1. Phase Space

2. Phase Curve  1-D Harmonic motion can be plotted as velocity vs position. Momentum instead of velocityMomentum instead of velocity  For one set of initial conditions there is a phase curve. Ellipse for simple harmonicEllipse for simple harmonic Spiral for damped harmonic.Spiral for damped harmonic. Undamped Damped

3. Phase Portrait  A series of phase curves corresponding to different energies make up a phase portrait. Velocity for Lagrangian systemVelocity for Lagrangian system Momentum for Hamiltonian systemMomentum for Hamiltonian system E < 2 E = 2 E > 2

4. Phase Flow  A region of phase space will evolve over time. Large set of pointsLarge set of points Consider conservative systemConsider conservative system  The region can be characterized by a phase space density. q p

5. Differential Flow  The change in phase space can be viewed from the flow. Flow in Flow out  Sum the net flow over all variables. q p

6. Liouville’s Theorem  Hamilton’s equations can be combined. Simplify phase space expressionSimplify phase space expression  This gives the total time derivative of the phase space density. Conserved over timeConserved over time