by Gaetano Vilasi (University of Salerno, Italy)

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.

• 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