On foundation of the generalized Nambu mechanics
This article outlines basic principles of canonical formalism for Nambu mechanics (a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973). For instance, the canonical Nambu bracket for a triple of classical observables \(f_ i: \mathbb{R}^ 3\to \mathbb{R}\) is given by \(\{f_ 1, f_ 2, f_ 3\}= {{\partial(f_ 1, f_ 2, f_ 3)} \over {\partial(x_ 1, x_ 2, x_ 3)}}\). The following fundamental identity is satisfied: \[ \{\{f_ 1, f_ 2, f_ 3\},f_ 4, f_ 5\}+ \{f_ 3,\{f_ 1, f_ 2, f_ 4\},f_ 5\}+ \{f_ 3, f_ 4,\{f_ 1, f_ 2, f_ 5\}\}= \{f_ 1, f_ 2,\{f_ 3, f_ 4, f_ 5\}\}.\tag \(*\) \] This resembles Jacobi identity for the usual Poisson brackets. Besides, skew symmetry and Leibniz rule: \(\{f_ 1, f_ 2, f_ 3, f_ 4\}= f_ 1\{f_ 2, f_ 3, f_ 4\}+ f_ 2\{f_ 1, f_ 3, f_ 4\}\) are also satisfied. Equations of motion for given two ``Hamiltonians'' \(H_ 1\), \(H_ 2\) are the following, by definition: \(df/dt= \{H_ 1, H_ 2, f\}\). More generally, the author defines a Nambu-Poisson manifold of order \(n\) as a manifold \(X\) together with a map \(\{.,\dots\}: A^{\oplus^ n}\to A\), where \(A\) is the ring of \(C^ \infty\)-functions on \(X\), and where \(\{.,\dots\}\) is skew symmetric and satisfies Leibniz rule as well as identity \((*)\): \[ \begin{multlined} \{\{f_ 1,\dots, f_{n-1}\}, f_{n+1},\dots, f_{2n-1}\}+ \{f_ n, \{f_ 1,\dots, f_{n-1}, f_{n+1}\}, f_{n+2},\dots, f_{2n-1}\}+\cdots\\ +\{f_ n,\dots, f_{2n-2}, \{f_ 1,\dots, f_{n-1}, f_{2n-1}\}\}= \{f_ 1,\dots, f_{n-1}, \{f_ n,\dots, f_{2n-1}\}\}. \end{multlined} \] Then many basic concepts and properties of Hamiltonian mechanics are generalized in the context of Nambu mechanics (integrals of motion, action principle). Finally quantization procedures are also described (generalizing Feynman's path integral, and canonical approach to quantization). It is, in my opinion, a reasonably self-contained and interesting article.
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