Article outline

Scientific research on the gas dynamics of particles whose velocities are linear functions of spatial coordinates is closely related to the study of the dynamics of liquid and gas self-gravitating ellipsoids. The relevance of this issue is due to its importance for cosmogony and astrophysics, in particular, the importance of the results obtained for the theory of the shapes of celestial bodies. These studies are based on a long history of studying static equilibrium figures, dating back to Newton's ”Beginnings“ and related to scientific interest in the shape of the Earth. A large number of scientists have conducted research on this topic, pointing out the practical and scientific significance of this type of solutions.

It is known that 11 submodels have been obtained for the equations of ideal gas dynamics with a linear velocity field solution. Submodels are systems of ordinary differential equations for unknown functions included in the definition of a linear velocity field. The submodels have a high order, so it is not possible to integrate them completely. It is possible to find exact solutions, but only under certain assumptions related to the specific shape of the linear velocity field. In this regard, the question arose, how can these submodels be simplified? It is a well-known fact that the equations of ideal gas dynamics with a general equation of state admit 11 equivalence transformations forming the Galileo group. Namely, time shifts and in all three coordinates, rotations around the coordinate axes, Galilean shifts, uniform stretching in time and coordinates. This group corresponds to the 11-dimensional Lie algebra of first-order differentiation operators, constructed according to a well-known rule using the Lie equation. Thus, the group analysis apparatus will be used to simplify the system of high-order ordinary differential equations. The mathematical formula defining the linear velocity field consists of a matrix (linearity matrix) and a vector (vector of inhomogeneity). Therefore, the question remains how exactly the matrix and vector defining the linear velocity field are transformed. Moreover, since the classification of submodels took place according to the rank of a special symmetric matrix, which is defined through the linearity matrix, it is necessary to know the transformations corresponding to the 11 known ones. This article solves this problem for one of the submodels, the case with zero angular velocity. In the future, it is planned to transfer this method to other, more complex submodels.

The submodel in question is written in Eulerian and Lagrangian variables. Therefore, equivalence transformations for the Eulerian and Lagrangian descriptions are written in the paper. For each of the five main types of transformations (time shifts and in all three coordinates, rotations around the coordinate axes, Galilean shifts, and uniform stretching in time and coordinates), laws for the transformation of matrices and vectors included in the equations of the submodel, as well as matrices and vectors in the Lagrangian description, are obtained. The first-order differentiation operators of the corresponding Lie algebra are written using the obtained transformations.

After switching from a submodel in Eulerian coordinates to a submodel in lagrange coordinates, the submodel is integrated once. Constant integrations arise, namely the constant matrix and vector. The resulting equivalence transformations are applied to these constant matrices and vectors in order to simplify them. For example, a constant integration matrix is symmetric, and if a rotation transformation is applied to it that is not performed by special orthogonal matrices, it becomes symmetric. This means that the number of integration constants has decreased. And after applying the stretching transformation to a constant vector, one of its elements can be taken as a unit.