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经典力学导论：从变分法到最优控制（影印版）

样章
作者：
Mark Levi 著
定价：
135.00元
版面字数：
540.00千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2026-01-06
ISBN：
978-7-04-065971-9
物料号：
65971-00
出版时间：
2026-03-27
读者对象：
学术著作
一级分类：
自然科学
二级分类：
力学
三级分类：
应用力学
暂无
  • 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
    • §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
    • §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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