Analytical Mechanics: An Introduction
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Analytical Mechanics is the investigation of motion with the rigorous tools of mathematics. Rooted in the works of Lagrange, Euler, Poincare (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 flavor 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.
1. Geometric and Kinematic Foundations of Lagrangian Mechanics2. Dynamics: General Laws and the Dynamics of a Point Particle3. One-dimensional Motion4. The Dynamics of Discrete Systems. Lagrangian Formalism5. Motion in a Central Field6. Rigid Bodies: Geometry and Kinematics7. The Mechanics of Rigid Bodies: Dynamics8. Analytical Mechanics: Hamiltonian Formalism9. Analytical Mechanics: Variational Principles10. Analytical Mechanics: Canonical Formalism11. Analytical Mechanics: Hamilton-Jacobi Theory and Integrability12. Analytical Mechanics: Canonical Perturbation Theory13. Analytical Mechanics: An Introduction to Ergodic Theory and to Chaotic Motion14. Statistical Mechanics: Kinetic Theory15. Statistical Mechanics: Gibbs Sets16. langrangian Formalism in Continuum Mechanics Appendices
