nambu mechanics

In mathematics, Nambu dynamics 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 particular we have a differential manifold M, for some integer N ≥ 2, we have a smooth N-linear map from n copies of

C^\infty(M)

to itself such that it is completely antisymmetric and {h₁,...,h_N-1,.} acts as a derivation {h₁,...,h_N-1,fg}={h₁,...,h_N-1,f}g+f{h₁,...,h_N-1,g} and the generalized Jacobi identities

\{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\}+\,\cdots

\cdots\,+\{g_1,...,g_{N-1},\{f_1,...,f_{N-1},g_N\}\}

i.e. {f_1,...,f_{N-1},.} acts as a (generalized) derivation over the n-fold product {.,...,.}. There are N − 1 Hamiltonians, H₁,..., H_N-1 generating a time flow

\frac{d}{dt}f=\{f,H_1,...,H_{N-1}\}

The case where N = 2 gives a Poisson manifold. Quantizing Nambu dynamics leads to interesting structures.

Nambu Mechanics

See also