Poisson brackets with the Hamiltonian of the system, and with each other, vanish). In finite dimensions, if the phase space is symplectic (i.e., the center of the Poisson algebra consists only of constants), it must have even dimension , and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is . The leaves of the foliation are totally isotropic with respect to the symplectic form and such a maximal isotropic foliation is called Lagrangian. All autonomous Hamiltonian systems (i.e. those for which the Hamiltonian and Poisson brackets are not explicitly time-dependent) have at least one invariant; namely, the Hamiltonian itself, whose value along the flow is the energy. If the energy level sets are compact, the leaves of the Lagrangian foliation are tori, and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical -form are called the action variables, and the resulting canonical coordinates are called action-angle variables (see below). There is also a distinction between complete integrability, in the Liouville sense, and partial integrability, as well as a notion of superintegrability and maximal superintegrability. Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. When the number of independent Poisson commuting invariants is less than maximal (but, in the case of autonomous systems, more than one), we say the system is partially integrable. When there exist further functionally independent invariants, beyond the maximal number that can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is less than n, we say the system is superintegrable. If there is a regular foliation with one-dimensional leaves (curves), this is called maximally superintegrable. Action-angle variables When a finite-dimensional Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are tori. There then exist, as mentioned above, special sets of canonical coordinates on the phase space known as action-angle variables, such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the torus. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables. The Hamilton–Jacobi approach In canonical transformation theory, there is