Lagrange, Hamilton-Dirac, and Nambu mechanics

In light of a recent work on the subject, the Nambu mechanics for Nambu triplets is compared with Lagrange, Hamilton, and Hamilton-Dirac mechanics. In order to have the proper tool to deal with this, we reexamined the subtle problem of which are the physical variables of the Hamilton-Dirac mechanics of phase-space constrained systems. We assert that in any pair of physically equivalent theories the basic causal variables of both theories must be in one-to-one correspondence, and that the same happens with the basic canonically allowed variables. Then we go back to the comparison of Nambu mechanics with the other mechanics. The proposed Hamiltonian embeddings of Nambu mechanics are correct; however, if the above assertion is also right, then the Nambu mechanics is physically inequivalent to any of the other mechanics. Therefore, it can, in principle, offer a proper description of new physical systems.