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Physics · Glossary

What is Action variable?

Definition 2.14 University Physics — Year 3 · Chapter 2 — Hamiltonian Mechanics

For a one-degree-of-freedom system oscillating on a closed phase trajectory, the action variable is the enclosed area divided by 2π2\pi:

I=12π∮p  ⁣dq.I = \frac{1}{2\pi}\oint p\,\dd q .
Adiabatic invariance. Left: as is slowly doubled, the phase ellipse changes shape at constant area, so E = I doubles. Right: hoisting the string of a swinging pendulum feeds energy into the swing in just the proportion that keeps I = E/ fixed.
Adiabatic invariance. Left: as ω\omega is slowly doubled, the phase ellipse changes shape at constant area, so E=ωIE = \omega I doubles. Right: hoisting the string of a swinging pendulum feeds energy into the swing in just the proportion that keeps I=E/ωI = E/\omega fixed.
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