Nambu mechanics

Remarks

The bracket is antisymmetric in its three arguments, satisfies the Leibniz rule in each argument, and satisfies a generalized Jacobi identity, the fundamental identity. Since $\{H_1, H_1, H_2\} = \{H_2, H_1, H_2\} = 0$, both $H_1$ and $H_2$ are first integrals. The vector field $\nabla H_1 \times \nabla H_2$ has divergence $0$, so its flow preserves volume. On $\R^n$ the construction uses $n - 1$ Hamiltonians and the $n \times n$ Jacobian determinant.

Example: Euler's equations of a free rigid body with principal moments $I_1, I_2, I_3 > 0$ and angular momentum $L \in \R^3$. Take $H_1 = \frac{1}{2} \abs{L}^2$ and $H_2 = \frac{1}{2} \big(L_1^2 / I_1 + L_2^2 / I_2 + L_3^2 / I_3\big)$. Then $\nabla H_1 = L$ and $\nabla H_2 = \omega$ with $\omega_i = L_i / I_i$, and the system is $\dot L = L \times \omega$; its first component is $I_1 \dot \omega_1 = (I_2 - I_3) \omega_2 \omega_3$.

Non-example: the damped system $\dot x = -x$ on $\R^3$ is not a Nambu system for any $H_1, H_2$, since its divergence is $-3 \ne 0$.