Nambu mechanics

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,

${\displaystyle \{h_{1},\ldots ,h_{N-1},\cdot \}:C^{\infty }(M)\times \cdots C^{\infty }(M)\rightarrow C^{\infty }(M),}$

which acts as a derivation

${\displaystyle \{h_{1},\ldots ,h_{N-1},fg\}=\{h_{1},\ldots ,h_{N-1},f\}g+f\{h_{1},\ldots ,h_{N-1},g\},}$

whence the Filippov Identities (FI),[2] (evocative of the Jacobi identities, but unlike them, not antisymmetrized in all arguments, for N ≥ 2 ):

${\displaystyle \{f_{1},\cdots ,~f_{N-1},~\{g_{1},\cdots ,~g_{N}\}\}=\{\{f_{1},\cdots ,~f_{N-1},~g_{1}\},~g_{2},\cdots ,~g_{N}\}+\{g_{1},\{f_{1},\cdots ,f_{N-1},~g_{2}\},\cdots ,g_{N}\}+\dots }$ ${\displaystyle +\{g_{1},\cdots ,g_{N-1},\{f_{1},\cdots ,f_{N-1},~g_{N}\}\},}$

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,

${\displaystyle {\frac {d}{dt}}f=\{f,H_{1},\ldots ,H_{N-1}\},}$

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.

Nambu mechanics can be extended to fluid dynamics, where the resulting Nambu brackets are non-canonical and the Hamiltonians are identified with the Casimir of the system, such as enstrophy or helicity[3][4]

Quantizing Nambu dynamics leads to intriguing structures[5] which coincide with conventional quantization ones when superintegrable systems are involved—as they must.

• Hamiltonian mechanics
• Symplectic manifold
• Poisson manifold
• Poisson algebra
• Integrable system
• Conserved quantity
• Hamiltonian Fluid Mechanics

• 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.CS1 maint: ref=harv (link)
• Filippov, V. T. (1986). "n-Lie Algebras". Sib. Math. Journal. 26 (6): 879–891. doi:10.1007/BF00969110.CS1 maint: ref=harv (link)
• Nambu, Y. (1973). "Generalized Hamiltonian dynamics". Physical Review. D7 (8): 2405–2412. Bibcode:1973PhRvD...7.2405N. doi:10.1103/PhysRevD.7.2405.CS1 maint: ref=harv (link)
• Nevir, P.; Blender, R. (1993). "A Nambu representation of incompressible hydrodynamics using helicity and enstrophy". J. Phys. A. 26 (22): 1189–1193. Bibcode:1993JPhA...26L1189N. doi:10.1088/0305-4470/26/22/010.CS1 maint: ref=harv (link)
• Blender, R.; Badin, G. (2015). "Hydrodynamic Nambu mechanics derived by geometric constraints". J. Phys. A. 48 (10): 105501. arXiv:1510.04832. Bibcode:2015JPhA...48j5501B. doi:10.1088/1751-8113/48/10/105501.CS1 maint: ref=harv (link)
• Blender, R.; Badin, G. (2017). "Construction of Hamiltonian and Nambu Forms for the Shallow Water Equations". Fluids. 2: 24. arXiv:1606.03355. doi:10.3390/fluids2020024.CS1 maint: ref=harv (link)