经典力学导论：从变分法到最优控制（影印版）

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目录
- 前辅文
- Preface
- Chapter 1. One Degree of Freedom
  - §1. The setup
  - §2. Equations of motion
  - §3. Potential energy
  - §4. Kinetic energy
  - §5. Conservation of total energy
  - §6. The phase plane
  - §7. Lagrangian equations of motion
  - §8. The variational meaning of the Euler–Lagrange equation
  - §9. Euler–Lagrange equations — general theory
  - §10. Noether’s theorem/Energy conservation
  - §11. Hamiltonian equations of motion
  - §12. The phase flow
  - §13. The divergence
  - §14. A lemma on moving domains
  - §15. Divergence as a measure of expansion
  - §16. Liouville’s theorem
  - §17. The “uncertainty principle” of classical mechanics
  - §18. Can one hear the shape of the potential?
  - §19. A dynamics-statics equivalence
  - §20. Chapter summary
  - §21. Problems
- Chapter 2. More Degrees of Freedom
  - §1. Newton’s laws
  - §2. Center of mass
  - §3. Newton’s second law for multi-particle systems
  - §4. Angular momentum, torque
  - §5. Rotational version of Newton’s second law; conservation of the angular momentum
  - §6. Circular motion: angular position, velocity, acceleration
  - §7. Energy and angular momentum of rotation
  - §8. The rotational – translational analogy
  - §9. Potential force fields
  - §10. Some physical remarks
  - §11. Conservation of energy
  - §12. Central force fields; conservation of angular momentum
  - §13. Kepler’s problem
  - §14. Kepler’s trajectories are conics: a short proof
  - §15. Motion in linear central fields
  - §16. Linear vibrations: derivation of the equations
  - §17. A nonholonomic system
  - §18. The modal decomposition of vibrations
  - §19. Lissajous’ figures and Chebyshev’s polynomials
  - §20. Invariant 2-tori in R4
  - §21. Rayleigh’s quotient and a physical interpretation
  - §22. The Coriolis and the centrifugal forces
  - §23. Miscellaneous examples
  - §24. Problems
- Chapter 3. Rigid Body Motion
  - §1. Reference frames, angular velocity
  - §2. The tensor of inertia
  - §3. The kinetic energy
  - §4. Dynamics in the body frame
  - §5. Euler’s equations of motion
  - §6. The tennis racket paradox
  - §7. Poinsot’s description of free rigid body motion
  - §8. The gyroscopic effect — an intuitive explanation
  - §9. The gyroscopic torque
  - §10. Speed of precession
  - §11. The gyrocompass
  - §12. Problems
- Chapter 4. Variational Principles of Mechanics
  - §1. The setting
  - §2. Lagrange’s equations
  - §3. Examples
  - §4. Hamilton’s principle
  - §5. Hamilton’s principle <＝> Euler–Lagrange equations
  - §6. Advantages of Hamilton’s principle
  - §7. Maupertuis’ principle — some history
  - §8. Maupertuis’ principle on an example
  - §9. Maupertuis’ principle — a more general statement
  - §10. Discussion of the Maupertuis principle
  - §11. Problems
- Chapter 5. Classical Problems of Calculus of Variations
  - §1. Introduction and an overview
  - §2. Dido’s problem — a historical note
  - §3. A special class of Lagrangians
  - §4. The shortest way to the smallest integral
  - §5. The brachistochrone problem
  - §6. Johann Bernoulli’s solution of the brachistochrone problem
  - §7. Geodesics in Poincaré’s metric
  - §8. The soap film, or the minimal surface of revolution
  - §9. The catenary: formulating the problem
  - §10. Minimizing with constraints — Lagrange multipliers
  - §11. Catenary — the solution
  - §12. An elementary solution for the catenary
  - §13. Problems
- Chapter 6. The Conditions of Legendre and Jacobi for a Minimum
  - §1. Conjugate points
  - §2. The Legendre and the Jacobi conditions
  - §3. Quadratic functionals: the fundamental theorem
  - §4. Sufficient conditions for a minimum for a general functional
  - §5. Necessity of the Legendre condition for a minimum
  - §6. Necessity of the Jacobi condition for a minimum
  - §7. Some intuition on positivity of functionals
  - §8. Problems
- Chapter 7. Optimal Control
  - §1. Formulation of the problem
  - §2. The Maximum Principle
  - §3. A geometrical explanation of the Maximum Principle
  - §4. Example 1: a smooth landing
  - §5. Example 2: stopping a harmonic oscillator
  - §6. Huygens’s principle vs. Maximum Principle
  - §7. Background on linearized and adjoint equations
  - §8. Problems
- Chapter 8. Heuristic Foundations of Hamiltonian Mechanics
  - §1. Some fundamental questions
  - §2. The main idea
  - §3. The Legendre transform, the Hamiltonian, the momentum
  - §4. Properties of the Legendre transform
  - §5. The Hamilton–Jacobi equation
  - §6. Noether’s theorem
  - §7. Conservation of energy
  - §8. Poincaré’s integral invariants
  - §9. The generating function
  - §10. Hamilton’s equations
  - §11. Hamiltonian mechanics as the “spring theory”
  - §12. The optical-mechanical analogy
  - §13. Hamilton–Jacobi equation leading to the Schrödinger equation
  - §14. Examples and Problems
- Bibliography
- Index

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