nLab Nambu mechanics

under construction

Idea

What is called Nambu mechanics is a generalization of the formulation of classical mechanics (prequantum geometry) via Poisson brackets to bracket-operations of arity $(n+1)$ higher than 2 – the Nambu brackets –, with an “(n+1)-Lie algebra”-structure (see the discussion there for distinction with proper Lie n-algebras).

Examples

The membrane

One example that naturally gives rise to Nambu mechanics is the relativistic membrane, see at membrane matrix model and at BLG model.

Related concepts

• higher Poisson structure

  • Poisson n-algebra

  • Poisson bracket Lie n-algebra

• membrane

References

General

Nambu mechanics and Nambu brackets were introduced in

• Yoichiro Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D (3) 7 (1973), p. 2405-2412 MR455611 doi Zbl

Detailed analysis is in

• Tamiaki Yoneya, Generalized Hamilton-Jacobi theory of Nambu Mechanics (arXiv:1612.08509)

• Tamiaki Yoneya, Lectures on Higher-Gauge Symmetries from Nambu Brackets and Covariantized M(atrix) Theory, lectures delivered in the workshop “Higher Structure in String Theory and M-theory”, TFC Thematic Program, Fundamental Problems on Quantum Physics, Tohoku University (March 7-11, 2016), (arXiv:1612.08513)

Review:

• Thomas L. Curtright, Cosmas K. Zachos, Classical and Quantum Nambu Mechanics, Phys. Rev. D 68 085001 (2003) [arXiv:hep-th/0212267, doi:10.1103/PhysRevD.68.085001]

• Wikipedia, Nambu mechanics

Interpretation in higher symplectic geometry

Comments on interpreting Nambu mechanics in 2-plectic geometry (and in view of the IKKT matrix model) appear in

• Christian Saemann, Patricia Ritter, Lie 2-algebra models (arXiv:1308.4892)

For background on this see also the discussion at 3-Lie algebra on how these are given by Lie 2-algebras (as metric Lie 2-algebras)