Two-dimensional dynamical systems which admit Lie and Noether symmetries

Abstract

We consider a dynamical system moving in a Riemannian space and prove two theorems which relate the Lie point symmetries and Noether symmetries of the equation of motion, with the special projective group and the homothetic group of the space respectively. These theorems are used to classify the two-dimensional Newtonian dynamical systems, which admit Lie point/Noether symmetries. The results of the study, i.e. expressions of forces/potentials, Lie symmetries, Noether vectors and Noether integrals are presented in the form of tables for easy reference and convenience. Two cases are considered, Hamiltonian and non-Hamiltonian systems. The results are used to determine the Lie/Noether symmetries of two different systems. The Kepler–Ermakov system, which in general is non-conservative, and the conservative system with potential similar to the Hènon–Heiles potential. As an additional application, we consider the scalar field cosmologies in a FRW background with no matter, and look for the scalar field potentials for which the resulting cosmological models are integrable. It is found that the only integrable scalar field cosmologies are defined by the exponential and the unified dark matter potential. It is to be noted that in all aforementioned applications the Lie/Noether symmetry vectors are found by simply reading the appropriate entry in the relevant tables.