## 2 Mathematical Problem Statement
### The Boundary Conditions  
Each body in the Cartesian coordinate system \(\{x_{j}\}\) corresponds to a computational domain \(\mathrm{D}^{i}(\mathbf{x},t)\) with boundaries \(\mathrm{G}^{i}(\mathbf{y},t)\) (see Figure 1). Here, \(\mathbf{x}=\mathbf{x}(t)\) is the position vector of the material particle and \(\mathbf{y}\) are the boundary points. In the general case, the computational domains \(\mathrm{D}^{j}(\mathbf{x},t)\) change in time and can be multi-connected. The initial conditions for the \(i\)-th body at \(t=0\) in the region \(\mathrm{D}^{i}(\mathbf{x},t)\) are of the form:  
\[\rho^{i}(\mathbf{x},0)=\rho^{i0}(\mathbf{x}),\ u^{i}_{j}(\mathbf{x},0)=u^{i0}_{ j}(\mathbf{x}),\ s_{ij}=P=e=0,\]  
where \(\rho^{i0}(\mathbf{x})\) and \(u^{i0}_{j}(\mathbf{x})\) are the given initial distributions of the material density and the velocity vector over the area \(\mathrm{D}^{i}(\mathbf{x},t)\). To formulate the boundary conditions, let us introduce the following notations:  
* \(\mathbf{n}^{i}\) the vector of the outward normal to the boundary \(\mathrm{G}^{i}(\mathbf{y},t)\) of the domain \(\mathrm{D}^{i}(\mathbf{x},t)\);
* \(F^{i}(\mathbf{y},t)\) the vector of external surface forces on the boundary \(\mathrm{G}^{i}(\mathbf{y},t)\);
* \(W^{i}(\mathbf{y},t)\) the vector of velocity at the boundary. \(\mathrm{G}^{i}(\mathbf{y},t)\). The following conditions can be specified on any boundary of the domain \(\mathrm{G}^{i}(\mathbf{y},t)\):  
* kinematic \[\mathbf{u}^{i}(\mathbf{y},t)=W^{i}(\mathbf{y},t),\]
* dynamic \[\sigma^{i}_{kl}(\mathbf{y},t)\ n^{i}_{l}=F^{i}_{k}(\mathbf{y},t),\ (k,l=1,\ 2,\ 3),\] where \(\sigma^{i}_{kl}\) are the components of the stress tensor on the boundary of the domain \(\mathrm{G}^{i}(\mathbf{y},t)\), which typically needs to be determined;
* mixed \[\mathbf{u}^{i}(\mathbf{y},t)=W^{i}(\mathbf{y},t),\ \mathbf{y}\in\mathrm{G}^{i \alpha}(\mathbf{y},t),\ \sigma^{i}_{kl}(\mathbf{y},t)n^{i}_{l}=F^{i}_{k}(\mathbf{y},t),\ \mathbf{y}\in\mathrm{G}^{i \beta}(\mathbf{y},t),\ \mathrm{G}^{i}(\mathbf{y},t)=\mathrm{G}^{i\alpha}(\mathbf{y},t)\cup \mathrm{G}^{i\beta}(\mathbf{y},t).\] The contact surface between two bodies \(\mathrm{G}^{ij}(\mathbf{z},t)\) is defined as the set of points \(\mathbf{z}\) that satisfy the condition. \[\mathrm{G}^{ij}(\mathbf{z},t)=\mathrm{G}^{i}(\mathbf{y},t)\cap\mathrm{G}^{j}( \mathbf{y},t).\] Certain compatibility conditions must be satisfied on the contact boundary between the bodies for the vectors of velocities \(\mathbf{u}^{i}(\mathbf{z},t)\) and \(\mathbf{u}^{j}(\mathbf{z},t)\), as well as the components of the stress tensor \(\sigma^{i}_{kl}\) and \(\sigma^{j}_{kl}\). Specific types of conditions on the contact boundaries will be stated below. To simplify the algorithm for calculating the motion of the boundaries, we will use the external surface force vectors \(F^{i}(\mathbf{y},t)\). The reaction forces, which are determined during the problem-solving process, will be denoted by \(\mathbf{R}^{i}(\mathbf{z},t)\). Figure 1: Boundary conditions. Let us assume that at each point \(\mathbf{z}\) on the contact surface \(\mathrm{G}^{ij}(\mathbf{z},t)\), there exists a common normal. In this case,  
\[\mathbf{n}^{i}=-\mathbf{n}^{j}.\]  
Let us decompose the vector \(\mathbf{A}(\mathbf{z},t)\) at point \(\mathbf{z}\) into its normal \(\mathbf{A}_{n}\) and tangential \(\mathbf{A}_{t}\) components  
\[\mathbf{A}=\mathbf{A}_{n}+\mathbf{A}_{t}. \tag{9}\]  
These components can be calculated using the formulas. \[\mathbf{A}_{n}=(\mathbf{A},\mathbf{n})\cdot\mathbf{n},\ \ \ \mathbf{A}_{t}= \mathbf{n}\times(\mathbf{n}\times\mathbf{A}).\]  
Let us replace the action of body \(i\) on body \(j\) at point \(\mathbf{z}\) with the reaction force vector \(\mathbf{R}^{i}(\mathbf{z},t)\), and, correspondingly, the action of body \(j\) on body \(i\) with the reaction force vector \(\mathbf{R}^{j}(\mathbf{z},t)\).