One-dimensional optimal system for 2D Rotating Ideal Gas

TL;DR

This paper derives the one-dimensional optimal system for a 2D rotating ideal gas described by PDEs, analyzing Lie symmetries for rotating and nonrotating cases, and reducing the system to solvable ODEs.

Contribution

It determines the Lie symmetries and optimal systems for rotating and nonrotating ideal gas equations, highlighting differences in algebraic structure.

Findings

Nonrotating system admits 8 Lie point symmetries.

Rotating system admits 7 Lie point symmetries.

The systems are not algebraically equivalent for  > 2.

Abstract

We derive the one-dimensional optimal system for a system of three partial differential equations which describe the two-dimensional rotating ideal gas with polytropic parameter $\gamma >2.$ The Lie symmetries and the one-dimensional optimal system are determined for the nonrotating and rotating systems. We compare the results and we found that when there is no Coriolis force the system admits eight Lie point symmetries, while the rotating system admits seven Lie point symmetries. Consequently the two systems are not algebraic equivalent as in the case of $\gamma =2~$ which was found by previous studies. For the one-dimensional optimal system we determine all the Lie invariants, while we demonstrate our results by reducing the system of partial differential equations into a system of first-order ordinary differential equations which can be solved by quadratures.