2 1 Parametrically driven harmonic oscillator A harmonic oscillator subject to a parametric driving can be defined by the Hamiltonian a p2 m H(p,q t) (w(t)) q, w(t) wo (1 asin(t)), a 1 Solve the equations of motion for this system in the adiabatic limit to w(t) W 0 t Solve the equations of motion assuming that at t to, the trajectory remains continuous, that is, does not jump, lim to or(t) lim to o r(t), and calculate the difference in total energy and in action (phase space area enclosed in the orbit) across to How does the difference in action (if any) depend on the phase of the oscillator at to when the frequency changes Sketch the trajectory before and after to for different values of this phase

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2.1 Parametrically driven harmonic oscillator A harmonic oscillator subject to a parametric driving can be defined by the Hamiltonian a p2 m H(p,q;t) (w(t))

2.1 Parametrically driven harmonic oscillator A harmonic oscillator subject to a parametric driving can be defined by the Hamiltonian a p2 m H(p,q;t) (w(t)) q, w(t) = wo (1+asin(t)), |a|

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