Contents

• Essentials of Classical Mechanics
  • 1. Formalism and generalized coordinates
    • 1.1 Analytic and vector mechanics
    • 1.2 Definitions and notation
    • 1.3 Problems
  • 2. Calculus of variations
    • 2.1 Virtual displacement
    • 2.2 Variation of a function
    • 2.3 Stationary value of a function
    • 2.4 Constraints and Lagrange multipliers
    • 2.5 Stationary value of an integral
    • 2.6 Stationary value of an integral that does not explicitly depend on x
    • 2.7 Stationary value of an integral with auxiliary conditions
    • 2.8 Problems
    • 2.9 Solutions
  • 3. Virtual work and D'Alembert's principle
    • 3.1 Virtual work
    • 3.2 D'Alembert's principle
    • 3.3 Conservation of energy from D'Alembert's principle
    • 3.4 Inertial forces in accelerated coordinate systems
    • 3.5 Problems
  • 4. The Lagrangian
    • 4.1 The Lagrangian from D'Alembert's Principle
    • 4.2 The Lagrangian from Hamilton's Principle
    • 4.3 Properties of the Lagrangian
    • 4.4 The generalized momentum
    • 4.5 The Routhian and eliminating cyclic coordinates
    • 4.6 Problems
  • 5. Conservations laws and symmetries
    • 5.1 Conserved generalized momenta
    • 5.2 Continuous Transformations
    • 5.3 Noether's theorem
    • 5.4 Translation symmetry
    • 5.5 Rotational symmetry
    • 5.6 Time symmetry
    • 5.7 Problems
  • 6. Applications of the Lagrangian formalism
    • 6.1 Central Force Problem
    • 6.2 Small oscillations and coupled oscillators
  • 7. The Hamiltonian
    • 7.1 The Legendre Transform
    • 7.2 Legendre Transform of the Lagrangian
    • 7.3 The Canonical equations
    • 7.4 Canonical equations from Hamilton's principle
    • 7.5 Symplectic notation
    • 7.6 Phase space, the phase space fluid and Liouville's Theorem
    • 7.7 Poisson Brackets
    • 7.8 Problems
  • 8. Canonical Transformations
    • 8.1 Types of transformations
    • 8.2 Poisson Brackets and canonical transformations
    • 8.3 Infinitesimal canonical transformations
    • 8.4 Hamilton-Jacobi Equation
    • 8.5 Action-angle variables
    • 8.6 Problems
  • 9. Continuous Media and the Theory of Fields
    • 9.1 The stretched string
    • 9.2 The variational principle applied to a one dimensional Lagrangian density
    • 9.3 The variational principle with three-dimensional fields
    • 9.4 Problems