Let a $d$-dimensional Hamiltonian system (i.e. $2d$-dimensional phase space) be given. Then the existence of $d-1$ observables $c_1,\dots,c_{d-1}$ that are in involution with each other and with the Hamiltonian means that there are $d$ constants of motion along each orbit - the Hamiltonian itself and the value of $c_1,\dots,c_{d-1}$.
Systems for which such a maximal system of observables in involution exists are called Liouville integrable, and the Liouville-Arnold theorem says that for every such system there are action-angle coordinates, in which the phase space is foliated into $d$-dimensional tori whose angular coordinates are the "angles", and each torus is labeled by the values of $H,c_1,\dots,c_{d-1}$ on it.
Now, on a torus labeled by specific values for $H,c_1,\dots,c_{d-1}$, choose any one motion on it and parametrize it by the coordinate $q^k$ (split it into parts if $q^k$ can't injectively parametrize the curve, which will in general be the case). This expresses $p_k$ along the trajectory as a function of $q^k$. It is now a fact that the integral
$$ \int p_k(q^k)\mathrm{d}q^k$$
does not depend on the motion chosen, only on the torus we are on. The geometric meaning of this integral may be seen as the area of the projection of the motion onto the $p_k$-$q^k$-plane. Therefore, it would be more prudent to write $J_k = \frac{1}{2\pi}\int p_k(q^k)\mathrm{d}q^k$ as a function of $H$ and $c_1,\dots,c_{d-1}$ instead.