Mathematical Physics

  1. Mathematical Formulation of Classical Mechanics
    1. Lagrangian Mechanics
      1. Configuration Space
        1. Generalized Coordinates
          1. Degrees of Freedom
            1. Constraints and Holonomic Systems
            2. Lagrangian Function
              1. Construction from Kinetic and Potential Energy
                1. Lagrangian for Systems with Constraints
                2. Principle of Least Action
                  1. Action Functional
                    1. Hamilton's Principle
                      1. Variational Derivation of Motion
                      2. Lagrange's Equations
                        1. Derivation from Variational Principle
                          1. Generalized Forces
                            1. Non-Conservative Systems
                            2. Symmetries and Conservation Laws
                              1. Cyclic Coordinates
                                1. Conservation of Energy
                                  1. Conservation of Momentum
                                    1. Conservation of Angular Momentum
                                      1. Noether's Theorem in Mechanics
                                    2. Hamiltonian Mechanics
                                      1. Phase Space Formulation
                                        1. Generalized Coordinates and Momenta
                                          1. Phase Space Structure
                                            1. Canonical Variables
                                            2. Legendre Transformation
                                              1. From Lagrangian to Hamiltonian
                                                1. Canonical Momenta
                                                  1. Hamiltonian Function
                                                  2. Hamilton's Equations
                                                    1. Canonical Equations of Motion
                                                      1. Phase Space Trajectories
                                                        1. Conservation of Energy
                                                        2. Poisson Brackets
                                                          1. Definition and Properties
                                                            1. Fundamental Poisson Brackets
                                                              1. Time Evolution of Observables
                                                                1. Jacobi Identity
                                                                2. Canonical Transformations
                                                                  1. Definition and Conditions
                                                                    1. Generating Functions
                                                                      1. Examples of Canonical Transformations
                                                                      2. Hamilton-Jacobi Theory
                                                                        1. Hamilton-Jacobi Equation
                                                                          1. Action-Angle Variables
                                                                            1. Integrable Systems
                                                                            2. Liouville's Theorem
                                                                              1. Conservation of Phase Space Volume
                                                                                1. Statistical Interpretation
                                                                              2. Symplectic Geometry
                                                                                1. Symplectic Vector Spaces
                                                                                  1. Symplectic Form
                                                                                    1. Canonical Basis
                                                                                      1. Lagrangian Subspaces
                                                                                      2. Symplectic Manifolds
                                                                                        1. Definition and Examples
                                                                                          1. Darboux's Theorem
                                                                                          2. Hamiltonian Vector Fields
                                                                                            1. Flow Generated by Hamiltonians
                                                                                              1. Symplectic Diffeomorphisms
                                                                                              2. Moment Maps
                                                                                                1. Definition and Properties
                                                                                                  1. Applications to Reduction

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                                                                                              1. Foundations in Mathematical Methods

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                                                                                              3. Mathematical Formulation of Quantum Mechanics