Editing Adiabatic invariant (section)

=== Adiabatic invariance of ''J'' ===

The Hamiltonian is a function of ''J'' only, and in the simple case of the harmonic oscillator,
<math display="block">
H = \omega J.
</math>

When ''H'' has no time dependence, ''J'' is constant. When ''H'' is slowly time-varying, the rate of change of ''J'' can be computed by re-expressing the integral for ''J'':
<math display="block">
J = \int_0^{2\pi} p \frac{\partial x}{\partial \theta} \,d\theta.
</math>

The time derivative of this quantity is
<math display="block">
\frac{dJ}{dt} = \int_0^{2\pi} \left(\frac{dp}{dt} \frac{\partial x}{\partial \theta} +
 p \frac{d}{dt} \frac{\partial x}{\partial \theta}\right) \,d\theta.</math>

Replacing time derivatives with theta derivatives, using <math>d\theta = \omega \, dt,</math> and setting <math>\omega := 1</math> without loss of generality (<math>\omega</math> being a global multiplicative constant in the resulting time derivative of the action) yields 
<math display="block">
\frac{dJ}{dt} = \int_0^{2\pi} \left(\frac{\partial p}{\partial \theta} \frac{\partial x}{\partial \theta} +
 p \frac{\partial}{\partial \theta} \frac{\partial x}{\partial \theta}\right) \,d\theta.
</math>

So as long as the coordinates ''J'', <math>\theta</math> do not change appreciably over one period, this expression can be integrated by parts to give zero. This means that for slow variations, there is no lowest-order change in the area enclosed by the orbit. This is the adiabatic invariance theorem{{snd}} the action variables are adiabatic invariants.

For a harmonic oscillator, the area in phase space of an orbit at energy ''E'' is the area of the ellipse of constant energy,
<math display="block">
E = \frac{p^2}{2m} + \frac{m\omega^2 x^2}{2}.
</math>

The ''x'' radius of this ellipse is <math>\sqrt{2E/\omega^2m},</math> while the ''p'' radius of the ellipse is <math>\sqrt{2mE}</math>. Multiplying, the area is <math>2\pi E/\omega</math>. So if a pendulum is slowly drawn in, such that the frequency changes, the energy changes by a proportional amount.