\frac{\partial B}{\partial q_{\mbox{\tiny(k)}}}\right)~,
\label{PB-n}\end{aligned}$$ and the Hamilton’s equation of motion for any function $f=f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d f}{dt} = \{f, H\}_{\mbox{\tiny{PB}}}~,
\label{H-eqf-n}\end{aligned}$$ where $H=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is the Hamiltonian of the system. On the other hand, the Nambu bracket of $\tilde{A}$, $\tilde{B}$, and $\tilde{C}$ is defined as $$\begin{aligned}
\{\tilde{A}, \tilde{B}, \tilde{C}\}_{\mbox{\tiny{NB}}} 
\equiv \sum_{{\rm k}=1}^{n} 
\frac{\partial (\tilde{A}, \tilde{B}, \tilde{C})}
{\partial (x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})}~,
\label{NB-n}\end{aligned}$$ where $x_{\mbox{\tiny(k)}} = x_{1\mbox{\tiny(k)}}$, $y_{\mbox{\tiny(k)}} = x_{2\mbox{\tiny(k)}}$, and $z_{\mbox{\tiny(k)}} = x_{3\mbox{\tiny(k)}}$. Then the Nambu equation for any function $\tilde{f}=
\tilde{f}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, \cdots, 
z_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \{\tilde{f}, \tilde{H}, \tilde{G}\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf-n}\end{aligned}$$ Here $\tilde{H}=
\tilde{H}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, 
\cdots, z_{\mbox{\tiny({\it n})}})
=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, \cdots, 
q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is the Hamiltonian and $\tilde{G}=\tilde{G}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, 
\cdots, z_{\mbox{\tiny({\it n})}})
= \sum_{\rm k} \tilde{G}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})$ is the sum of the induced constraints that satisfy the conditions $$\begin{aligned}
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial x_{\mbox{\tiny(k)}}} = 
\frac{\partial(y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~,~~
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial y_{\mbox{\tiny(k)}}} = 
\frac{\partial(z_{\mbox{\tiny(k)}}, x_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~,~~
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial z_{\mbox{\tiny(k)}}} = 
\frac{\partial(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~.
\label{xyzG-n}\end{aligned}$$ Note that the induced constraints are defined so as to be zero, $\tilde{G}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})
=G_{\mbox{\tiny(k)}}(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})=0$, and the Hamiltonian 