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 the 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 inFRW 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. 1 Of course it is possible to look for a metric for which a given set of Γ i jk are the connection coefficients, or, even avoid the metric altogether. However we shall not attempt this in the present work. For such an attempt see [7].

Constraints and analytical solutions off(R)theories of gravity using Noether symmetries

“…In the context of f (R) models, following the general methodology of [36] (see also the references therein), the Noether symmetries are computed for 6 modified gravity models that contain also a dark matter component. The main results of the current paper can be summarized in the following statements (see sections 3 and 4):…”

“…In particular, the scope of the current article is (a) to investigate which of the available f (R) models admit extra Lie and Noether point symmetries, and (b) for these models to solve the system of the resulting field equations and derive analytically (for the first time to our knowledge) the main cosmological functions (the scale factor, the Hubble expansion rate etc.). We would like to remind the reader that a fundamental approach to derive the Lie and Noether point symmetries for a given dynamical problem living in a Riemannian space has been published recently by Tsamparlis & Paliathanasis [36] (a similar analysis can be found in [49][50][51][52][53][54][55]). …”

Invariant solutions and Noether symmetries in hybrid gravity

“…Therefore, in order to search for special forms of the potential V (φ), where the Lagrangian admits Noether point symmetries, we will apply the geometric approach developed in [42]. Since the Lagrangian is time-independent, it admits the Noether symmetry ∂ t with the Hamiltonian as a conservation law, that is…”

“…This feature provides integrals of motions capable of reducing the related dynamical system and then getting exact solutions. For the determination of the Noether symmetries of the classical Lagrangian, we will apply the geometric procedure outlined in [42], where the Noether symmetries of the Lagrangian are connected to the collineations of the second order tensor which is defined by the kinematic part of the Lagrangian. Therefore the Noether symmetry is not only a criterion for the integrability of the system but also a geometric criterion since it allows to select the free functions of the theory.…”

N = 2 Integrable Systems and First Integrals Constrained by Scaling Symmetries

“…The existence of point symmetries in addition to ∂ ∂t heavily depends on the form of the Lagrangian, and some generic classification of systems admitting symmetries have been carried out (see e.g. [10] and references therein). Here we are only interested on scaling symmetries in connection with the first integrals.…”