We find the Lie point symmetries for non–relativistic two-dimensional charged particle motion. These symmetries comprise a quasi–inva-riance transformation, a time–dependent rotation, a time–dependent spatial translation and a dilatation. The associated electromagnetic fields satisfy a system of first–order linear partial differential equations. This system is solved exactly, yielding four clas...

We present some remarks on the existence and the properties of Lie point symmetries of nite dimensional dynamical systems expressed either in Newton-Lagrange or in Hamilton form. We show that the only Lie symmetries admitted by Newton-Lagrange-type problems are essentially linear symmetries, and construct the most general problem admitting such a symmetry. In the case of Hamilton problems, we d...

A review of the role of symmetries in solving differential equations is presented. After showing some recent results on the application of classical Lie point symmetries to problems in fluid draining, meteorology, and epidemiology of AIDS, the nonclassical symmetries method is presented. Finally, it is shown that iterations of the nonclassical symmetries method yield new nonlinear equations, wh...

The different natures of approximate symmetries and their corresponding first integrals/invariants are delineated in the contexts of both Lie symmetries of ordinary differential equations and Noether symmetries of the Action Integral. Particular note is taken of the effect of taking higher orders of the perturbation parameter. Approximate symmetries of approximate first integrals/invariants and...

In this paper we establish the correspondence between Bogoyavlenskij symmetries [1, 2] of the MHD equilibrium equations and Lie point transformations of these equations. We show that certain non-trivial Lie point transformations (that are obtained by direct application of Lie method) are equivalent to Bogoyavlenskij symmetries. PACS Codes: 05.45.-a , 02.30.Jr, 02.90.+p, 52.30.Cv.

As we will later see, a crucial role in model building is played by symmetries. You are already familiar with symmetries and with some of their consequences. For example, Nature seems to have the symmetry of the Lorentz group which implies conservation of energy, momentum and angular momentum. In order to understand the interplay between symmetries and interactions, we need a mathematical tool ...

The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie–Bäcklund generator is extended to include any Lie–Bäcklund generator. Also, it is shown that the Lie algebra of Lie–Bäcklund symmetries of a conserved vector of a system is a subalgebra of the Lie–Bäcklund symmetries of the system. Moreover, we in...

In this article we consider nonholonomic deformations of disk solutions in general relativity to generic off–diagonal metrics defining knew classes of exact solutions in 4D and 5D gravity. These solutions possess Lie algebroid symmetries and local anisotropy and define certain generalizations of manifolds with Killing and/ or Lie algebra symmetries. For Lie algebroids, there are structures func...

We introduce a new class of symmetries, that strictly includes Lie symmetries, for which there exists an algorithm that lets us reduce the order of an ordinary differential equation. Many of the known order-reduction processes, that are not consequence of the existence of Lie symmetries, are a consequence of the invariance of the equation under vector fields of the new class. These vector field...