Nambu mechanics

From Wikipedia, the free encyclopedia

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 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

{f₁,...,f_{N − 1},{g₁,...,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

The case where N = 2 gives a Poisson manifold.

Quantizing Nambu dynamics leads to interesting structures.

[edit] See also

• Hamiltonian mechanics
• symplectic manifold
• Poisson manifold
• Nambu-Poisson manifold
• Poisson algebra