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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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