Symplectic Geometry

Contents
Preface
Notation
Chapter 1. Symplectic geometry in Euclidean space
	1.1 Some information from matrix group theory
		1.1.1 Lie groups and algebras
		1.1.2 The complete linear groups GL(n, R) and GL(n, C) and their Lie algebras
		1.1.3 The special linear groups SL(n, R) and SL(n, C)
		1.1.4 The orthogonal group O(n) and the special orthogonal group SO(n)
		1.1.5 The unitary group U(n) and the special unitary group SU(n)
		1.1.6 Connected components of matrix groups
		1.1.7 The realification operation and complex structures
	1.2 Groups of symplectic transformations of a linear space
		1.2.1 Symplectic linear transformations
		1.2.2 The noncompact groups Sp(n, R) and Sp(n, C)
		1.2.3 The compact group Sp(n)
		1.2.4 The relation between symplectic groups and other matrix groups
	1.3 Lagrangian manifolds
		1.3.1 Real Lagrangian manifolds in a symplectic linear space
		1.3.2 Complex Lagrangian Grassmann manifolds
		1.3.3 Real Lagrangian Grassmann manifolds
Chapter 2. Symplectic geometry on smooth manifolds
	2.1 Local structure of symplectic manifolds
		2.1.1 Local symplectic coordinates
		2.1.2 Hamiltonian vector fields
		2.1.3 The Poisson bracket
		2.1.4 Darboux' theorem
	2.2 Embeddings of symplectic manifolds
		2.2.1 Embeddings of symplectic manifolds in R^{2N}
		2.2.2 Embeddings of symplectic manifolds in CP^N
		2.2.3 Examples of symplectic manifolds
Chapter 3. Hamiltonian systems with symmetries on symplectic manifolds
	3.1 Liouville's theorem
		3.1.1 Integrals of Hamiltonian systems
		3.1.2 Complete involutive sets of functions
	3.2 Hamiltonian systems with noncommutative symmetries
		3.2.1 Finite-dimensional Lie subalgebras in a space of functions on a symplectic manifold
		3.2.2 A theorem on integration of systems with noncommutative symmetries
		3.2.3 Connections between systems with commutative and noncommutative symmetries
		3.2.4 Noncommutative integration in those cases when the sets of integrals do not form an algebra
		3.2.5 Integration in quadratures of systems with noncommutative integrals
		3.2.6 The canonical form of the Poisson bracket in a neighbourhood of a singular point. The case of degenerate Poisson brackets
		3.2.7 Noncommutative integrability and its connection with canonical submanifolds and isotropic tori
		3.2.8 Solvable Lie algebras of functions on symplectic manifolds and integration of mechanical systems corresponding to them
	3.3 Dynamical systems generated by sectional operators
		3.3.1 General plan of construction of sectional operators
		3.3.2 Construction of a many-parameter family of exterior 2-forms on orbits of stationary groups of symmetric spaces
Chapter 4. Geodesic flows on two~dimensional Riemann surfaces
	4.1 Completely integrable geodesic flows on a sphere and a torus
		4.1.1 Geodesic flow of a two-dimensional Riemannian metric
		4.1.2 A necessary and sufficient condition for the existence of an additional polynomial integral quadratic in the momenta
		4.1.3 Description of Riemannian metrics on a sphere and a torus that admit an additional integral
		4.1.4 Geometric properties of metrics on a sphere that admits an additional integral
	4.2 Nonintegrability of analytic geodesic flows on surfaces of genus g>1
	4.3 Nonintegrability of the problem of n centres for n>2
	4.4 Morse-type theory of integrable Hamiltonian systems. Connections between integrability of systems, existence of stable periodic solutions and the one-dimensional homology group of surfaces of constant energy
Chapter 5. Effective methods of constructing completely integrable systems on Lie algebras. Dynamics of multi-dimensional rigid body
	5.1 Left-invariant Hamiltonian systems on Lie groups and the Euler equations on Lie algebras
		5.1.1 Symplectic structure and left-invariant Hamiltonians
		5.1.2 Quadratic Hamiltonians associated with the displacement of the argument on Lie algebras
		5.1.3 Properties of the general Euler equations
