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Lie Systems, Lie Symmetries and Reciprocal Transformations

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Lie Systems, Lie Symmetries and Reciprocal Transformations LIE SYSTEMS, LIE SYMMETRIES AND RECIPROCAL TRANSFORMATIONS. Departamento de F´ısicaFundamental Area´ F´ısicaTe´orica Universidad de Salamanca MEMORIA PARA OPTAR AL T´ITULO DE DOCTOR arXiv:1508.00726v1 [math-ph] 4 Aug 2015 Cristina Sard´onMu~noz PhD Thesis, 2015 D~na. Pilar Garc´ıaEst´evez, Catedr´atica de la Universidad de Salamanca y D. Javier de Lucas Ara´ujo, ayudante Doctor de la Universidad de Varsovia, CERTIFICAN: Que el trabajo de investigaci´onque se recoge en la siguiente memoria titulado \Sistemas de Lie, simetr´ıasde Lie y transformaciones rec´ıprocas", presentada por D~na. Cristina Sard´onMu~noz para optar al t´ıtulode doctor y la Menci´onde \Doctorado Internacional", ha sido realizada en su totalidad bajo su direcci´ony autorizan su presentaci´on. Abril, 2015 D~na.PILAR GARC´IA ESTEVEZ´ Catedr´atica Universidad de Salamanca D. JAVIER DE LUCAS ARAUJO´ Ayudante Doctor Universidad de Varsovia My spine just squinted and my eye is weak I can't find peace in a form of speech My views change color red hot, ice cold Black ain't a color, happiness ain't gold It's hard for me to feel normal easy to feel free It's hard for you to understand if you can't feel me I'm a municipal and the mundane I try to keep weird keep away from the same I haven't read an outline on how to pass youth Rather than get passed, I pass doobs I'm out, I'm in, it's hard to live in this given culture All the hers and hymns All of the rats, snakes, and youth vultures around My heart's out of shape and my head's in a cast I am tired, I am weary I could sleep for a thousand years A thousand dreams that would awake me different colors, made of tears. -The Growlers and The Velvet Underground- La esencia de todo arte bello, es la gratitud -Friederich Nietzsche Cuando expresamos nuestra gratitud, no debemos olvidar que la mayor apreciaci´on no va en las palabras, sino en las acciones, parafraseando a J.F. Kennedy. Las personas han sido muchas, las acciones fueron m´as.Intentar hacer un resumen de cuatro a~nosde trabajo y apoyo en un simple folio, son incompatibles. Sin embargo, en un breve y futil intento, siguiendo un orden cronol´ogico,me dispongo a agradecer debidamente: A la Universidad de Salamanca, por cubrir econ´onicamente este trabajo y a Pilar Garc´ıaEst´evez por aceptarme como alumna. Al Departamento de F´ısicaFundamen- tal y todos sus integrantes, por la grata acogida y hacerme sentir parte de ellos. En especial, a Jose Mar´ıaCerver´opor sus sabias aportaciones en la investigaci´onaso- ciada a esta tesis. A mis dos directores de tesis, Pilar Garc´ıaEst´evez y Javier de Lucas Ara´ujopor depositar su confianza en m´ıcomo futura investigadora y tener la paciencia suficiente para formar a una novata, muy despistada. La discusi´oncient´ıfica,las explicaciones, las demostraciones y consejos en temas distendidos y/o personales, han sido de lo m´asnecesarios y bienvenidos. Pero sobre todo, es de digno agradecimiento final, el esfuerzo final. La elaboraci´onde este manuscrito y correcciones. A la C´atedrade M´etodos Matem´aticosen F´ısicade la Universidad de Varsovia, al IMPAN (Instituto de Matem´aticas de la Academia Polaca de las Ciencias) y al Polish National Science Center, por todo tipo de acondicionamiento y apoyo econ´omicopara realizar mi estancia internacional. En particular, querr´ıaagradecer la financiaci´ondel proyecto HARMONIA1 y al Profesor Grabowski por la aceptaci´on de mi colaboraci´ony resolver la burocracia pertinente. 1DEC-2012/04/M/ST1/00523 ii A la Universidad Roma Tr´een Roma, por aceptarme en una estancia breve y gratificante para la discusi´oncon los profesores de mi m´assuma admiraci´on,Decio Levi, Orlando Ragnisco y Pavel Winternitz, con quienes espero mantener un estre- cho contacto cient´ıficode ahora en adelante. A Pep´ınCari~nena,por transmitirme sus conocimientos y contribuir a mi formaci´onen mis estancias en la Universidad de Zaragoza. A la red de Geometr´ıa,Mec´anicay Control por permitirme ser miembro de ella y sufragar mis gastos en viajes nacionales, que han contribu´ıdomuy prove- chosamente durante mi doctorado, permiti´endomeconocer a un gran n´umerode cient´ıficosde renombre internacional. A Angel´ Ballesteros por invitarme a la Universidad de Burgos y a Alfonso Blasco por trabajar conmigo. A mi colega Francisco Jos´eHerranz, colega en sen- tido cient´ıfico,porque en el plano personal le considero un gran amigo, que me ha servido de consejero, apoyo incondicional y el mejor compa~nerode viajes interna- cionales. Adem´asde lo obvio, por