Conservation laws and the equivalence group

Abstract

Abstract The paper generalizes differential conservation laws for the two-dimensional hydrodynamic Euler equations and their equivalent conservation laws for families of streamlines previously derived by the author. Three-dimensional analogues of these results, i.e., differential conservation laws for the three-dimensional hydrodynamic Euler equations, are presented. All conservation laws are divergent identities of the form div F = 0 and are found in two equivalent forms: in “mathematical-physical” form, where the vector field F is expressed in terms of quantities included in the Euler equations and their partial derivatives, and in geometric form. The second form represents conservation laws for families of streamlines where the vector field F under the divergence sign is expressed in terms of the classical geometric characteristics of curves — their Frenet unit vectors τ, ν, and β (the unit tangent, principal normal, and binormal vectors), the first curvature k, and the second curvature ϰ. Special attention is paid to the geometric interpretation of the obtained conservation laws and their constituent expressions. Issues related to the equivalence group of the eikonal equation and other equations are discussed. All results were obtained using the general vector and geometric formulas (differential conservation laws and other formulas), obtained by the author for families of arbitrary smooth curves, families of arbitrary smooth surfaces, and arbitrary smooth vector fields.