# Integrable cosmological potentials

Integrable cosmological potentials The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field $$\varphi$$ φ , which possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint $$H=0$$ H = 0 , is considered. Necessary and sufficient conditions for the existence of the first-, second- and third-degree integrals are derived. These conditions have the form of ODEs for the cosmological potential $$V(\varphi )$$ V ( φ ) . In the case of linear and quadratic integrals we find general solutions of the ODEs and construct the corresponding integrals explicitly. A new wide class of Hamiltonians that possess a cubic integral is derived. The corresponding potentials are represented in parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described, and sporadic superintegrable cases are discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Letters in Mathematical Physics Springer Journals

# Integrable cosmological potentials

, Volume 107 (9) – May 8, 2017
28 pages

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Publisher
Springer Netherlands
Copyright
Copyright © 2017 by Springer Science+Business Media Dordrecht
Subject
Physics; Theoretical, Mathematical and Computational Physics; Complex Systems; Geometry; Group Theory and Generalizations
ISSN
0377-9017
eISSN
1573-0530
D.O.I.
10.1007/s11005-017-0962-y
Publisher site
See Article on Publisher Site

### Abstract

The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field $$\varphi$$ φ , which possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint $$H=0$$ H = 0 , is considered. Necessary and sufficient conditions for the existence of the first-, second- and third-degree integrals are derived. These conditions have the form of ODEs for the cosmological potential $$V(\varphi )$$ V ( φ ) . In the case of linear and quadratic integrals we find general solutions of the ODEs and construct the corresponding integrals explicitly. A new wide class of Hamiltonians that possess a cubic integral is derived. The corresponding potentials are represented in parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described, and sporadic superintegrable cases are discussed.

### Journal

Letters in Mathematical PhysicsSpringer Journals

Published: May 8, 2017

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