{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T17:52:48Z","timestamp":1649008368536},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"3-4","license":[{"start":{"date-parts":[[2011,11,14]],"date-time":"2011-11-14T00:00:00Z","timestamp":1321228800000},"content-version":"unspecified","delay-in-days":8352,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proceedings of the Royal Society of Edinburgh: Section A Mathematics"],"published-print":{"date-parts":[[1989]]},"abstract":"<jats:title>Synopsis<\/jats:title><jats:p>We discuss the C<jats:sup>\u221e<\/jats:sup> complete integrability of Hamiltonian systems of type <jats:italic>q<\/jats:italic> = \u2014grad <jats:italic>V<\/jats:italic>(<jats:italic>q<\/jats:italic>) = <jats:italic>F<\/jats:italic>(<jats:italic>q<\/jats:italic>), in which the closure of the cone generated (with nonnegative coefficients) by the vectors <jats:italic>F<\/jats:italic>(<jats:italic>q<\/jats:italic>), <jats:italic>q<\/jats:italic> \u03f5 \u211d<jats:sup>n<\/jats:sup>, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential <jats:italic>V<\/jats:italic>(<jats:italic>q<\/jats:italic>)=\u03a3<jats:sup>N<\/jats:sup><jats:sub>i=1<\/jats:sub> exp(<jats:italic>f<\/jats:italic><jats:sub>i<\/jats:sub>\u2223 <jats:italic>q<\/jats:italic>) is a special case of the considered systems ifthe cone <jats:italic>C<\/jats:italic>(<jats:italic>f<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,f<jats:sub>N<\/jats:sub>)={\u03a3<jats:sup>N<\/jats:sup><jats:sub>i=1<\/jats:sub>c<jats:sub>i<\/jats:sub>f<jats:sub>i<\/jats:sub>,<jats:italic>c<\/jats:italic><jats:sub>i<\/jats:sub>\u22670} does notcontain a line. In any number of degrees of freedom, if <jats:italic>C<\/jats:italic>(<jats:italic>f<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,f<jats:sub>N<\/jats:sub>) has amplitude not too large (ang (<jats:italic>f<\/jats:italic><jats:sub>i<\/jats:sub>, <jats:italic>f<\/jats:italic><jats:sub>j<\/jats:sub> \u2266\u03c0\/2i,j=1,2,\u2026, N), the first integrals are C<jats:sup>\u221e<\/jats:sup> functions. In two degrees of freedom, without restriction on the amplitude of the cone, C<jats:sup>\u221e<\/jats:sup>-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C<jats:sup>\u221e<\/jats:sup>-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.<\/jats:p>","DOI":"10.1017\/s0308210500024161","type":"journal-article","created":{"date-parts":[[2011,11,14]],"date-time":"2011-11-14T13:19:44Z","timestamp":1321276784000},"page":"293-314","source":"Crossref","is-referenced-by-count":3,"title":["A class of C<sup>\u221e<\/sup>-integrable Hamiltonian systems"],"prefix":"10.1017","volume":"113","author":[{"given":"W. M.","family":"Oliva","sequence":"first","affiliation":[]},{"given":"M. S. A. C.","family":"Castilla","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2011,11,14]]},"reference":[{"key":"S0308210500024161_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/0378-4371(87)90194-4"},{"key":"S0308210500024161_ref009","first-page":"233","volume-title":"CIME Summer School Bressanone 1978","author":"Moser","year":"1980"},{"key":"S0308210500024161_ref005","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.9.1924"},{"key":"S0308210500024161_ref004","doi-asserted-by":"publisher","DOI":"10.1063\/1.526411"},{"key":"S0308210500024161_ref003","unstructured":"3 Chow S.-N. , Lin X.-B. and Lu K. . 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