Integrable cosmological potentials
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The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field \(\varphi \), which possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint \(H=0\), 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 \(V(\varphi )\). 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.
KeywordsLiouville integrability Integrals of motion Nonlinear ODEs
Mathematics Subject Classification34A34 37J35 37K10 70H06
We thank P. Fré, S. O. Krivonos, D. Sternheimer and A. I. Zobnin for useful discussions. We are grateful to an anonymous referee for useful comments and references. V. S. thanks JINR (Dubna) for the hospitality extended during his visits. V. S. was supported by RFBI Grant No. 16-01-00289. The work of A. S. was partially supported by RFBR Grants No. 15-52-05022 Arm-a and No. 16-52-12012-NNIO-a, DFG Grant Le-838/12-2 and by the Heisenberg-Landau program.
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