Classical Mechanics

  1. Introduction to Advanced Formulations
    1. Limitations of Newtonian Mechanics
      1. Coordinate-Dependent Formulations
        1. Constraint Handling Difficulties
        2. The Principle of Least Action
          1. Statement and Physical Meaning
            1. Nature's Optimization
            2. Action Integral
              1. Lagrangian Integration
              2. Calculus of Variations
                1. Euler-Lagrange Derivation
              3. Lagrangian Mechanics
                1. Generalized Coordinates
                  1. Definition and Selection
                    1. Degrees of Freedom
                      1. Constraint Reduction
                      2. The Lagrangian
                        1. L = T - V Definition
                          1. Kinetic Energy Expression
                            1. Potential Energy Expression
                            2. Euler-Lagrange Equations
                              1. Derivation from Least Action
                                1. Application to Simple Systems
                                  1. Free Particle
                                    1. Harmonic Oscillator
                                      1. Pendulum
                                    2. Advantages of Lagrangian Approach
                                      1. Coordinate Independence
                                        1. Constraint Handling
                                      2. Hamiltonian Mechanics
                                        1. Generalized Momenta
                                          1. Definition and Calculation
                                            1. Canonical Momentum
                                            2. The Hamiltonian
                                              1. H = T + V for Conservative Systems
                                                1. Legendre Transformation
                                                2. Hamilton's Equations of Motion
                                                  1. Formulation and Interpretation
                                                    1. Phase Space Description
                                                    2. Comparison with Lagrangian Approach
                                                      1. First-Order vs. Second-Order Equations
                                                        1. Phase Space vs. Configuration Space
                                                        2. Conservation Laws and Symmetries
                                                          1. Noether's Theorem Introduction
                                                            1. Canonical Transformations