{"id":7,"date":"2017-01-11T15:06:31","date_gmt":"2017-01-11T15:06:31","guid":{"rendered":"http:\/\/users.utu.fi\/hietarin\/?page_id=2"},"modified":"2017-01-17T13:45:55","modified_gmt":"2017-01-17T13:45:55","slug":"liouville","status":"publish","type":"page","link":"https:\/\/users.utu.fi\/hietarin\/home\/liouville\/","title":{"rendered":"Liouville integrability"},"content":{"rendered":"<p>&nbsp;<\/p>\n<h3>Liouville integrable classical systems<\/h3>\n<ul>\n<li>J. Hietarinta: <strong><em>Direct methods for the search of the second invariant<\/em>,<\/strong> <a href=\"http:\/\/dx.doi.org\/10.1016\/0370-1573(87)90089-5\"> Physics Reports <b>147<\/b>, 87-154 (1987). <\/a><\/li>\n<li>B. Dorizzi, B. Grammaticos, J. Hietarinta, A. Ramani and F. Schwarz: <em>New integrable three dimensional quartic potentials<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(86)90377-4\"> Phys. Lett. A <b>116<\/b>, 432-436 (1986).<\/a><\/li>\n<li>A. Ramani, B. Dorizzi, B. Grammaticos and J. Hietarinta: <em>Linearization on a submanifold of integrable Hamiltonians with polynomial potentials<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1016\/0167-2789(86)90174-0\">Physica D <b>18<\/b>, 171-179 (1986).<\/a><\/li>\n<li>J. Hietarinta: <em>How to construct integrable Fokker-Planck and electromagnetic Hamiltonians from ordinary integrable Hamiltonians<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1063\/1.526865\">J. Math. Phys. <b>26<\/b>, 1970-1975 (1985).<\/a><\/li>\n<li>B. Grammaticos, B. Dorizzi, A. Ramani and J. Hietarinta: <em>Extending integrable Hamiltonian systems from 2 to N dimensions<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(85)90259-2\">Phys. Lett. A <b>109<\/b>, 81-84 (1985).<\/a><\/li>\n<li>A. Ramani, J. Hietarinta, B. Dorizzi and B. Grammaticos: <em>Complete Painlev\u00e9 analysis for coupled quartic oscillators in N dimensions<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(85)90515-8\">Phys. Lett. A <b>108<\/b>, 55-58 (1985).<\/a><\/li>\n<li>J. Hietarinta, B. Grammaticos, B. Dorizzi, and A. Ramani: <strong><em>Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems<\/em><\/strong>,<a href=\"http:\/\/dx.doi.org\/10.1103\/PhysRevLett.53.1707\"> Phys. Rev. Lett. <b>53<\/b>, 1707-1710 (1984).<\/a><\/li>\n<li>J. Hietarinta: <strong><em>New integrable Hamiltonians with transcendental invariants<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1103\/PhysRevLett.52.1057\"> Phys. Rev. Lett. 52, 1057-1060 (1984).<\/a><\/strong><\/li>\n<li>J. Hietarinta: <em>Integrable families of Henon-Heiles type Hamiltonians and a new duality<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1103\/PhysRevA.28.3670\"> Phys. Rev. A <b>28<\/b>, 3670-3672 (1983).<\/a><\/li>\n<li>J. Hietarinta: <em>A search for integrable two-dimensional hamiltonian systems with polynomial potential<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(83)90178-0\"> Phys. Lett. A <b>96<\/b>, 273-278 (1983).<\/a><\/li>\n<\/ul>\n<h3>Pure quantum integrability<\/h3>\n<ul>\n<li>J. Hietarinta: <em>Pure quantum integrability<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1016\/S0375-9601(98)00535-0\">Phys. Lett. A <b> 246<\/b>, 97-104 (1998).<\/a><\/li>\n<li>J. Hietarinta: <em>Solvability in quantum mechanics and classically superfluous invariants<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1088\/0305-4470\/22\/5\/002\">J. Phys. A: Math. Gen. <b>22<\/b>, L143-L147 (1989).<\/a><\/li>\n<\/ul>\n<h3>Quantum versions of classically integrable systems<\/h3>\n<ul>\n<li>J. Hietarinta and B. Grammaticos: <em>On the hbar^2-correction terms in quantum mechanics<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1088\/0305-4470\/22\/9\/020\">J. Phys. A: Math. Gen. <b>22<\/b>, 1315-1322 (1989).<\/a><\/li>\n<li>J. Hietarinta: <em>Classical versus quantum integrability<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1063\/1.526373\"> J. Math. Phys. <b>25<\/b>, 1833-1840 (1984).<\/a><\/li>\n<li>J. Hietarinta, B. Dorizzi, B. Grammaticos et A. Ramani: <em>Sur l&#8217;integrabilite quantique des systemes hamiltoniens classiquement integrables<\/em>, C. R. Acad. Sc. Paris, Serie II, <b>297<\/b>, 791-794 (1983).<\/li>\n<li>J. Hietarinta: <em>Quantum integrability is not a trivial consequence of classical integrability<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(82)90213-4\"> Phys. Lett. A <b>93<\/b>, 55-57<br \/>\n(1982).<\/a><\/li>\n<\/ul>\n<h6>Miscellaneous<\/h6>\n<ul>\n<li>J. Hietarinta: <em>Nambu tensors and commuting vector fields<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1088\/0305-4470\/30\/3\/001\">J. Phys. A: Math. Gen. <b> 30<\/b>, L27-L33 (1997).<\/a><\/li>\n<li>A. Elipe, J. Hietarinta and S. Tompaidis: <em>Comment on a paper by S. Kasperczuk, Celest. Mech. 58: 387-391 (1994)<\/em>,<a href=\"http:\/\/dx.doi.org\/ 10.1007\/BF00692087\"> Celest. Mech. <b> 62<\/b>, 191-192 (1995).<\/a><\/li>\n<li>J. Hietarinta, H. Yoshida, A. Ramani and B. Grammaticos: <em>On the nonintegrability of a class of Hamiltonian systems<\/em>, Europhys. Lett. <b>5<\/b>, 195-198 (1988).<\/li>\n<li>H. Yoshida, A. Ramani, B. Grammaticos and J. Hietarinta: <em>On the non-integrability of some generalized Toda lattices<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1016\/0378-4371(87)90194-4\">Physica A <b>144<\/b>, 310-321 (1987).<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp; Liouville integrable classical systems J. Hietarinta: Direct methods for the search of the second invariant, Physics Reports 147, 87-154 (1987). B. Dorizzi, B. Grammaticos, J. Hietarinta, A. Ramani and F. Schwarz: New integrable three dimensional quartic potentials, Phys. Lett. A 116, 432-436 (1986). A. Ramani, B. Dorizzi, B. Grammaticos and J. Hietarinta: Linearization on &hellip; <a href=\"https:\/\/users.utu.fi\/hietarin\/home\/liouville\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Liouville integrability&#8221;<\/span><\/a><\/p>\n","protected":false},"author":178,"featured_media":0,"parent":43,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-7","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/7","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/users\/178"}],"replies":[{"embeddable":true,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/comments?post=7"}],"version-history":[{"count":8,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/7\/revisions"}],"predecessor-version":[{"id":223,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/7\/revisions\/223"}],"up":[{"embeddable":true,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/43"}],"wp:attachment":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/media?parent=7"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}