Classical Motion and Phase Space for a Harmonic Oscillator

This Demonstration illustrates the classical harmonic motion of a particle governed by the Hamiltonian H=(ω/2)(((OverTilde[p])^2)+((OverTilde[x])^2)), where the scaled variables are defined as  (OverTilde[x])≡(Sqrt[m ω])x, (OverTilde[p])≡(p/(Sqrt[m ω])). Here (OverTilde[x])(t) and (OverTilde[p])(t) are obtained by solving Hamilton's equations of motion, subject to the initial conditions (Subscript[(OverTilde[x]), 0])=(OverTilde[x])(0)  and (Subscript[(OverTilde[p]), 0])=(OverTilde[p])(0). The three panels animate synchronously: (1) the motion of the particle in the potential; (2) the phase space trajectory; and (3) the time series of (OverTilde[x])(t) and (OverTilde[p])(t). In the upper two panels, the points with minimum (turning points) and maximum momentum are labeled with blue and green x's, respectively.

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Contributed by: Porscha McRobbie and Eitan Geva

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