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.

See also

• Hamiltonian mechanics
• symplectic manifold
• Poisson manifold
• Poisson algebra

References

Y.Nambu, Physical Review D,7, 2405 (1973)