= Solution
<Hamilton's equations> are
$$
\boxed{\ \dot q_j=\frac{\partial H}{\partial p_j},
\qquad
\dot p_j=-\frac{\partial H}{\partial q_j}.\ }
$$
A time-independent function $F:M\to\mathbb R$ is a <first integral> when it is constant along every Hamiltonian trajectory. Since
$$
\frac{dF}{dt}=\{F,H\},
$$
this is equivalent to $\{F,H\}=0$.
The $2n$-dimensional Hamiltonian system is <Liouville integrability>[integrable] when it possesses $n$ first integrals $F_1=H,F_2,\ldots,F_n$ that are functionally independent on a dense open set and in mutual involution:
$$
\{F_j,F_k\}=0
\qquad(1\leq j,k\leq n).
$$
Back to article page
