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Invariant Algebraic Surfaces for Generalized Raychaudhuri Equations

  • Claudia VallsEmail author
Article

Abstract

We consider a generalized Raychaudhuri equation,
$$\begin{array}{llll} \dot x = -\frac 1 2 x^2 -\alpha x -2(y^2 +z^2 -w^2)-2 \beta,\\ \dot y = -(\alpha +x) y -\gamma,\\ \dot z = -(\alpha +x) z -\delta,\\ \dot w = -(\alpha +x) w, \end{array}$$
where α, β, γ, δ are real parameters. This model has appeared in modern string cosmology. We study the algebraic invariants of this model for all values of the parameters \({\alpha,\beta,\gamma,\delta\in \mathbb{R}}\) . We prove that when γ = δ = 0 the system is integrable and for any other values of the parameters γ, δ, α, β we characterize all the invariant surfaces of this system. In particular we characterize all the polynomial and proper rational first integrals.

Keywords

Arbitrary Function Homogeneous Polynomial Polynomial Vector Raychaudhuri Equation Invariant Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Departamento de MatemáticaInstituto Superior Técnico, Universidade Técnica de LisboaLisboaPortugal

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