**:** ..
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**:** 84
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**:** 2020
**:** .. // . 84. .: , 2020. .51-65. DOI: https://doi.org/10.25728/ubs.2020.84.3

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** (.):** ideal polytropic gas, motion with uniform deformation, equilibria.

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** (.):** We consider a two-dimensional system of equations of an ideal polytropic gas on a rotating plane, which arises in the dynamics of the atmosphere. The problem is very difficult in the general case, however, it admits solutions with a linear profile of velocity (corresponding to motion with uniform deformation), which can be found by solving a quadratically nonlinear system of ordinary differential equations. The system has two families of equilibria: the first family is one-parametric (corresponds to a vortex) and the second one is two-parametric (corresponds to a shift flow), the latter is always unstable. The stability of equilibria means the stability of stationary solutions of the original system in the class of perturbations with linear profile of velocity. The article considers a one-parametric family of equilibria, which corresponds to a stationary vortex motion, the parameter is responsible for the vortex intensity and changes over the real axis. Intervals of the parameter where equilibrium is unstable and where it is stable in the sense of Lyapunov were found earlier. However, they did not cover the entire real axis. For the remaining parameter values, the matrix of linearization has three pairs of purely imaginary complex conjugate eigenvalues, therefore the study of stability by conventional methods is difficult. We investigate the matter in Lagrangian coordinates. Estimates that provide intervals of guaranteed stability are constructed. The stability issue is completely resolved for a gas with one, two, and three degrees of freedom.

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