  Focusing on the algebraical analysis of two various kinds of one-dimensional
G-dynamics ${{\hat{w}}^{\left( cl \right)}}$ and ${{\hat{w}}^{\left(
ri\right)}}$ separately induced by different Hamiltonian operators $\hat{H} $
are the keypoints. In this work, it's evidently proved that an identity
${{\hat{w}}^{\left( cl \right)}}{{u}^{-1/2}}\equiv0$ always holds for any $u>0$
based on the formula of one-dimensional G-dynamics ${{\hat{w}}^{\left( cl
\right)}}$. We prove that the G-dynamics ${{\hat{w}}^{\left( cl \right)}}$ and
${{\hat{w}}^{\left( ri\right)}}$ obey Leibniz identity if and only if
${{\hat{w}}^{\left( cl \right)}}1=0$ and ${{\hat{w}}^{\left( ri\right)}}1=0$,
respectively. \par In accordance with the G-dynamics ${{\hat{w}}^{\left( cl
\right)}}$, we investigate the unique eigenvalues equation ${{\hat{w}}^{\left(
cl \right)}}L\left( u,t,\lambda \right)=-\sqrt{-1}\lambda L\left( u,t,\lambda
\right)$ of the G-dynamics with a precise geometric eigenfunction $L\left(
u,t,\lambda \right)={{u}^{-1/2}}{{e}^{{{\lambda }}t}},~u>0$ as time $t\in
\left[ 0,T \right]$ develops and the equation of energy spectrum is then
induced. The non-Hermitian Hamiltonian operators are studied as well, we obtain
a series of ODE with their special solutions, and we prove multiplicative
property of the geometric eigenfunction. The coordinate derivative and time
evolution of the G-dynamics are respectively considered. Seeking the invariance
of G-dynamics ${{\hat{w}}^{\left( cl \right)}}$ under coordinate transformation
is considered, so that we think of one-dimensional G-dynamics
${{\hat{w}}^{\left( cl \right)}}$ on coordinate transformation, it gives the
rule of conversion between two coordinate systems. As a application, some
examples are given for such rule of conversion. Meanwhile, we search the
conditions that quantum geometric bracket vanishes and a specific case follows.
