First order systems of ordinary differential equations having a given four-dimensional Lie group as their symmetry group

Abstract

We determine systems of the first order ordinary differential equations such that their group of symmetries contains the four-dimensional filiform Lie subgroup. The solutions of the obtained systems of first order ordinary differential equations are computed using the infinitesimal generators of their symmetries. We also determine the equivalent second order systems and compare them with the equation of geodesics in an arbitrary coordinate frame of a Riemannian space it turns out that the second order systems describe motions of a particle moving in a fluid. The motion is caused by a viscous force depending only on the velocity. We also investigate Kundt structures in the four-dimensional Lie algebra g_4.1. We determine the corresponding three-dimensional abelian and non-abelian subalgebras of the Lie algebra g_4.1. We investigate the left invariant Lorentz metrics corresponding to these subalgebras and find out which pairs of the Lie algebras and their corresponding left invariant Lorentz metrics are of Kundt type.