Liouville’s theorem

Classical Mechanics (Fall, 2007)

Lecture Playlist

  1. State diagrams and the nature of physical laws

  2. Newton's laws, principle of least action

  3. Euler-Lagrange equations, symmetry and conservation laws

  4. Symmetry and conservation Laws

  5. Lagrangians and Hamiltonians

  6. Hamilton's equations

  7. Liouville’s theorem

  8. Motion in an electromagnetic field

  9. Poisson brackets formulation

November 26, 2007
Prove Liouville’s theorem from Hamilton’s equations.
Liouville’s theorem can be thought of as information conservation. The laws of mechanics are equivalent to the rules governing state transition.
Ignored degrees of freedom, such as friction, can result in multiple paths to the same final state. Classical mechanics is deterministic so that points do not condense in phase space. The flow through phase space is incompressible. Consider equations of motion, the Lagrangian, and Hamilton's equations, of a particle in a magnetic field.
Topics: 
  • Liouville theorem
  • Phase space
  • Magnetic field
  • Incompressible flow in phase space.
References