Symmetries and Related Physical Balances for Discontinuous Flow Phenomena within the Framework of Lagrange Formalism

By rigorous analysis, it is proven that from discontinuous Lagrangians, which are invariant with respect to the Galilean group, Rankine–Hugoniot conditions for propagating discontinuities can be derived via a straight forward procedure that can be considered an extension of Noether’s theorem. The use of this general procedure is demonstrated in particular for a Lagrangian for viscous flow, reproducing the well known Rankine–Hugoniot conditions for shock waves.

New equation of nonrelativistic physics and theory of dark matter

Two infinite sets of Galilean invariant equations are derived using the irreducible representations of the orthochronous extended Galilean group. It is shown that one set contains the Schrödinger equation, which is the fundamental equation for ordinary matter, and the other set has a new asymmetric equation, which is proposed to be the fundamental equation for dark matter. Using this new equation, a theory of dark matter is developed and its profound physical implications are discussed. This theory explains the currently known properties of dark matter and also predicts a detectable gravitational radiation.

The chain of embedded invariant submodels for conic motions

The equations of continuum mechanics are invariant in relation to the Galilean group generalized by extention. Its 11-dimensional Lie algebra has many subalgebras, which form the optimal system of dissimilar subalgebras. Subalgebras from the optimal system form the graph of embedded subalgebras. There are many chains of subalgebras in the graph. We consider the chain of embedded subalgebras containing operators of space and time translation, the rotation and uniform extension of all independent variables for the models of the continuous medium mechanics. We choose concordant invariants for each subalgebra from the chain. The chain of invariant submodels is constructed in a cylindrical coordinates based on chosen invariants. It is proved that solutions of a submodel constructed on a subalgebra of higher dimension will be part of solutions of submodels constructed on subalgebra of smaller dimensions for the considered chain. Thus, the chain of embedded invariant submodels is constructed by the example of equations of ideal gas dynamics.

Galilean symmetry in generalized Abelian Schrödinger–Higgs models with and without gauge field interaction

We consider a generalization of non-relativistic Schrödinger–Higgs Lagrangian by introducing a nonstandard kinetic term. We show that this model is Galilean invariant, we construct the conserved charges associated to the symmetries and realize the algebra of the Galilean group. In addition, we study the model in the presence of a gauge field. We also show that the gauged model is Galilean invariant. Finally, we explore relations between the twin models and their solutions.