Article outline

The differential equations of a motion of the ideal gas with varying entropy and the spatial thermodynamics are an overdetermined system. If values the thermodynamic parameters lie on a curved line or one of the parameters is a constant then the overdetermined system is the same for any state equation. In the plane case all solutions were obtained by two ways in the spatial Lagrange coordinates. The first way the system is led to involution when we obtain the finite number differential consequences no generated new. In the second way we produce an infinite overdetermined system of differential equations for auxiliary values that were obtained by integration of differential consequences of the initial system. Here we correct the second way obtaining of eight types of the exact solutions maximum depending on one essential arbitrary function of one argument and several constants. The classification was led to within infinite admitted group of transformations. The examples of the smooth movement of a gas particle for any time were given for each type of the solutions.

Objective. The invariant plane submodel of the ideal gas dynamics with the constant temperature are considered. In the spatial Lagrange coordinates the overdetermined system of differential equations leads to the linear vector system and one more nonlinear equation. The objective of the paper is obtaining differential consequences which may be integrated by the time and to represent them in the type of infinite overdetermined chain of differential equations for auxiliary functions. All representations of solutions for auxiliary functions must be obtained. The solutions of the initial system must be found for each representation.

Methods. In depending on the representation of the auxiliary functions we obtained the representation of the desired values in the type of linear combinations of independent expressions: polynomial by time, harmonic, linear growing harmonic, biharmonic, exponential growing harmonic. The decomposition of the basic equations by independent expressions gives the overdetermined system of equations.

Results. The overdetermined system of equations may be contradictory or generates eight types of exact solutions which may be depending from an arbitrary function of one argument and several constants. The motion of each particle may be represented though its initial position for any moment of time. The examples of the smooth motions for gas particles are given for each type of solutions. The trajectories maybe parabolas, curves with very able convexity, circles, curves oscillating alongside the straight line, linearly growing spirals, finite part of the power curve with the periodic vibrations on it, hyperbolas, exponentially growing spirals in the spatial distributed thermodynamic medium.

Conclusions. The plane gas dynamics with constant temperature was considered. In the spatial Lagrange coordinates the model were reduced to the linear vector system of equations and one nonlinear equation overdetermining the system. The differential consequences are presented in the type of infinite overdetermined sequence of differential equations for the auxiliary functions which generate eight types of exact solutions. The smooth motions for gas particles are given for any time.