equation for any function $\tilde{f}(x_1, \cdots, x_{2n}) = 
f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \sum_{{\rm k}=1}^{n} 
\frac{\partial(f, H)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}
= \frac{1}{2} \sum_{{\rm k}=1}^{n} 
\sum_{{a, b}=1}^{2n} \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x_a, x_b)}
\frac{\partial(x_a, x_b)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})} 
= \sum_{{a, b}=1}^{2n} \tilde{g}_{ab} 
\frac{\partial(\tilde{f}, \tilde{H})}{\partial(x_a, x_b)}~,
\label{H-eq(m=0)}\end{aligned}$$ where $\tilde{H}=\tilde{H}(x_1, \cdots, x_{2n}) = 
H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ and $\tilde{g}_{ab}$ is defined as $$\begin{aligned}
\tilde{g}_{ab}(x_1, \cdots, x_{2n}) 
= g_{ab}(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
\equiv \frac{1}{2}\sum_{{\rm k}=1}^{n} 
\frac{\partial(x_a, x_b)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}
= \frac{1}{2}\{x_a, x_b\}_{\mbox{\tiny{PB}}}~.
\label{Gab}\end{aligned}$$ The $\tilde{g}_{ab}$ plays the role of a metric tensor, because it transforms under a change of variables $x_a \to x'_a$ as follows: $$\begin{aligned}
\tilde{g}'_{ab}(x'_1, \cdots, x'_{2n}) 
= \sum_{{c, d}=1}^{2n} \frac{\partial x'_a}{\partial x_c} 
\frac{\partial x'_b}{\partial x_d} ~\tilde{g}_{cd}(x_1, \cdots, x_{2n})~.
\label{G'ab}\end{aligned}$$ In the case in which $\tilde{g}_{ab}$ depends on $x_a$, neither the transformation $(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
\to (x_1, \cdots, x_{2n})$ nor the time evolution of $x_a$ is a canonical transformation. The latter means that the Liouville theorem in general does not hold for the dynamics of $x_a$. This fact reminds us of the superiority of canonical variables.

Now let us proceed to the case with $m \ge 1$. The equation for 