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.
Part 1 Analytical mechanics: the Lagrangian coordinates; Hamiltonian systems; transformation theory; the integration methods. Part 2 Basic ideas of differential geometry: manifolds and tangent spaces; differential forms; integration theory; Lie groups and Lie algebras. Part 3 Geometry and physics: symplectic manifolds and Hamiltonian systems; the orbits method; classical electrodynamics. Part 4 Integrable field theories: "KdV" equation; general structures; meaning and existence of recursion operators; miscellanea; integrability of fermionic dynamics.