	5.2 A brief summary of classical results on the root decomposition of complex semisimple Lie algebras
	5.3 Analogs of multidimensional rigid body motion for semisimple Lie algebras
		5.3.1 The sectional decomposition of an algebra coincides with Cartan's decomposition
		5.3.2 Various types of sectional operators. Complex metrics. Normal nilpotent metrics. Normal solvable metrics
		5.3.3 Compact series of metrics
		5.3.4 Normal series of metrics
	5.4 Construction of integrals of the Euler equations corresponding to complex, compact and normal dynamics of multi-dimensional rigid body
		5.4.1 Integrals of a complex left-invariant metrics
		5.4.2 Integrals of a compact left-invariant metrics
		5.4.3 Integrals of a normal left-invariant metrics
		5.4.4 Involutoriness of integrals
	5.5 Complete integrability of the Euler equations for "symmetrical" multi-dimensional rigid body
		5.5.1 Complex integrable cases
		5.5.2 Compact integrable cases
		5.5.3 Normal integrable cases
		5.5.4 Integrability of the Euler equations on singular orbits
	5.6 Quadratic integrals of the Euler equations
	5.7 Integrability of geodesic flows of left-invariant metrics of the form \varphi_{abD} on semisimple groups and geodesic flows on symmetric spaces
		5.7.1 Geodesic flow on T*G
		5.7.2 G-invariant geodesic flows on T*(G/H)
		5.7.3 Geodesic flows of general form on symmetric surfaces
Chapter 6. A brief review of the theory of topological classification of integrable nondegenerate Hamiltonian equations with two degrees of freedom
	6.1 Formulation of the problem
		6.1.1 Example: classical Hamiltonian equations of the motion of a rigid body
		6.1.2 Integrability or nonintegrability as a manifestation of symmetry or randomness in system evolution
		6.1.3 Examples of physical and mechanical systems integrable in the Liouville sense
		6.1.4 Classification of all integrable nondegenerate Hamiltonian systems (integrable Hamiltonians) with two degrees of freedom
	6.2 Smooth functions typical on smooth manifolds
		6.2.1 Morse simple functions
		6.2.2 Simple atoms and simple molecules
		6.2.3 Complex Morse functions
		6.2.4 Complex atoms and complex molecules
	6.3 Bott's functions as "typical" int~grals of integrable systems
		6.3.1 Bott's functions
		6.3.2 Integrals which are "typical" in the Hamiltonian physics
	6.4 Rough and fine topological equivalence of integrable systems
	6.5 Theorem of rough and fine classification of integrable Hamiltonian systems with two degrees of freedom. Applications in physics and mechanics
		6.5.1 Formulation of the main theorem
		6.5.2 Relation between invariants W, W* and the topology of an integrable system. Substantial interpretation of atoms and molecules
	6.6 Method of computing topological invariants for specific physical integrable Hamiltonians
	6.7 A brief historical commentary
	6.8 Class (H) of isoenergy three-dimensional integrable manifolds. "Five faces" of this class
		6.8.1 Class (H) of the isoenergy 3-surfaces
		6.8.2 Class (Q) of three-dimensional manifolds glued from two types of blocks
		6.8.3 Class (W) of Waldhausen manifolds (graph-manifolds)
		6.8.4 The class (S)
		6.8.5 The class (T) of isointegrable manifolds corresponding to Hamiltonians with tame integrals
		6.8.6 The class (R) of manifolds glued from round handles
		6.8.7 Theorem on the coincidence of five classes
	6.9 Application of the topological classification theory of integrable systems to geodesic flows on a 2-sphere and 2-torus
		6.9.1 Hypothesis on geodesic flows
		6.9.2 Integrable geodesic flows on a 2-sphere and a 2-torus
		6.9.3 Complexity of integrable geodesic flows on a 2-sphere and a 2-torus
		6.9.4 Hypothesis: linearly-quadratically integrable metrics "approximate" any nondegenerate integrable Riemannian metric on a 2-torus
	6.10 Topological classification of classical cases of integrability in the dynamics of a heavy rigid body
References
	1-18
	19-36
	37-52
	53-68
	69-88
	89-107
	108-125
	126-138
	139-151
	152-166
Subject Index