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基本説明
Presents core of Analytical Mechanics, and some of its most relevant applications e.g. to Astronomy, Statistical Mechanics, Continuum Mechanics.
Full Description
Analytical Mechanics is the investigation of motion with the rigorous tools of mathematics. Rooted in the works of Lagrange, Euler, Poincaré (to mention just a few), it is a very classical subject with fascinating developments and still rich of open problems. It addresses such fundamental questions as: Is the solar system stable? Is there a unifying 'economy' principle in mechanics? How can a point mass be described as a 'wave'? And has remarkable applications to many branches of physics (Astronomy, Statistical Mechanics, Quantum Mechanics).
This book was written to fill a gap between elementary expositions and more advanced (and clearly more stimulating) material. It takes up the challenge to explain the most relevant ideas (generally highly non-trivial) and to show the most important applications using a plain language and 'simple' mathematics, often through an original approach. Basic calculus is enough for the reader to proceed through the book. New mathematical concepts are fully introduced and illustrated in a simple, student-friendly language. More advanced chapters can be omitted while still following the main ideas. Anybody wishing to go deeper in some direction will find at least the flavour of recent developments and many bibliographical references. The theory is always accompanied by examples. Many problems are suggested and some are completely worked out at the end of each chapter. The book may effectively be used (and has been used at several Italian Universities) for undergraduate as well as for PhD courses in Physics and Mathematics at various levels.
Contents
1. Geometric and Kinematic Foundations of Lagrangian Mechanics ; 2. Dynamics: General Laws and the Dynamics of a Point Particle ; 3. One-dimensional Motion ; 4. The Dynamics of Discrete Systems. Lagrangian Formalism ; 5. Motion in a Central Field ; 6. Rigid Bodies: Geometry and Kinematics ; 7. The Mechanics of Rigid Bodies: Dynamics ; 8. Analytical Mechanics: Hamiltonian Formalism ; 9. Analytical Mechanics: Variational Principles ; 10. Analytical Mechanics: Canonical Formalism ; 11. Analytical Mechanics: Hamilton-Jacobi Theory and Integrability ; 12. Analytical Mechanics: Canonical Perturbation Theory ; 13. Analytical Mechanics: An Introduction to Ergodic Theory and to Chaotic Motion ; 14. Statistical Mechanics: Kinetic Theory ; 15. Statistical Mechanics: Gibbs Sets ; 16. langrangian Formalism in Continuum Mechanics ; Appendices
