Hamiltonian Dynamics

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. The book is both a textbook and a monograph. It is partially based on a two-semesters course on Analytical Mechanics, Differential Geometry, Symplectic Manifolds, Integrable Systems, and Electrodynamics, given by the author to third-year students in Physics and Mathematics.

As a textbook, it provides a systematic and self-consistent formulation of the Hamiltonian Dynamics in a first moment in a rigorous coordinates language, and after in the modern one of the Differential Geometry. It also supplies the reader with powerful mathematical methods of Theoretical Physics, especially Gauge Theories and General Relativity.

I. Analytical mechanics. 1. The Lagrangian coordinates -- 2. Hamiltonian systems -- 3. Transformation theory -- 4. The integration methods -- II. Basic ideas of differential geometry. 5. Manifolds and tangent spaces -- 6. Differential forms -- 7. Integration theory -- 8. Lie groups and Lie algebras -- III. Geometry and physics. 9. Symplectic manifolds and Hamiltonian systems -- 10. The orbits method -- 11. Classical electrodynamics -- IV. Integrable field theories. 12. KdV equation -- 13. General structures -- 14. Meaning and existence of recursion operators -- 15. Miscellanea -- 16. Integrability of fermionic dynamics Preface; Introduction; Contents; I Analytical Mechanics; 1 The Lagrangian Coordinates; 1.1 A Primer for Various Formulations of Dynamics; 1.2 Constraints; 1.3 Degrees of Freedom and Lagrangian Coordinates; 1.4 The Calculus of Variations and the Lagrange Equations; 1.5 Remarks on Lagrange's Equations; 2 Hamiltonian Systems; 2.1 The Legendre Transformation; 2.2 The Hamilton Equations; 2.3 The Poisson Bracket and the Jacobi-Poisson Theorem; 2.4 A More Compact Form of The Hamiltonian Dynamics; 2.5 The Variational Principle for the Hamilton Equations; 3 Transformation Theory Textbook and monographs featuring material suitable for and based on a two-semester course on analytical mechanics, differential geometry, sympletic manifolds and integrable systems. Deals with the advanced research topic of completely intregable dynamics, with finitely and infinitely many degrees of freedom, presenting mathematical methods of theoretical physics.