Let $X$ be a smooth $n$-dimensional manifold, let $T^*X \setminus 0$ denote the cotangent bundle with the zero section removed, and let $p \in C^\infty(T^*X \setminus 0; \mathbb{R})$. Let $H_p$ be the Hamilton vector field of $p$ with respect to the canonical symplectic structure on $T^*X$. Then $H_p p = 0$ on $T^*X \setminus 0$.

Consequently, if $I \subset \mathbb{R}$ is an interval and $\gamma: I \to T^*X \setminus 0$ is an integral curve of $H_p$, then $p(\gamma(s))$ is constant on $I$. In particular, if $s_0 \in I$ and $\gamma(s_0) \in \{p = 0\}$, then $\gamma(s) \in \{p = 0\}$ for every $s \in I$.