On Symmetry Group Extension and Vorticity Conservation in Continuum Mechanics

A.Golubiatnikov (Faculty of Mechanics and Mathematics,Moscow State University, Moscow, Russia)

e-mail: golubiat@mech.math.msu.su

Examples of many Lagrange's hydrodynamic systems show that often in medium motion the conservation of the curl of certain covector takes place. Moreover this covector can be unequal to the medium velocity. The absence of initial vorticity of such covector leads to the local reduction of the number of sought function and can be used for construction of wide classes of "potential" solution.  It is shown that group nature of this conservation law corresponds to the invariance of the Lagrangian with respect to equiaffine (volume-preserving) transformation group of independent Lagrange variables. In general case the Lagrange function can be always reduced to such form by a special extension of sought function set, moreover without presence of unnecessary information.  As applications, the models of inhomogeneous incompressible fluid, adiabatic motion of gas with varying entropy, magnetic hydrodynamics, nonlinear theory of elasticity and some other media are considered. Some exact solutions, having such potentiality properties, are also presented.  This work is supported by the RFBR (grant 02-01-00613).


Section : 12