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Nambu dynamics is an interesting generalization of Hamiltonian mechanics. 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.
Nambu came up with 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 <math>C^\infty(M)<math> to itself such that it is completely antisymmetric and {h1,...,hN-1,.} acts as a derivation
{h1,...,hN-1,fg}={h1,...,hN-1,f}g+f{h1,...,hN-1,g}
and the generalized Jacobi identities
- <math>\{f_1,...,f_{N-1},\{g_1,...,g_N\}\}<math>
- <math>=\{\{f_1,...,f_{N-1},g_1\},g_2,...,g_N\}+\{g_1,\{f_1,...,f_{N-1},g_2\},...,g_N\}+\,\cdots<math>
- <math>\cdots\,+\{g_1,...,g_{N-1},\{f_1,...,f_{N-1},g_N\}\}<math>
i.e. {f_1,...,f_{N-1},.} acts as a (generalized) derivation over the n-fold product {.,...,.}.
There are N − 1 Hamiltonians, H1,..., HN-1 generating a time flow
<math>\frac{d}{dt}f=\{f,H_1,...,H_{N-1}\}<math>
The case where N = 2 gives a Poisson manifold.
Quantizing Nambu dynamics leads to interesting structures.
See also
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