Recently a cubic Galileon cosmological model was derived by the assumption that the field equations are invariant under the action of point transformations. The cubic Galileon model admits a second conservation law which means that the field equations form an integrable system. The analysis of the critical points for this integrable model is the main subject of this work. To perform the analysis, we work on dimensionless variables different from that of the Hubble normalization. New critical points are derived while the gravitational effects which follow from the cubic term are studied. 95.35.+d, 95.36.+x

A class of generalized Galileon cosmological models, which can be described by a point-like Lagrangian, is considered in order to utilize Noether's Theorem to determine conservation laws for the field equations. In the Friedmann-Lemaître-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Due to the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability. 95.35.+d, 95.36.+x

We investigate the wave equation in Bianchi type III space-time. We construct a Lagrangian of the model, calculate and classify the Noether symmetry generators, and construct corresponding conserved forms. A reduction of the underlying equations is performed to obtain invariant solutions.

Regular perturbative Lagrangians that admit approximate Noether symmetries and approximate conservation laws are studied. Specifically, we investigate the connection between approximate Noether symmetries and collineations of the underlying manifold. In particular we determine the generic Noether symmetry conditions for the approximate point symmetries and we find that for a class of perturbed Lagrangians, Noether symmetries are related to the elements of the Homothetic algebra of the metric which is defined by the unperturbed Lagrangian. Moreover, we discuss how exact symmetries become approximate symmetries. Finally, some applications are presented.which describes a dynamical system changes under the action of a point transformation such that the Action integral is invariant, the dynamical system is also invariant under the action of the same point transformation. Moreover, a conservation law corresponds to this point transformation according to Noether's second theorem.Usually, when we refer to symmetries, we consider the exact symmetries. However, for perturbative dynamical systems the context of symmetries is extended and the so-called approximate symmetries are defined [6,7,8,9,10,11,12,13]. In this work we are interested in the application of Noether's theorem for approximate symmetries on some regular perturbative Lagrangians. Approximate Noether symmetries [14,15] provide approximate first integrals, functions which can be used as conservation laws until a specific step in numerical integrations. This kind of approximate conservation laws have played an important role for the study of chaotic systems -for an extended discussion we refer the reader to an application in Galactic dynamics [16,17,18].Whilst recently, the advent of automated software algorithms has made light work of calculating symmetries [19]. Such programs are often limited by models involving many variables or higher-order perturbations. This problem, in part, has fueled the need to write this paper. Here, we take a compound problem, whereby scientists have previously relied on numerical techniques for analysis, and instead frame it in the context of an analytical scheme. We present a set of conditions that may be specialized for appropriate Lagrangian functions that necessarily contains a perturbation. Inspired by the approach of Tsamparlis and Paliathanasis [20,21,22], we show how those conditions can be solved with the use of some theorems from differential geometry. Indeed, geometric based theories have far reaching applications [23,24,25].Specifically, in this paper, the Noether conditions, or symmetry determining system of equations, are formulated by contemplating point transformations in ascending order of the perturbation parameter ε. To illustrate the advantages of such a formulation, the general conditions are applied to the perturbations of oscillator type equations corresponding to n dimensions. Moreover, we discuss the admitted approximate conserved quantities for symmetries of first-order up-to n th -order. This paper assum...

In this paper, we complement the analysis involving the “fundamental” solutions of the Klein–Gordon equation in de Sitter space–times given by Yagdjian and A. Galstian (Comm. Math. Phys. 285, 293 (2009); Discrete and Continuous Dynamical Systems S, 2(3), 483 (2009)). Using the symmetry generators, we classify and reduce the underlying equations and show how this process may lead to exact solutions by quadratures.