## 2 Formation of Mathematical Physics Equation Sets
### Determination of Deformation Function of Collision Interface
#### 2.3.2 Determination of the Deformation Curve in Non-Contact Area  
(i) Boundary point curve equation of contact area of collision process interface  
The sphere moves from \(O_{1}\) to \(O_{2}\) in micro time \(\Delta t\), the boundary point in the contact area of a sphere and half-space are \(B\), \(C^{\prime}\), in the coordinate system \(\omega_{0}B_{0}r_{0}\); the sphere's motion(coordinate) state parameters are \(u_{0}\), \(\delta\), \(\omega_{0}\), \(r_{0}\); the delta state parameters are \(\Delta u_{0}\), \(\Delta\delta\), \(\Delta\omega_{0}\), \(\Delta r_{0}\)--see Figure 3.  
Let the curve equation of sphere curve be \(\omega_{0}=\omega_{0}(r_{0})\), the boundary points \(B\) and \(C^{\prime}\) lies on curve \(\widehat{BC^{\prime}}\), in the coordinate system \(\omega Br\), let \(\widehat{BC^{\prime}}\) equation be \(\omega_{1}=\omega_{1}(r)\). In order to obtain the curve at any moment in the non-contact area, we first assume that it remains unchanged. After obtaining its equation, the rationality and uniqueness of the assumed curve will be proven according to the physical and mathematical logic. Based on the assumption, we have \(B_{-1}\widehat{C}_{-1}\cong C^{\prime}\widehat{C}_{-1}\cong B_{-1}^{-}B\), and let the equation of the curve be \(\omega_{2}=\omega_{2}(r)\), then:  
\[\left\{ \begin{array}{l}\omega_{0}=\omega_{0}(r_{0})\\ \omega_{1}=\omega_{1}(r)\\ \omega_{2}=\omega_{2}(r)\end{array} \right. \tag{18}\]  
According to the known condition: the sphere's equation of the sphere determines the contact boundary point should satisfy Equation (7); we find that:  
\[\Delta\omega_{0}=\frac{r_{0}\Delta r_{0}}{\sqrt{R^{2}-{r_{0}}^{2}}} \tag{19}\]  
Because the point \(C^{\prime}\) is also the contact boundary point, it satisfies \(\Delta\delta=\Delta\omega_{0}=\Delta u_{0}\) (shown in the previous context), then:  
\[\Delta\omega_{1}+\Delta\omega_{0}=0 \tag{20}\]  
Figure 3: The deformation curve of collision interface in non-contact area. This can also be written as:  
\[\Delta\omega_{1}=-\Delta\omega_{0}=\frac{-r_{0}\Delta r_{0}}{\sqrt{R^{2}-{r_{0}}^{ 2}}}=\frac{d\omega_{1}(r)}{dr}\Delta r \tag{21}\]  
In the coordinate system \(\omega Br^{\prime}\), \(\Delta r_{0}=\Delta r\), then:  
\[\frac{d\omega_{1}(r)}{dr}=\frac{-r_{0}}{\sqrt{R^{2}-{r_{0}}^{2}}}\Rightarrow \Delta\omega_{1}=-\frac{r_{0}\Delta r_{0}}{\sqrt{R^{2}-{r_{0}}^{2}}}=\frac{-r _{0}\Delta r_{0}}{\sqrt{R^{2}-{r_{0}}^{2}}} \tag{22}\]  
According to the initial condition, when \(r_{0}=0\), it can be derived that: \(\omega_{0}=\omega_{1}=0\). The equation of curve \(\widetilde{BC^{\prime}}\) in the coordinate system \(\omega_{0}B_{0}r_{0}\) can be expressed as:  
\[\left(R+\omega_{1}\right)^{2}+{r_{0}}^{2}=R^{2} \tag{23}\]  
For any boundary point, Equation (24) can be satisfied:  
\[\left\{ \begin{array}{c}\left(R+\omega_{1}\right)^{2}+{r_{0}}^{2}=R^{2}\\ \left(R-\omega_{0}\right)^{2}+{r_{0}}^{2}=R^{2}\end{array} \right.\Rightarrow \omega_{1}=-\omega_{0} \tag{24}\]  
The result shows that the curve \(\omega_{1}\) is decided by \(\left(R+\omega_{1}\right)^{2}+{r_{0}}^{2}=R^{2}\), and is symmetrical with \(\left(R-\omega_{0}\right)^{2}+{r_{0}}^{2}=R^{2}\) with \(\omega_{0}(\delta)\) as the symmetry axis. (ii) Deformation Curve Equation of Interface's Non-contact Area  
For any moment, the non-contact area curve \(\omega_{2}\), to satisfy the continuity of deformation assumption, must be tangent to the small sphere's curve \(\omega_{0}\) at point B, i.e., \(d\omega_{0}/dr_{0}=d\omega_{2}/dr\), which can be written as:  
\[\omega_{0}^{\prime}(r_{0})=\omega_{2}^{\prime}(r)|_{r=0} \tag{25}\]  
The maximum contact boundary point from \(B\) to \(C^{\prime}\), on the curve \(\omega_{2}\) from point \(B_{-1}\) to point \(B\), it can be proven that:  
\[\left\{ \begin{array}{c}\Delta\omega_{2}=-\left.\Delta\omega_{2}^{\prime}(r) \right|_{r=0}\Delta r=-\Delta\delta\\ \Delta\delta=\left.\omega_{0}^{\prime}(r)\right|_{r=0}\Delta r_{0}=\Delta \delta\end{array} \right.