Projective transformations and symmetries of differential equation

Aminova, A. V.

doi:10.1070/sm1995v186n12abeh000090

Cited by 56 publications

(78 citation statements)

References 12 publications

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“…Retaining terms of first-order in and neglecting O( 2 ), the first approximate symmetry generators are given by (13). In the second approximation, that is when we retain terms quadratic in , this equation possesses no non-trivial second-order approximate symmetry generators, but the first-order approximate symmetry generators are still retained.…”

Section: Symmetries and Approximate Symmetries Of The Schwarzschild Mmentioning

confidence: 99%

“…Hence, for our purposes, Lie symmetries as embodied in isometries and in Noether's theorem, should provide us the desired approach to define "approximate symmetries". Now, there is a connection between isometries and the symmetries of differential equations (DEs) through the geodesic equations [13,14]. We propose that the Baikov-GazizovIbragimov concept of "approximate symmetry" [15,16] could be extended and adapted for the purpose of defining energy in GR via "approximate isometries" through the above-mentioned connection between the symmetries of geometry and the symmetries of differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Second-Order Approximate Symmetries of the Geodesic Equations for the Reissner-Nordström Metric and Re-Scaling of Energy of a Test Particle

Hussain¹

2007

SIGMA

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Abstract. Following the use of approximate symmetries for the Schwarzschild spacetime by A.H. Kara, F.M. Mahomed and A. Qadir (Nonlinear Dynam., to appear), we have investigated the exact and approximate symmetries of the system of geodesic equations for the Reissner-Nordström spacetime (RN). For this purpose we are forced to use second order approximate symmetries. It is shown that in the second-order approximation, energy must be rescaled for the RN metric. The implications of this rescaling are discussed.

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Section: Symmetries and Approximate Symmetries Of The Schwarzschild Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Second-Order Approximate Symmetries of the Geodesic Equations for the Reissner-Nordström Metric and Re-Scaling of Energy of a Test Particle

Hussain¹

2007

SIGMA

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“…The above matters have been discussed extensively in a series of interesting papers by Aminova [5], [2], [3], [4] who has given an answer. Furthermore in a recent work [6] they have considered the KVs of the metric and their relation to the Lie symmetries of the system of affinely parameterized geodesics of maximally symmetric spaces of low dimension.…”

Section: Motivationmentioning

confidence: 99%

Lie symmetries of geodesic equations and projective collineations

Tsamparlis

Paliathanasis

2010

Nonlinear Dyn

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“…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]). …”

Section: Introductionmentioning

confidence: 99%

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

2011

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We perform a detailed study of the modified gravity f (R) models in the light of the basic geometrical symmetries, namely Lie and Noether point symmetries, which serve to illustrate the phenomenological viability of the modified gravity paradigm as a serious alternative to the traditional scalar field approaches. In particular, we utilize a model-independent selection rule based on first integrals, due to Noether symmetries of the equations of motion, in order to identify the viability of f (R) models in the context of flat FLRW cosmologies. The Lie/Noether point symmetries are computed for six modified gravity models that include also a cold dark matter component. As it is expected, we confirm that all the proposed modified gravity models admit the trivial first integral namely energy conservation. We find that only the f (R) = (R b − 2Λ) c model, which generalizes the concordance Λ cosmology, accommodates extra Lie/Noether point symmetries. For this f (R) model the existence of non-trivial Noether (first) integrals can be used to determine the integrability of the model. Indeed within this context we solve the problem analytically and thus we provide for the first time the evolution of the main cosmological functions such as the scale factor of the universe and the Hubble expansion rate.PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x

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Projective transformations and symmetries of differential equation

Cited by 56 publications

References 12 publications

Second-Order Approximate Symmetries of the Geodesic Equations for the Reissner-Nordström Metric and Re-Scaling of Energy of a Test Particle

Second-Order Approximate Symmetries of the Geodesic Equations for the Reissner-Nordström Metric and Re-Scaling of Energy of a Test Particle

Lie symmetries of geodesic equations and projective collineations

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

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