$$\begin{aligned}
\frac{\partial (\tilde{A}_1, \tilde{A}_2, \cdots, \tilde{A}_N)}
{\partial (x_1, x_2, \cdots, x_N)}
= \frac{1}{2(N-2)!} \sum_{i_1, i_2, i_3, \cdots i_{N}=1}^{N}
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{A}_1, \tilde{A}_2)}{\partial (x_{i_1}, x_{i_2})}
\frac{\partial (\tilde{A}_3, \cdots, \tilde{A}_{N})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}~.
\label{J(gN)}\end{aligned}$$

By the use of Eq. (\[xiGb\]), it is shown that the Poisson bracket between any of $N-2$ functions $G_b(q,p)=\tilde{G}_b(x_1, x_2, \cdots, x_{N})$ and an arbitrary function $u(q,p)=\tilde{u}(x_1, x_2, \cdots, x_{N})$ vanishes such that $$\begin{aligned}
\{G_b, u\}_{\mbox{\tiny{PB}}} 
&=& \frac{1}{2} \sum_{i_1, i_2=1}^{N} 
\frac{\partial (\tilde{G}_b, \tilde{u})}{\partial (x_{i_1}, x_{i_2})}
\{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}
\nonumber \\
&=& \frac{1}{2(N-2)!} \sum_{i_1, i_2, i_3, \cdots i_{N}=1}^{N} 
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{G}_b, \tilde{u})}{\partial (x_{i_1}, x_{i_2})}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}
\nonumber \\
&=& \frac{\partial (\tilde{G}_b, \tilde{u}, \tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{1}, x_{2}, x_{3}, \cdots, x_{N})} = 0~.
\label{Gbu=0}\end{aligned}$$ Hence $G_b$ are constants. We can eliminate the constants by redefining $G_b$, and the resulting $\tilde{G}_b(x_1,x_2,\cdots, x_{N})=0$ can be regarded as [*induced constraints*]{}, which are associated with enlarging the phase space from $(q, p)$ to $(x_1,x_2,\cdots, x_{N})$.

In this way, [*Hamiltonian systems can be formulated as Nambu systems by the use of $N$ variables $x_i=x_i(q, p)$ $(i=1, 2, \cdots, N)$. The variables form an $N$-plet, and the $N-1$ Hamiltonians are given 