= 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 $\partial p/\partial E=1/H_p=1/\dot q$. Thus
$$
\frac{dI}{dE}=\frac1{2\pi}\oint\frac{dq}{\dot q}=\frac T{2\pi},\qquad
\boxed{\frac{dH}{dI}=\frac{2\pi}{T}.}
$$
The 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 to
$$
H(I,\lambda)=\int_0^I\frac{2\pi}{T(J,\lambda)}\,dJ.
$$
If the parameter change decreases $T(J,\lambda)$ for every intervening action $J$, this integrand increases pointwise. At the adiabatically fixed positive $I$, \b[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.
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