 Integrable Hamiltonian system

An autonomous Hamiltonian system on a $2n$-dimensional symplectic manifold is Liouville integrable when it has $n$ conserved functions $F_1,\dots ,F_n$ in mutual Poisson commutation, with independent differentials on the regular region. One normally takes $F_1=H$. The involution condition is $\{F_i,F_j\}=0$; independence excludes counting powers of one integral of motion as different integrals.

The Liouville–Arnold theorem turns a compact connected regular common level into an invariant torus. Nearby action-angle variables give $H=H(I)$ and ${\dot{\theta }}_j=\partial H/\partial I_j$, so the angular motion is linear.^[ref] Compactness and regularity matter: separatrices and singular common levels need not be tori. The uncoupled harmonic oscillators in Liouville integrability of the isotropic harmonic oscillator give an explicit example.