Nambu bracket, the Nambu equation for any function ${f}={f}(x_1, x_2, \cdots, x_N)$ can be written as $$\begin{aligned}
\frac{d {f}}{dt} = \{{f}, {H}_1, {H}_2, \cdots, {H}_{N-1}\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf-N}\end{aligned}$$

Hidden Nambu structure {#Hidden Nambu structure}
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Here let us describe a Hamiltonian system with a canonical doublet $(q, p)$ by means of $N$ variables $x_i=x_i(q, p)$ $(i=1, \cdots, N)$.

### Formulation {#Formulation}

First we study the case with $N=2$, for completeness. We assume that $x=x_1(q, p)$ and $y=x_2(q, p)$ satisfy $\{x, y\}_{\mbox{\tiny{PB}}} \ne 0$. In this case, the equation for a function $\tilde{f}(x, y) = f(q, p)$ is written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)}
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)}
\frac{\partial(x, y)}{\partial(q, p)} 
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)} \{x, y\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(N=2)}\end{aligned}$$ where $\tilde{H}(x, y) = H(q, p)$. If $\{x, y\}_{\mbox{\tiny{PB}}}=1$, the transformation $(q, p) \to (x, y)$ is the canonical transformation, and $(x, y)$ are canonical variables.

Next we study the case with $N=3$. We assume that variables $x=x_1(q, p)$, $y=x_2(q, p)$, and $z=x_3(q, p)$ satisfy at least two of the conditions $\{x, y\}_{\mbox{\tiny{PB}}} \ne 0$, $\{y, z\}_{\mbox{\tiny{PB}}} \ne 0$, and $\{z, x\}_{\mbox{\tiny{PB}}} \ne 0$. In this case, the equation for a function $\tilde{f}(x, y, z) = f(q, p)$ is written as $$\begin{aligned}
