Mathematical formulation for the numerical solution of the internal ballistics of classical weapons in Lagrangian coordinates

Today, most design processes are carried out using specialized tools in the form of advanced computer software, but in the case of designing weapons (especially internal ballistic) it is not generally available or is even classified. On the other hand, there is no analytical system of non-linear differential equations describing the phases of the shot phenomenon in barrel weapons. Therefore, in order to obtain more accurate solutions to the system of equations, it is optimal to use numerical methods. The paper presents a theoretical description of the problem of solving the equations of internal ballistics. Based on this analysis, it was assumed that the solution to the main problem of internal ballistics formulated in Lagrange coordinates gives correct results and enables obtaining more accurate and realistic analytical solutions than using classical solutions. The utilitarian aim was to create a new program to support the design of classic gun tube weapons.