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Phase Space

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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

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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

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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

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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

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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

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Revision Previous lecture was about Generating Function Approach Derivation of Conservation Laws via Lagrangian via Hamiltonian.

Revision Previous lecture was about Generating Function Approach Derivation of Conservation Laws via Lagrangian via Hamiltonian.

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