redundant variable in this case. Second, $\tilde{H}(x,y,z)$ is not uniquely determined as a function of $x$, $y$, and $z$, i.e., we can add a term $\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ to $\tilde{H}(x,y,z)$, where $\tilde{\lambda}(x,y,z)$ is some function. If a Hamiltonian $\tilde{H}(x,y,z)$ satisfies $\tilde{H}(x,y,z)=H(q,p)$ and Eq. (\[H-eqf(N=3)\]), another Hamiltonian $\tilde{H}(x,y,z)+\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ also satisfies them. This is because the additional term $\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ always vanishes on the Nambu bracket.

It is straightforward to extend the above formulation to the case with general $N(\ge 3)$. We assume that at least $N-1$ of $\{x_i, x_j\}_{\mbox{\tiny{PB}}}$ $(i, j=1, \cdots, N)$ do not vanish. In this case, the equation for any function $\tilde{f}(x_1, \cdots, x_N) = f(q, p)$ is written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)} 
= \frac{1}{2} \sum_{i, j=1}^{N} 
\frac{\partial (\tilde{f}, \tilde{H})}{\partial (x_{i}, x_{j})}
\{x_{i}, x_{j}\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(gN)}\end{aligned}$$ where $\tilde{H}(x_1, \cdots, x_N) = H(q, p)$.

Introducing functions $\tilde{G}_b=\tilde{G}_b(x_1, \cdots, x_N)$ $(b=1, \cdots, N-2)$ that satisfy the conditions $$\begin{aligned}
\frac{1}{(N-2)!} \sum_{i_3 \cdots i_{N}=1}^{N}
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}
= \{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}~,
\label{xiGb}\end{aligned}$$ Eq. (\[H-eq(gN)\]) is rewritten as the Nambu equation in the form of Eq. (\[N-eqf-N\]), $$\begin{aligned}
\frac{d \tilde{f}}{dt} = 
\{\tilde{f}, \tilde{H}, \tilde{G}_1, \cdots, \tilde{G}_{N-2}\}_{\mbox{\tiny{NB}}}~,
\label{H-eqf(gN)}\end{aligned}$$ where we use the formula concerning Jacobians, 