This volume develops classical mechanics using analytical methods.
This volume develops classical mechanics using analytical methods. It studies motion of particles and systems under forces, emphasizing geometric and variational formulations.
Part I. Foundations
Chapter 1. Kinematics
1.1 Position, velocity, acceleration 1.2 Coordinate systems 1.3 Relative motion 1.4 Examples 1.5 Applications
Chapter 2. Newtonian Mechanics
2.1 Newton’s laws 2.2 Force and mass 2.3 Equations of motion 2.4 Examples 2.5 Applications
Chapter 3. Work and Energy
3.1 Work 3.2 Kinetic and potential energy 3.3 Conservation of energy 3.4 Power 3.5 Examples
Part II. Systems of Particles
Chapter 4. Many-Particle Systems
4.1 Center of mass 4.2 Momentum 4.3 Angular momentum 4.4 Conservation laws 4.5 Examples
Chapter 5. Constraints
5.1 Holonomic constraints 5.2 Non-holonomic constraints 5.3 Degrees of freedom 5.4 Examples 5.5 Applications
Chapter 6. Collisions and Scattering
6.1 Elastic collisions 6.2 Inelastic collisions 6.3 Scattering theory (overview) 6.4 Applications 6.5 Examples
Part III. Lagrangian Mechanics
Chapter 7. Principle of Least Action
7.1 Variational principles 7.2 Action functional 7.3 Euler–Lagrange equations 7.4 Examples 7.5 Applications
Chapter 8. Lagrangian Formulation
8.1 Generalized coordinates 8.2 Lagrange equations 8.3 Constraints 8.4 Examples 8.5 Applications
Chapter 9. Symmetries and Conservation Laws
9.1 Noether’s theorem 9.2 Symmetry groups 9.3 Conserved quantities 9.4 Examples 9.5 Applications
Part IV. Hamiltonian Mechanics
Chapter 10. Hamiltonian Formulation
10.1 Legendre transform 10.2 Hamilton’s equations 10.3 Phase space 10.4 Examples 10.5 Applications
Chapter 11. Canonical Transformations
11.1 Definitions 11.2 Generating functions 11.3 Invariants 11.4 Examples 11.5 Applications
Chapter 12. Hamilton–Jacobi Theory
12.1 Hamilton–Jacobi equation 12.2 Separation of variables 12.3 Integrability 12.4 Examples 12.5 Applications
Part V. Advanced Topics
Chapter 13. Stability and Small Oscillations
13.1 Equilibrium points 13.2 Linearization 13.3 Normal modes 13.4 Applications 13.5 Examples
Chapter 14. Rigid Body Dynamics
14.1 Rotational motion 14.2 Moment of inertia 14.3 Euler equations 14.4 Applications 14.5 Examples
Chapter 15. Continuous Systems (Overview)
15.1 Fields and continua 15.2 Wave motion 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Geometric and Analytical Methods
Chapter 16. Symplectic Geometry
16.1 Symplectic manifolds 16.2 Hamiltonian flows 16.3 Poisson brackets 16.4 Applications 16.5 Examples
Chapter 17. Integrable Systems
17.1 Definitions 17.2 Action-angle variables 17.3 Examples 17.4 Applications 17.5 Connections
Chapter 18. Perturbation Theory
18.1 Small perturbations 18.2 Approximation methods 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Applications
Chapter 19. Celestial Mechanics
19.1 Two-body problem 19.2 Three-body problem (overview) 19.3 Orbital dynamics 19.4 Applications 19.5 Examples
Chapter 20. Engineering Systems
20.1 Mechanical systems 20.2 Control systems 20.3 Vibrations 20.4 Applications 20.5 Examples
Chapter 21. Computational Mechanics
21.1 Numerical integration 21.2 Simulation methods 21.3 Stability 21.4 Software tools 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Nonlinear dynamics 22.2 Chaos in mechanics 22.3 Geometric mechanics 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Stability questions 23.2 Integrability classification 23.3 Computational challenges 23.4 Physical modeling limits 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of classical mechanics 24.2 Key contributors 24.3 Evolution of analytical mechanics 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Classical formulas reference B. Conservation law summary C. Proof techniques checklist D. Numerical method templates E. Cross-reference to other MSC branches