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Integrable cosmological potentials
The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field φ , 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 e...
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| Published in: | Letters in mathematical physics 2017-09, Vol.107 (9), p.1741-1768 |
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| Main Authors: | , |
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| Language: | English |
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| container_end_page | 1768 |
| container_issue | 9 |
| container_start_page | 1741 |
| container_title | Letters in mathematical physics |
| container_volume | 107 |
| creator | Sokolov, V. V. Sorin, A. S. |
| description | The problem of classification of the Einstein–Friedman cosmological Hamiltonians
H
with a single scalar inflaton field
φ
, 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
(
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. |
| doi_str_mv | 10.1007/s11005-017-0962-y |
| format | article |
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H
with a single scalar inflaton field
φ
, 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
(
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.</description><identifier>ISSN: 0377-9017</identifier><identifier>EISSN: 1573-0530</identifier><identifier>DOI: 10.1007/s11005-017-0962-y</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Complex Systems ; Economic models ; Functions (mathematics) ; Geometry ; Group Theory and Generalizations ; Hamiltonian functions ; Integrals ; Legendre functions ; Mathematical analysis ; Mathematical and Computational Physics ; Physics ; Physics and Astronomy ; Theoretical</subject><ispartof>Letters in mathematical physics, 2017-09, Vol.107 (9), p.1741-1768</ispartof><rights>Springer Science+Business Media Dordrecht 2017</rights><rights>Copyright Springer Science & Business Media 2017</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-db8bec6e0661b4a13250fd14db00b38338a825411ae85dec44a2d2ba1fd91ceb3</citedby><cites>FETCH-LOGICAL-c316t-db8bec6e0661b4a13250fd14db00b38338a825411ae85dec44a2d2ba1fd91ceb3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids></links><search><creatorcontrib>Sokolov, V. V.</creatorcontrib><creatorcontrib>Sorin, A. S.</creatorcontrib><title>Integrable cosmological potentials</title><title>Letters in mathematical physics</title><addtitle>Lett Math Phys</addtitle><description>The problem of classification of the Einstein–Friedman cosmological Hamiltonians
H
with a single scalar inflaton field
φ
, 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
(
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.</description><subject>Complex Systems</subject><subject>Economic models</subject><subject>Functions (mathematics)</subject><subject>Geometry</subject><subject>Group Theory and Generalizations</subject><subject>Hamiltonian functions</subject><subject>Integrals</subject><subject>Legendre functions</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Theoretical</subject><issn>0377-9017</issn><issn>1573-0530</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kDtPwzAUhS0EEqHwA9gqmA332nEeI6qgVKrEArPlV6JUaRzsdMi_x1UYWJjOcM93ju4h5B7hCQHK54hJBAUsKdQFo_MFyVCUnILgcEky4GVJ63S-JjcxHiAxTEBGHnbD5NqgdO_Wxsej733bGdWvRz-5YepUH2_JVZPE3f3qiny9vX5u3un-Y7vbvOyp4VhM1OpKO1M4KArUuUKe8huLudUAmlecV6piIkdUrhLWmTxXzDKtsLE1Gqf5ijwuuWPw3ycXJ3nwpzCkSok1hxSYepILF5cJPsbgGjmG7qjCLBHkeQq5TCHTr_I8hZwTwxYmJu_QuvAn-V_oB2xZYNk</recordid><startdate>20170901</startdate><enddate>20170901</enddate><creator>Sokolov, V. V.</creator><creator>Sorin, A. S.</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20170901</creationdate><title>Integrable cosmological potentials</title><author>Sokolov, V. V. ; Sorin, A. S.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-db8bec6e0661b4a13250fd14db00b38338a825411ae85dec44a2d2ba1fd91ceb3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Complex Systems</topic><topic>Economic models</topic><topic>Functions (mathematics)</topic><topic>Geometry</topic><topic>Group Theory and Generalizations</topic><topic>Hamiltonian functions</topic><topic>Integrals</topic><topic>Legendre functions</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Sokolov, V. V.</creatorcontrib><creatorcontrib>Sorin, A. S.</creatorcontrib><collection>CrossRef</collection><jtitle>Letters in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Sokolov, V. V.</au><au>Sorin, A. S.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Integrable cosmological potentials</atitle><jtitle>Letters in mathematical physics</jtitle><stitle>Lett Math Phys</stitle><date>2017-09-01</date><risdate>2017</risdate><volume>107</volume><issue>9</issue><spage>1741</spage><epage>1768</epage><pages>1741-1768</pages><issn>0377-9017</issn><eissn>1573-0530</eissn><abstract>The problem of classification of the Einstein–Friedman cosmological Hamiltonians
H
with a single scalar inflaton field
φ
, 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
(
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.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s11005-017-0962-y</doi><tpages>28</tpages></addata></record> |
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| ispartof | Letters in mathematical physics, 2017-09, Vol.107 (9), p.1741-1768 |
| issn | 0377-9017 1573-0530 |
| language | eng |
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| subjects | Complex Systems Economic models Functions (mathematics) Geometry Group Theory and Generalizations Hamiltonian functions Integrals Legendre functions Mathematical analysis Mathematical and Computational Physics Physics Physics and Astronomy Theoretical |
| title | Integrable cosmological potentials |
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