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See: Math exercises

Confirm that a harmonic oscillator preserves symplectic area

Consider a one-dimensional harmonic oscillator. Consider two extremes: when it has minimum potential energy and maximum kinetic energy, and when it has maximum potential energy and minimum kinetic energy. Allow for a small delta of variation in each case:

• {$\triangle q \sim 0$} when the maximum extension of the oscillator ranges from {$p_{MIN}$} to {$p_{MAX}$}
• {$\triangle p \sim 0$} when the maximum momentum of the oscillator ranges from {$q_{MIN}$} to {$q_{MAX}$}

The area in phase space at the extreme is given by:

{$\int_{S} p \: dq = \int_{q_{MIN}}^{q_{MAX}} p_{MAX} dq + \int_{q_{MAX}}^{q_{MIN}} p_{MIN} (-dq)=(p_{MAX} - p_{MIN})(q_{MAX} - q_{MIN})$}

Consider the four corner points and the four edges as constraints and what happens to them under the equations of motion for a harmonic oscillator.

How can we know that it is only the extreme points - the boundary conditions - that matter?