa1 Deformations: An Informal Description, Critical Points of Functions, Simple Singularities, The Platonics, Miniversal Deformations, Singularities of Wave Fronts and Caustics, The Classification of Singularities of Wave Fronts and Caustics in Small Dimensions, Boundary Singularities, Weyl Groups and Simple Fronts, Metamorphoses of Wave Fronts and Caustics, Fronts in the Problem of Going Around an Obstacle
6. Lagrangian and Legendre Cobordisms.
The Maslov Index, The Quasiclassical Asymptotics of the Solutions of the Schrodinger Equation, The Morse Index and the Maslov Index, The Maslov Index of Closed Curves, The Lagrangian Grassmann Manifold and the Universal Maslov Class, Cobordisms of Wave Fronts on the Plane, Cobordisms, The Lagrangian and the Legendre Boundary, The Ring of Cobordism Classes, Vector Bundles with a Trivial Complexification, Cobordisms of Smooth Manifolds, The Legendre Cobordism Groups as Homotopy Groups, The Lagrangian Cobordism Groups, Characteristic Numbers, Characteristic Classes of Vector Bundles, The Characteristic Numbers of Cobordism Classes, Complexes of Singularities, Coexistence of Singularities
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