Nambu mechanics explained

In mathematics, Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem. This was soon generalized to flows generated by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu–Poisson manifolds with more than one Hamiltonian.

Nambu bracket

Specifically, consider a differential manifold, for some integer ; one has a smooth -linear map from copies of to itself, such that it is completely antisymmetric: the Nambu bracket,

\{h _{1,\ldots,h} _{N-1} , ⋅ \} : C ^{infty(M)} x … C ^{infty(M)} → C ^{infty(M),}

which acts as a derivation

\{h _{1,\ldots,h} _{N-1} ,fg\} = \{h _{1,\ldots,h} _{N-1} ,f\}g + f\{h _{1,\ldots,h} _{N-1} ,g\},

whence the Filippov Identities (FI) (evocative of the Jacobi identities,but unlike them, not antisymmetrized in all arguments, for):

\{ f _{1, …} , ~f _{N-1} ,~ \{ g _{1, … ,~} g _{N\}\}} = \{ \{ f _{1,} … , ~ f _{N-1} ,~g _{1\},~g} _{2, … ,~g} _{N\}+\{g} _{1,} \{f _{1, … ,f} _{N-1} , ~g _{2\}, … ,g} _{N\}+...} + \{g _{1, … ,} g _{N-1} ,\{f _{1, … ,f} _{N-1} ,~g _{N\}\},}

so that