Symmetries of fluid dynamics with polytropic exponent
Résumé
The symmetries of the general Euler equations of fluid dynamics with polytropic exponent are determined using the Kaluza-Klein type framework of Duval et $\\it{al}$. In the standard polytropic case the recent results of O\'Raifeartaigh and Sreedhar are confirmed and generalized. Similar results are proved for polytropic exponent $\\gamma=-1$, which corresponds to the dimensional reduction of $d$-branes. The relation between the duality transformation used in describing supernova explosion and Cosmology is explained.