Sussman Gerald Jay, Wisdom Jack. Structure and Interpretation of Classical Mechanics

2nd edition. — The MIT Press, 2001. — 527 p. — ISBN10: 0262194554.This textbook takes an innovative approach to the teaching of classical mechanics, emphasizing the development of general but practical intellectual tools to support the analysis of nonlinear Hamiltonian systems. The development is organized around a progressively more sophisticated analysis of particular natural systems and weaves examples throughout the presentation. Explorations of phenomena such as transitions to chaos, nonlinear resonances, and resonance overlap to help the student to develop appropriate analytic tools for understanding. Computational algorithms communicate methods used in the analysis of dynamical phenomena. Expressing the methods of mechanics in a computer language forces them to be unambiguous and computationally effective. Once formalized as a procedure, a mathematical idea also becomes a tool that can be used directly to compute results. The student actively explores the motion of systems through computer simulation and experiment. This active exploration is extended to the mathematics. The requirement that the computer be able to interpret any expression provides strict and immediate feedback as to whether an expression is correctly formulated. The interaction with the computer uncovers and corrects many deficiencies in understanding.Lagrangian Mechanics
Configuration Spaces
Generalized Coordinates
The Principle of Stationary Action
Computing Actions
The Euler–Lagrange Equations
Derivation of the Lagrange Equations
Computing Lagrange’s Equations
How to Find Lagrangians
Coordinate Transformations
Systems with Rigid Constraints
Constraints as Coordinate Transformations
The Lagrangian Is Not Unique
Evolution of Dynamical State
Conserved Quantities
Conserved Momenta
Energy Conservation
Central Forces in Three Dimensions
The Restricted Three-Body Problem
Noether’s Theorem
Abstraction of Path Functions
Constrained Motion
Coordinate Constraints
Derivative Constraints
Nonholonomic Systems
Summary
Projects
Rigid Bodies
Rotational Kinetic Energy
Kinematics of Rotation
Moments of Inertia
Inertia Tensor
Principal Moments of Inertia
Vector Angular Momentum
Euler Angles
Motion of a Free Rigid Body
Computing the Motion of Free Rigid Bodies
Qualitative Features
Euler’s Equations
Axisymmetric Tops
Spin-Orbit Coupling
Development of the Potential Energy
Rotation of the Moon and Hyperion
Spin-Orbit Resonances
Nonsingular Coordinates and Quaternions
Motion in Terms of Quaternions
Summary
Projects
Hamiltonian Mechanics
Hamilton’s Equations
The Legendre Transformation
Hamilton’s Equations from the Action Principle
A Wiring Diagram
Poisson Brackets
One Degree of Freedom
Phase Space Reduction
Lagrangian Reduction
Phase Space Evolution
Phase-Space Description Is Not Unique
Surfaces of Section
Periodically Driven Systems
Computing Stroboscopic Surfaces of Section
Autonomous Systems
Computing Hénon–Heiles Surfaces of Section
Non-Axisymmetric Top
Exponential Divergence
Liouville’s Theorem
Standard Map

Projects
Phase Space Structure
Emergence of the Divided Phase Space
Linear Stability
Equilibria of Differential Equations
Fixed Points of Maps
Relations Among Exponents
Homoclinic Tangle
Computation of Stable and Unstable Manifolds
Integrable Systems
Poincaré–Birkhoff Theorem
Computing the Poincaré–Birkhoff Construction
Invariant Curves
Finding Invariant Curves
Dissolution of Invariant Curves
Summary
Projects
Canonical Transformations
Point Transformations
General Canonical Transformations
Time-Dependent Transformations
Abstracting the Canonical Condition
Invariants of Canonical Transformations
Generating Functions
F1 Generates Canonical Transformations
Generating Functions and Integral Invariants
Types of Generating Functions
Point Transformations
Total Time Derivatives
Extended Phase Space
Poincaré–Cartan Integral Invariant
Reduced Phase Space

Projects
Canonical Evolution
Hamilton–Jacobi Equation
Harmonic Oscillator
Hamilton–Jacobi Solution of the Kepler Problem
F2 and the Lagrangian
The Action Generates Time Evolution
Time Evolution is Canonical
Another View of Time Evolution
Yet Another View of Time Evolution
Lie Transforms
Lie Series
Exponential Identities

Projects
Canonical Perturbation Theory
Perturbation Theory with Lie Series
Pendulum as a Perturbed Rotor
Higher Order
Eliminating Secular Terms
Many Degrees of Freedom
Driven Pendulum as a Perturbed Rotor
Nonlinear Resonance
Pendulum Approximation
Reading the Hamiltonian
Resonance-Overlap Criterion
Higher-Order Perturbation Theory
Stability of the Inverted Vertical Equilibrium
Summary
Projects
Appendices:
Scheme
Our Notation
References
List of Exercises
Index