Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 15B / b / Solution

For a regular periodic energy orbit, differentiate the action integral with respect to its energy. On a branch $H(p,q)=E$, implicit differentiation gives $\partial p/\partial E=1/H_p=1/\dot{q}$. ThusThe integral follows the entire orbit, with its orientation; at turning points this formula is interpreted by the limiting integral from each branch. It is valid for a smooth family of nonseparatrix periodic orbits.

An adiabatic invariant means that the action variable remains constant to leading order when the external parameter changes on a timescale much longer than the orbital period. Denote that parameter by $\lambda$. With $H(0,\lambda )=0$, the displayed identity integrates toIf the parameter change decreases $T(J,\lambda )$ for every intervening action $J$, this integrand increases pointwise. At the adiabatically fixed positive $I$, the energy increases. This uses the phrase “with any $I$” as a statement about the family of orbits, not merely a period decrease at a single selected action; that weaker premise alone would not determine the integral.