Contents -- Preface -- Introduction -- 1. Dynamic equations -- 1.1 Preliminary. Fibre bundles over R -- 1.2 Autonomous dynamic equations -- 1.3 Dynamic equations -- 1.4 Dynamic connections -- 1.5 Non-relativistic geodesic equations -- 1.6 Reference frames -- 1.7 Free motion equations -- 1.8 Relative acceleration -- 1.9 Newtonian systems -- 1.10 Integrals of motion -- 2. Lagrangian mechanics -- 2.1 Lagrangian formalism on Q ! R -- 2.2 Cartan and Hamilton-De Donder equations -- 2.3 Quadratic Lagrangians -- 2.4 Lagrangian and Newtonian systems -- 2.5 Lagrangian conservation laws -- 2.5.1 Generalized vector fields -- 2.5.2 First Noether theorem -- 2.5.3 Noether conservation laws -- 2.5.4 Energy conservation laws -- 2.6 Gauge symmetries -- 3. Hamiltonian mechanics -- 3.1 Geometry of Poisson manifolds -- 3.1.1 Symplectic manifolds -- 3.1.2 Presymplectic manifolds -- 3.1.3 Poisson manifolds -- 3.1.4 Lichnerowicz-Poisson cohomology -- 3.1.5 Symplectic foliations -- 3.1.6 Group action on Poisson manifolds -- 3.2 Autonomous Hamiltonian systems -- 3.2.1 Poisson Hamiltonian systems -- 3.2.2 Symplectic Hamiltonian systems -- 3.2.3 Presymplectic Hamiltonian systems -- 3.3 Hamiltonian formalism on Q ! R -- 3.4 Homogeneous Hamiltonian formalism -- 3.5 Lagrangian form of Hamiltonian formalism -- 3.6 Associated Lagrangian and Hamiltonian systems -- 3.7 Quadratic Lagrangian and Hamiltonian systems -- 3.8 Hamiltonian conservation laws -- 3.9 Time-reparametrized mechanics -- 4. Algebraic quantization -- 4.1 GNS construction -- 4.1.1 Involutive algebras -- 4.1.2 Hilbert spaces -- 4.1.3 Operators in Hilbert spaces -- 4.1.4 Representations of involutive algebras -- 4.1.5 GNS representation -- 4.1.6 Unbounded operators -- 4.2 Automorphisms of quantum systems -- 4.3 Banach and Hilbert manifolds -- 4.3.1 Real Banach spaces -- 4.3.2 Banach manifolds.

4.3.3 Banach vector bundles -- 4.3.4 Hilbert manifolds -- 4.3.5 Projective Hilbert space -- 4.4 Hilbert and C -algebra bundles -- 4.5 Connections on Hilbert and C -algebra bundles -- 4.6 Instantwise quantization -- 5. Geometric quantization -- 5.1 Geometric quantization of symplectic manifolds -- 5.2 Geometric quantization of a cotangent bundle -- 5.3 Leafwise geometric quantization -- 5.3.1 Prequantization -- 5.3.2 Polarization -- 5.3.3 Quantization -- 5.4 Quantization of non-relativistic mechanics -- 5.4.1 Prequantization of T Q and V Q -- 5.4.2 Quantization of T Q and V Q -- 5.4.3 Instantwise quantization of V Q -- 5.4.4 Quantization of the evolution equation -- 5.5 Quantization with respect to di erent reference frames -- 6. Constraint Hamiltonian systems -- 6.1 Autonomous Hamiltonian systems with constraints -- 6.2 Dirac constraints -- 6.3 Time-dependent constraints -- 6.4 Lagrangian constraints -- 6.5 Geometric quantization of constraint systems -- 7. Integrable Hamiltonian systems -- 7.1 Partially integrable systems with non-compact invariant submanifolds -- 7.1.1 Partially integrable systems on a Poisson manifold -- 7.1.2 Bi-Hamiltonian partially integrable systems -- 7.1.3 Partial action-angle coordinates -- 7.1.4 Partially integrable system on a symplectic manifold -- 7.1.5 Global partially integrable systems -- 7.2 KAM theorem for partially integrable systems -- 7.3 Superintegrable systems with non-compact invariant submanifolds -- 7.4 Globally superintegrable systems -- 7.5 Superintegrable Hamiltonian systems -- 7.6 Example. Global Kepler system -- 7.7 Non-autonomous integrable systems -- 7.8 Quantization of superintegrable systems -- 8. Jacobi fields -- 8.1 The vertical extension of Lagrangian mechanics -- 8.2 The vertical extension of Hamiltonian mechanics -- 8.3 Jacobi fields of completely integrable systems.

9. Mechanics with time-dependent parameters -- 9.1 Lagrangian mechanics with parameters -- 9.2 Hamiltonian mechanics with parameters -- 9.3 Quantum mechanics with classical parameters -- 9.4 Berry geometric factor -- 9.5 Non-adiabatic holonomy operator -- 10. Relativistic mechanics -- 10.1 Jets of submanifolds -- 10.2 Lagrangian relativistic mechanics -- 10.3 Relativistic geodesic equations -- 10.4 Hamiltonian relativistic mechanics -- 10.5 Geometric quantization of relativistic mechanics -- 11. Appendices -- 11.1 Commutative algebra -- 11.2 Geometry of bre bundles -- 11.2.1 Fibred manifolds -- 11.2.2 Fibre bundles -- 11.2.3 Vector bundles -- 11.2.4 Affine bundles -- 11.2.5 Vector fields -- 11.2.6 Multivector fields -- 11.2.7 Differential forms -- 11.2.8 Distributions and foliations -- 11.2.9 Differential geometry of Lie groups -- 11.3 Jet manifolds -- 11.3.1 First order jet manifolds -- 11.3.2 Second order jet manifolds -- 11.3.3 Higher order jet manifolds -- 11.3.4 Differential operators and differential equations -- 11.4 Connections on fibre bundles -- 11.4.1 Connections -- 11.4.2 Flat connections -- 11.4.3 Linear connections -- 11.4.4 Composite connections -- 11.5 Differential operators and connections on modules -- 11.6 Differential calculus over a commutative ring -- 11.7 Infinite-dimensional topological vector spaces -- Bibliography -- Index.