Classical Mechanics

1. Hamiltonian Mechanics
   1. Introduction to Hamiltonian Mechanics
      1. Historical Context and Development
         1. Contributions of William Rowan Hamilton
         2. Contrast with Lagrangian Mechanics
            1. Differences in formulation
            2. Advantages for certain problems
   2. Core Concepts
      1. Phase Space
         1. Definition and Importance
         2. Multi-dimensional space representing system states
         3. Phase space diagrams and trajectories
      2. Hamiltonian Function
         1. Relationship with Energy
            1. Representation of total energy in mechanical systems
            2. Distinction between kinetic and potential energy
         2. Construction from the Lagrangian
            1. Legendre Transformation
            2. Symmetry between coordinates and momenta
   3. Hamilton’s Equations
      1. Derivation from the Hamiltonian
         1. Canonical equations of motion
         2. Poisson brackets
      2. Advantages over Lagrangian Mechanics
         1. Simplification for large systems
         2. Natural use in phase space
   4. Canonical Transformations
      1. Definition and Purpose
         1. Coordinate transformations that preserve Hamilton's form
         2. Generating functions
      2. Examples and Applications
         1. Simplifying complex Hamiltonians
         2. Action-angle coordinates in integrable systems
   5. Applications in Physics
      1. Quantum Mechanics
         1. Hamiltonian as an operator in quantum systems
         2. Schrödinger’s Equation
      2. Statistical Mechanics
         1. Role in thermodynamic ensembles
         2. Link to Liouville's Theorem
      3. Classical Mechanics Problems
         1. Simple harmonic oscillators
         2. Central force problems
         3. Rigid body dynamics
   6. Advanced Topics
      1. Symplectic Geometry
         1. Geometric underpinning of Hamiltonian mechanics
         2. Preservation of symplectic structure under time evolution
      2. Hamilton-Jacobi Theory
         1. Connection to wavefronts and geometrical optics
         2. Method for solving Hamilton's equations
            1. Action as a characteristic function
      3. Chaos and Non-linear Dynamics
         1. Observing chaotic behavior through phase space
         2. Sensitive dependence on initial conditions
   7. Computational Approaches
      1. Numerical Solutions
         1. Algorithms for solving Hamilton’s equations
         2. Stability of symplectic integrators
      2. Simulation of Complex Systems
         1. N-body problems in celestial mechanics
         2. Molecular dynamics simulations in chemical physics
   8. Theoretical Insights and Implications
      1. Conservation Laws
         1. Explanation of energy, momentum conservation in terms of symmetries
      2. Time Reversibility
         1. Symmetry in time under Hamiltonian dynamics
         2. Implications for thermodynamics and entropy