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# 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 $C^infty\left(M\right)$ 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

$\left\{f_1,...,f_\left\{N-1\right\},\left\{g_1,...,g_N\right\}\right\}$

$=\left\{\left\{f_1,...,f_\left\{N-1\right\},g_1\right\},g_2,...,g_N\right\}+\left\{g_1,\left\{f_1,...,f_\left\{N-1\right\},g_2\right\},...,g_N\right\}+,cdots$

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

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

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

The case where N = 2 gives a Poisson manifold.

Quantizing Nambu dynamics leads to interesting structures.

## References

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