transmitirme sus conocimientos y ense~narmeel perfeccionismo que pone en la investigaci´on. A Juan Domingo Lejarreta, por su metodismo y meticulosidad en sus c´alculos,ayud´andomeen la realizaci´ony com- probaci´onde los m´ıos.A Silvia Vilari~no,porque es admirable en su trabajo y ofrece verdadera confianza y profesionalidad y me ha ense~nadoa dar una buena charla y en la organizaci´onde mi desorden matem´atico. A Cuchi, que ha contribuido en la estructuraci´ony bonita presentaci´onde esta tesis. Adem´asde nuestra amistad. A los marsopas Alvaro,´ Cuchi, Teresa, Alberto y Edu, porque no s´olode ciencia vive el hombre, nos pegamos nuestras juergas. A Lorena, estimada amiga y compa~nenade promoci´on,de alegr´ıasy penas, de explo- raci´oninternacional, cuya amistad y su profesionalidad investigadora no han sido para m´ısino el acicate para intentar conseguir parecerme y mantenerme junto a ella. A Bea, estimada amiga y compa~nerade joven adolescencia, de instituto, de promoci´on,de alegr´ıasy penas, de exploraci´oninternacional, cuya amistad, profe- sionalidad y buen humor son el incentivo de j´ubiloy buenos ratos en mi vida, si me encuentro con ella. A Rafa, Aser y Luis, porque son mis mejores amigos. Encuentro en ellos la mayor afinidad de cualquier ´ındole.Porque me han ense~nadoque la vida puede ser simple pero insisto en hacerla complicada. Un diez por ciento es lo que nos ocurre, el noventa es c´omoactuamos. La despreocupaci´onha contribuido muy positivamente estos ´ultimoscuatro a~nos.Y que lo pasamos de muerte. Finalmente, me gustar´ıaagradecer a mis queridos padres el haberme acompa~nado siempre. Hoy me siento dichosa mujer, que gracias a ellos, no he sentido nunca la soledad en este mundo. Que su ejemplo de sacrificio me ha servido en esforzarme en la finalizaci´onde un doctorado y que me recuerden mi val´ıa,en per´ıodos de privaci´on de mi ocio y su comprensi´onen la ansiedad, han sido ineludibles en esta etapa. A todos vosotros, gracias de todo coraz´on. Contents 1 Introduction 1 1 The notion of integrability . .1 1.1 Integrability of dynamical systems . .2 1.2 Integrability of Hamiltonian systems . .2 2 Methods to guarantee integrability . .3 2.1 Lax pairs and the IST . .3 2.2 The Hirota bilinear method . .5 2.3 The Painlev´etest . .5 2.4 The singular manifold method . .9 2.5 Lie systems . .9 2.6 Lie symmetries . 13 2.7 Reciprocal transformations . 14 2 Geometric Fundamentals 17 1 Tangent spaces, vector fields and integral curves . 19 2 Time-dependent vector fields . 22 3 Higher-order tangent spaces and related notions . 25 4 Jet bundles . 26 5 Poisson algebras . 30 6 Symplectic, presymplectic and Poisson manifolds . 32 7 Dirac manifolds . 34 8 Jacobi manifolds . 37 3 Lie systems 39 1 The interest of Lie systems . 39 iv 2 Lie systems . 46 2.1 Lie systems and superposition rules . 46 2.2 Lie systems on Lie groups . 49 2.3 An algorithm to derive superposition rules . 50 2.4 New applications of Lie systems . 51 3 Lie{Hamilton systems . 66 3.1 On the necessity of Lie{Hamilton systems . 66 3.2 Lie{Hamilton structures . 69 3.3 t-independent constants of motion . 73 3.4 Symmetries, linearization and comomentum maps . 76 3.5 On t-dependent constants of motion . 79 3.6 Lie integrals . 81 3.7 Polynomial Lie integrals . 82 3.8 The coalgebra method, constants of motion and superposition rules . 86 4 Applications of the geometric theory of Lie{Hamilton systems . 90 4.1 The Ermakov system . 90 4.2 A superposition rule for Riccati equations . 91 4.3 Kummer{Schwarz equations in Hamilton form . 93 4.4 The n-dimensional Smorodinsky{Winternitz systems . 95 4.5 A trigonometric system . 98 5 Classification of Lie{Hamilton systems on the plane . 99 5.1 On general definitions and properties . 100 5.2 Lie{Hamilton algebras . 105 5.3 Local classification . 108 6 Applications of Lie{Hamilton systems on the plane . 116 6.1 sl(2; R)-Lie{Hamilton systems . 117 6.2 Lie{Hamilton biological models . 122 6.3 Other Lie{Hamilton systems on the plane . 129 6.4 Two-photon Lie{Hamilton systems . 137 6.5 h2-Lie{Hamilton systems . 139 7 Dirac{Lie systems . 142 7.1 Motivation for Dirac{Lie systems . 142 7.2 Dirac{Lie Hamiltonians . 145 7.3 Diagonal prolongations . 149 7.4 Superposition rules and t-independent constants of motion . 153 7.5 Bi{Dirac{Lie systems . 156 7.6 Dirac{Lie systems and Schwarzian{KdV equations . 159 8 Jacobi{Lie systems . 160 8.1 Jacobi{Lie systems . 160 8.2 Jacobi{Lie Hamiltonians . 162 v 8.3 Jacobi{Lie systems on low dimensional manifolds . 163 4 Lie symmetry for differential equations 167 1 What can a symmetry transformation achieve? . 167 2 The CLS for ODEs . 170 2.1 Algorithmic computation of Lie point symmetry for systems of ODEs .
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