Liouville s theorem

Classical Mechanics (Fall, 2011)

Lecture Playlist

  1. State diagrams and the nature of physical laws

  2. Newton's law, phase space, momentum and energy

  3. Lagrangian, least action, Euler-Lagrange equations

  4. Symmetry and conservation laws

  5. The Hamiltonian

  6. Hamilton's equations

  7. Liouville s theorem

  8. Poisson brackets

  9. Electric and magnetic fields 1

  10. Electric and magnetic fields 2

November 7, 2011

This lecture analyses the flow in phase space of multiple systems, and that flow is shown to be incompressible. Poisson brackets are introduced as yet another way to express classical mechanics formally.

Topics: 
  • Liouville's famous theorem
  • Review of Hamiltonian and energy conservation
  • Energy conservation and surfaces in phase space
  • Concept of flow in phase space
  • Compressible and incompressible flows, the divergence
  • Demonstration of Liouville's theorem
  • Liouville using a toy Hamiltonian. Topology of evolving phase space elements
  • The damped harmonic oscillator as a counterexample of Liouville
  • Definition of Poisson bracket
  • Poisson bracket and time derivative of any quantity
References