Journal
LETTERS IN MATHEMATICAL PHYSICS
Volume 107, Issue 9, Pages 1741-1768Publisher
SPRINGER
DOI: 10.1007/s11005-017-0962-y
Keywords
Liouville integrability; Integrals of motion; Nonlinear ODEs
Categories
Funding
- RFBI [16-01-00289]
- RFBR [15-52-05022 Arm-a, 16-52-12012-NNIO-a]
- DFG [Le-838/12-2]
- Heisenberg-Landau program
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The problem of classification of the Einstein-Friedman cosmological Hamiltonians H with a single scalar inflaton field phi, 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(phi). 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.
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