Introduction

1. Non-autonomous mechanics

1.1 Geometry of fibre bundles over R

1.2 Dynamic equations

1.3 Reference frames

1.4 Integrals of motion

1.5 Lagrangian mechanics

1.6 Lagrangian conservation laws

2. Hamiltonian mechanics

2.1 Geometry of Poisson manifolds

2.1.1 Symplectic manifolds

2.1.2 Poisson manifolds

2.1.3 Symplectic foliations

2.1.4 Group action on Poisson manifolds

2.2 Hamiltonian systems on Poisson manifolds

2.3 Non-autonomous Hamiltonian mechanics

2.4 Associated Lagrangian and Hamiltonian systems

2.5 Hamiltonian conservation laws

3. Partially integrable systems

3.1 Poisson partially integrable systems

3.2 Bi-Hamiltonian systems

3.3 Partial action-angle coordinates

3.4 Symplectic partially integrable systems

3.5 Global partially integrable systems

3.6 Completely integrable systems

3.7 KAM theorem for partially integrable systems

4. Superintegrable systems

4.1 Symplectic superintegrable systems

4.2 Global superintegrable systems

4.3 Superintegrable Hamiltonian systems

5. Superintegrable Kepler system

5.1 Lagrangian Kepler system

5.2 Hamiltonian Kepler system

5.3 Global Kepler system

6. Integrable non-autonomous systems

6.1 Completely integrable systems

6.2 Superintegrable systems

7. Quantized superintegrable systems

7.1 Symplectic geometric quantization

7.2 Leafwise geometric quantization

7.3 Quantization in action-angle variables

7.4 Quantized superintegrable so(3) system

8. Mechanics with parameters

8.1 Hamiltonian mechanics with parameters

8.2 Quantum mechanics with classical parameters

8.3 Berry phase in action-angle variables

8.3.1 Berry phase factor

8.3.2 Classical non-adiabatic holonomy operator

8.3.3 Quantum holonomy operator

9. Appendixes

9.1. Geometry of fibre bundles

9.1.1 Fibred manifolds and fibre bundles

9.1.2 Vector and affine bundles

9.1.3 Vector and multivector fields

9.1.4 Exterior and tangent-valued forms

9.2. Jet manifolds

9.2.1 First order jet manifolds

9.2.2 Higher order jet manifolds

9.2.3 Differential operators and equations

9.3. Connections on fibre bundles

9.3.1 Connections

9.3.2 Flat connections

9.3.3 Composite connections

9.4. Commutative geometry

9.4.1 Commutative algebra

9.4.2 Differential operators and connections on modules

9.4.3 Differential calculus over a commutative ring

9.5 Geometry of foliations

9.6 Differential geometry of Lie groups

Bibliography

Index

Sardanashvily G.A.

Theoretical and mathematical physicist, Ph.D. and D.Sc. in physics and mathematics, a principal research scientist at the Department of Theoretical Physics of Moscow State University (Russia). His research area covers geometric methods in field theory, quantum theory, mechanics and gravitation theory. He has published over 350 scientific works, including 26 books.