Nambu mechanics

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. In 1994, Leon Takhtajan revisited Nambu dynamics.

Specifically, consider a differential manifold M, for some integer N ≥ 2; one has a smooth N-linear map from N copies of C^∞ (M) to itself, such that it is completely antisymmetric: the Nambu bracket,

which acts as a derivation

whence the Filippov Identities (FI) (evocative of the Jacobi identities, but unlike them, not antisymmetrized in all arguments, for N ≥ 2 ):

${\displaystyle \{f_{1},\cdots ,~f_{N-1},~\{g_{1},\cdots ,~g_{N}\}\}=\{\{f_{1},\cdots ,~f_{N-1},~g_{1}\},~g_{2},\cdots ,~g_{N}\}+\{g_{1},\{f_{1},\cdots ,f_{N-1},~g_{2}\},\cdots ,g_{N}\}+\dots }$ ${\displaystyle +\{g_{1},\cdots ,g_{N-1},\{f_{1},\cdots ,f_{N-1},~g_{N}\}\},}$

so that {f₁, ..., f_N−1, •} acts as a generalized derivation over the N-fold product {. ,..., .}.

There are N − 1 Hamiltonians, H₁, ..., H_N−1, generating an incompressible flow,

The generalized phase-space velocity is divergenceless, enabling Liouville's theorem. The case N = 2 reduces to a Poisson manifold, and conventional Hamiltonian mechanics.