Poincaré canonical momenta and nambu mechanics

We show that if certain Poincaré-like integrals are conserved, then to each configuration coordinate of a system an entity can be associated that is an acceptable generalization of the notion of canonical momentum: In the particular case of standard mechanics, the canonical momenta are retrieved. Under certain general restrictions, the Poincaré momenta make sense for either mechanical or general systems for which we do not have (or are not aware of) entities (like the Lagrangian) that are generally used to define the momentum. The Poincaré momentum may also make sense for systems whose characteristics are difficult, or impossible, to reconcile with the notion of the usual canonical momentum. It is also relevant for certain cases where a Lagrangian exists, but it leads to a mixture of physical and unphysical entities. In particular, we show that while physical canonical momenta do not generally exist in the new Nambu mechanics (because of the dimensionality of state vector space), the Poincaré momenta exist, they are physical, and have the properties we could have expected for the mechanics.