coordinate $\alpha $ oscillates with $\Omega $ frequency, the Inglis mass $M$ is equal to $mAR_{0}^{2}$, and the nucleon mass: $m=938\ \rm{MeV}$.

For the time being we are not interested in studying the nucleon dynamics beyond the Woods-Saxon barrier:

$$V\left( r_j,R\left( \theta _j\right) \right) =\frac{V_0}{1+\exp \left[ \frac{r_j-R\left( \theta _j,\alpha \right) }a\right] },$$

and therefore we choose a deep well: $V_{0}=-1500\ \rm{MeV}$ and accordingly, a low value for the diffusivity coefficient: $a=0.01\ \rm{fm}$.

When considering the two-dimensional case, the frontier of the collective motion is described as a function of the collective variable $\alpha $ and of the Legendre polynomials $P_{L}\left( \cos \theta_{j}\right)$ [@burgio-95; @baldo-96; @baldo-98; @felea-09a]:

$$R_j = R\left( \theta _j,\alpha \right) =R_0\left[ 1+\alpha P_L\left( \cos \theta_j\right) \right].$$

The oscillation degree of the potential well $L$ is considered for the monopole $\left(0\right)$, dipole $\left(1\right)$ and quadrupole case $\left(2\right)$.

If the surface has a stationary behaviour, or whenever one takes into account the uncoupled Hamilton equations ([UCE]{}) for the particle:

$$\stackrel{\cdot }{r_j}=\frac{p_{r_j}}m,\ 
\stackrel{\cdot }{\theta _j}=\frac{p_{\theta _j}}{mr_j^2},\ 
\stackrel{\cdot }{p_{r_j}}=\frac{p_{\theta _j}^2}{mr_j^3}-\frac{\partial V}{\partial r_j},\ 
\stackrel{\cdot }{p_{\theta _j}}=-\frac{\partial V}{\partial R_j}\cdot \frac{\partial R_j}{\partial \theta _j},$$

and collective degrees of freedom (*d.o.f.*):

$$\stackrel{\cdot }{\alpha }=\frac{p_\alpha }M,\ 
\stackrel{\cdot }{p_\alpha}=-M\Omega ^2\alpha -\sum_{j=1}^{A}\left(\frac{\partial V}{\partial R_j} \cdot \frac{\partial R_j}{\partial \alpha }\right),$$

