I am interested in the following question. (It is a rephrased problem in Arnold's book "Mathematics methods of classical mechanics" (2nd ed. page 20)).
Given are potential function $U(x)$ such that $U(x)\to\infty$ as $|x|\to\infty$, and $E$ be a noncritical value of $U$. Let $$F(x,y)=\frac{y^2}{2}+U(x),$$ and $F^{-1}(E)$ be the level curve of $F$ (say, oriented clockwise). Denote $$S=\int_{F^{-1}(E)}ydx$$ and $$T=\int_{F^{-1}(E)}\frac{dx}{y}.$$
Question: show that $T=dS/dE$.
(I can prove it writing all these integrals as plain Riemann integrals and then doing a tedious differentiation, but I suspect there is some simpler way by playing with these 1-forms directly).