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.^[1]

Specifically, consider a differential manifold M, for some integer N ≥ 2; one has a smooth N-linear map from N copies of C ^∞ (M) to itself, such that it is completely antisymmetric: the Nambu bracket, {h₁, ..., h_N−1, .}, which acts as a derivation {h₁, ..., h_N−1,fg} = {h₁, ..., h_N−1, f} g + f {h₁, ..., h_N−1, g}; whence the Filippov Identities (FI),^[2] (evocative of the Jacobi identities, but unlike them, not antisymmetrized in all arguments, for N ≥ 2 ):

$\{ f_1,\cdots , ~f_{N-1},~ \{ g_1,\cdots,~ g_N\}\} = \{ \{ f_1, \cdots, ~ f_{N-1},~g_1\},~g_2,\cdots,~g_N\}+\{g_1, \{f_1,\cdots,f_{N-1}, ~g_2\},\cdots,g_N\}+\dots$ $+\{g_1,\cdots, g_{N-1},\{f_1,\cdots,f_{N-1},~g_N\}\},$

so that {f₁, ..., f_N−1, •} acts as a generalized derivation over the N-fold product {. ,..., .}.

There are N − 1 Hamiltonians, H₁, ..., H_N−1, generating an incompressible flow,

^d⁄_dt f= {f, H₁, ..., H_N−1}.

The generalized phase-space velocity is divergenceless, enabling Liouville's theorem. The case N = 2 reduces to a Poisson manifold, and conventional Hamiltonian mechanics.

For larger even N, the N − 1 Hamiltonians identify with the maximal number of independent invariants of motion (cf. Conserved quantity) characterizing a superintegrable system which evolves in N-dimensional phase space. Such systems are also describable by conventional Hamiltonian dynamics; but their description in the framework of Nambu mechanics is substantially more elegant and intuitive, as all invariants enjoy the same geometrical status as the Hamiltonian: the trajectory in phase space is the intersection of the N − 1 hypersurfaces specified by these invariants. Thus, the flow is perpendicular to all N − 1 gradients of these Hamiltonians, whence parallel to the generalized cross product specified by the respective Nambu bracket.

Quantizing Nambu dynamics leads to intriguing structures^[3] which coincide with conventional quantization ones when superintegrable systems are involved—as they must.

See also

• Hamiltonian mechanics
• symplectic manifold
• Poisson manifold
• Poisson algebra
• Integrable system
• Conserved quantity

References

1. ^ Nambu, Y. (1973). "Generalized Hamiltonian dynamics". Physical Review D7 (8): 2405–2412. Bibcode 1973PhRvD...7.2405N. doi:10.1103/PhysRevD.7.2405. 
2. ^ Filippov, V. T. (1986). "n-Lie Algebras". Sib. Math. Journal 26 (6): 879–891. doi:10.1007/BF00969110. 
3. ^ Curtright, T.; Zachos, C. (2003). "Classical and quantum Nambu mechanics". Physical Review D68 (8): 085001. arXiv:hep-th/0212267. Bibcode 2003PhRvD..68h5001C. doi:10.1103/PhysRevD.68.085001.