Fundamental principles of classical mechanics. A geometrical perspective (Q2925813)

This is a very good and useful book for those who are interested in geometric and topological language of classical mechanics. For me it looks like the author involves all geometric and topological notions that have mechanical meaning, and describes their mechanical interpretation. The author's idea is that the book is written first of all for a physical audience, but it will be also useful for mathematicians, experts in Global Analysis, looking for applications of their mathematical results in mechanics. The author says it can be considered as a bridge between physics and modern mathematics. With a little effort, the reader can use this book as a first introduction into the ideas and methods of Global Analysis, as well as into their applications to mechanics. The details about the content of the book are clear from the following list of Sections:NEWLINENEWLINENEWLINEPreface. 1. Vectors, Tensors, and Linear Transformations. 2. Exterior Algebra: Determinants, Oriented Frames and Oriented Volumes. 3. The Hodge-Star Operator and the Vector Cross Product. 4. Kinematics of Moving Frames: From the Angular Velocity to Gauge Fields. 5. Differentiable Manifolds: The Tangent and Cotangent Bundles. 6. Exterior Calculus: Differential Forms. 7. Vector Calculus by Differential Forms. 8. The Stokes Theorem. 9. Cartan's Method of Moving Frames: Curvilinear Coordinates in \(\mathbb{R}^3\). 10. Mechanical Constraints: The Frobenius Theorem.NEWLINENEWLINENEWLINE 11. Flows and Lie Derivatives. 12. Newton's Laws: Inertial and Non-inertial Frames. 13. Simple Applications of Newton's Laws. 14. Potential Theory: Newtonian Gravitation. 15. Centrifugal and Coriolis Forces. 16. Harmonic Oscilators: Fourier Transforms and Green's Functions. 17. Classical Model of the Atom: Power Spectra. 18. Dynamical Systems and their Stabilities. 19. Many-Particle Systems and the Conservation Principles. 20. Rigid-Body Dynamics: The Euler-Poisson Equations of Motion.NEWLINENEWLINENEWLINE21. Topology and Systems with Holonomic Constraints: Homology and de Rham Cohomology. 22. Connections on Vector Bundles: Affine Connection of Tangent Bundles. 23. The Parallel Translation of Vectors: The Foucault Pendulum. 24. Geometric Phases, Gauge Fields and the Mechanics of Deformable Bodies: ``The Falling Cat'' Problem. 25. Force and Curvature. 26. The Gauss-Bonnet-Chern Theorem and Holonomy. 27. The Curvature Tensor in Riemannian Geometry. 28. Frame Bundles and Principal Bundles, Connections of Principal Bundles. 29. Calculus of Variations, The Euler-Lagrange Equations, The First Variation of Arclength and Geodesics. 30. The Second Variation of Arclength, Index Forms and Jacobi Fields.NEWLINENEWLINENEWLINE31. The Lagrangian Formulation of Classical Mechanics: Hamilton's Principle of Least Action, Lagrange Multipliers of Constraint Motion. 32. Small Oscillations and Normal Modes. 33. The Hamiltonian Formalism of Classical Mechanics: Hamilton's Equations of Motion. 34. Symmetry and Conservation. 35. Symmetric Tops. 36. Canonical Transformations and the Symplectic Group. 37. Generating Functions and the Hamilton-Jacobi Equation. 38. Integrability, Invariant Tori, Action-Angle Variables. 39. Symplectic Geometry in Hamiltonian Dynamics, Hamiltonian Flows, and Poincaré-Cartan Integral Invariants. 40. Darboux's Theorem in Symplectic Geometry.NEWLINENEWLINENEWLINE41. The Kolmogorov-Arnold-Moser (KAM) Theorem. 42. The Homoclinic Tangle and Instability, Shifts and Subsystems. 43. The Restricted Tree-Body Problem.