## 2 Formation of Mathematical Physics Equation Sets
### Determination of Deformation Function of Collision Interface  
From Figure 1a, it can be known that the relationship between the _O-xyz_ coordinate system and the \(O^{\prime}\)-\(x^{\prime}y^{\prime}z^{\prime}\) coordinate system is:  
\[\left\{\begin{array}{c}x^{\prime}=x\\ y^{\prime}=y\\ z^{\prime}=z-R\end{array}\right. \tag{1}\]  
Let the vertical displacement of the sphere at any point \(A\) on the collision contact surface relative to the plane the contact boundary points forming after the collision be \(u\) and the vertical displacement of the boundary point be _\(\delta\)_, then according to the geometric relationship between \(u\) and \(u\)0 shown in Figure 1b, and equations can be obtained:  
\[x^{2}+y^{2}+(R-u_{0}+u)^{2}=R^{2} \tag{2}\]  
\[x^{\prime 2}+y^{\prime 2}+(R-u_{0}+u)^{2}=R^{2} \tag{3}\]  
From Equation (2) that when \(u\) = 0, it is the maximum boundary of the contact between the small sphere and the elastic half-space, and the value of its radius \(r\)0 (see Figure 1b) is:  
\[r_{0}=\sqrt{x^{2}+y^{2}}=\sqrt{2 Ru_{0}-u_{0}^{2}} \tag{4}\]  
From Equations (1) and (2), in the _O-xyz_ coordinate system, the expression of the vertical displacement of any point on the contact surface (_z_ = _R_) of the collision interface relative to the collision center is:  
\[u(x,y)=\left[u_{0}-(R-\sqrt{(R^{2}-x^{2}-y^{2})})\right] \tag{5}\]  
### Determination of Deformation Function of Collision Interface  
#### 2.3.1 Time Dependency of \(u\)0, \(\delta\), \(\omega_{0}\)  
Let \(r\) = \(\sqrt{x^{2}+y^{2}}\), it can be proven that:  
\[\omega=\omega(r)=R-\sqrt{R^{2}-r^{2}} \tag{6}\]  
Taking the sphere as the research object, let \(\omega_{0}=R-\sqrt{R^{2}-{r_{0}}^{2}}\), \(r\)0 is the maximum half width of contact, and \(u\)0 = \(\omega_{0}\), then:  
\[(R-\omega_{0})^{2}+{r_{0}}^{2}=R^{2} \tag{7}\]  
After a time \(t\), motion characteristics in the center of mass of the sphere and collision center increase _\(\Delta u_{0}\)_, \(\Delta\delta\), and \(\Delta\omega_{0}\) respectively, and equations can be obtained:  
\[\Delta\omega_{0}=\Delta u_{0}=\frac{r_{0}\Delta r_{0}}{\sqrt{R^{2}-{r_{0}}^{2}}} \tag{8}\]  
Moreover, the deformation equation in the _z_O_r coordinate system can be written as:  
\[(Z-u_{0}-\delta)^{2}+r^{2}=R^{2} \tag{9}\]  
Then:  
\[(Z-u_{0}-\delta)\Delta u_{0}=r\Delta r \tag{10}\]  
When \(Z\) = \(R\) + \(\delta\) and \(r\) = \(r\)0, the point (_z_, \(r\)0) is the contact point between the sphere and interface, substitute it into Equations (10)-(12), and we can derive:  
\[(R-u_{0})\Delta u_{0}=r_{0}\Delta r_{0} \tag{11}\]\[\Delta u_{0}=\frac{r_{0}}{R-u_{0}}\Delta r_{0}=\frac{r_{0}}{\sqrt{R^{2}-r_{0}^{2}}} \Delta r_{0} \tag{12}\]  
From Equation (9), it can be derived that:  
\[(Z-u_{0}-\delta)\Delta\delta=r\Delta r \tag{13}\]  
When \(Z=R+\delta\), it reaches the maximum contact boundary \(r=r_{0}\), it can be rewritten as:  
\[(R-u_{0})\Delta\delta=r_{0}\Delta r_{0} \tag{14}\]  
Deduced further and we can obtain:  
\[\Delta\delta=\frac{r_{0}}{R-u_{0}}\Delta r_{0}=\frac{r_{0}}{\sqrt{R^{2}-r_{0} ^{2}}}\Delta r_{0} \tag{15}\]  
From the derivation above, after any micro time \(\Delta t\), \(\Delta u_{0}=\Delta\delta=\Delta\omega_{0}\) can always be derived, also be written as:  
\[\frac{\Delta\omega_{0}}{\Delta t}=\frac{\Delta\delta}{\Delta t}\Rightarrow \lim_{\Delta t\to 0}\frac{\Delta\omega_{0}}{\Delta t}=\lim_{\Delta t\to 0}\frac{\Delta \delta}{\Delta t}\Rightarrow\frac{\Delta\omega_{0}}{\Delta t}-\frac{\Delta \delta}{\delta t}=0\Rightarrow\frac{\Delta(u_{0}-\delta)}{\Delta t}=0 \Rightarrow\omega_{0}-\delta=a \tag{16}\]  
where \(a\) is a constant.  
When \(t=0\), \(\omega_{0}=u_{0}=\delta\)= 0, and then \(a=0\); it can be proven that:  
\[\omega_{0}=u_{0}=\delta \tag{17}\]  
It also means that at any time point \(\delta=u_{0}=\omega_{0}\), as shown in Figure 2.  
Figure 2: The relative relationship of displacement (deformation) increment of rigid sphere motion, state in contact boundary.