This is both a textbook and a monograph. It is partially based on a two-semester course, held by the author for third-year students in physics and mathematics at the University of Salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems.
As a textbook, it provides a systematic and self-consistent formulation of Hamiltonian dynamics both in a rigorous coordinate language and in the modern language of differential geometry. It also presents powerful mathematical methods of theoretical physics, especially in gauge theories and general relativity.
As a monograph, the book deals with the advanced research topic of completely integrable dynamics, with both finitely and infinitely many degrees of freedom, including geometrical structures of solitonic wave equations.
Contents:
- Analytical Mechanics:
- The Lagrangian Coordinates
- Hamiltonian Systems
- Transformation Theory
- The Integration Methods
- Basic Ideas of Differential Geometry:
- Manifolds and Tangent Spaces
- Differential Forms
- Integration Theory
- Lie Groups and Lie Algebras
- Geometry and Physics:
- Symplectic Manifolds and Hamiltonian Systems
- The Orbits Method
- Classical Electrodynamics
- Integrable Field Theories:
- KdV Equation
- General Structures
- Meaning and Existence of Recursion Operators
- Miscellanea
- Integrability of Fermionic Dynamics
Readership: Physicists and mathematicians.
