The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation

[eng] We have divided this thesis in four parts: a) PART I: Exact symplectic geometry (introduction of the problems). This part contains the basic tools of symplectic geometry and outlines the four subjects that we have study along the thesis: the determination problem, the interpolation problem, th...

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Author: Haro, Àlex
Format: doctoral thesis
Status:Published version
Publication Date:1998
Country:España
Institution:Universidad de Barcelona
Repository:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/42094
Online Access:https://hdl.handle.net/2445/42094
http://www.tdx.cat/TDX-1030109-115035
http://hdl.handle.net/10803/2116
Access Level:Open access
Keyword:Topologia diferencial
Geometria diferencial
Òrbites
Differential topology
Differential geometry
Orbits
id ES_0f7603fe2a65b105b05855ca95ce41e7
oai_identifier_str oai:diposit.ub.edu:2445/42094
network_acronym_str ES
network_name_str España
repository_id_str
dc.title.none.fl_str_mv The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation
title The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation
spellingShingle The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation
Haro, Àlex
Topologia diferencial
Geometria diferencial
Òrbites
Differential topology
Differential geometry
Orbits
title_short The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation
title_full The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation
title_fullStr The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation
title_full_unstemmed The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation
title_sort The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation
dc.creator.none.fl_str_mv Haro, Àlex
author Haro, Àlex
author_facet Haro, Àlex
author_role author
dc.contributor.none.fl_str_mv Simó, Carles
Universitat de Barcelona. Departament de Matemàtica Aplicada i Anàlisi
dc.subject.none.fl_str_mv Topologia diferencial
Geometria diferencial
Òrbites
Differential topology
Differential geometry
Orbits
topic Topologia diferencial
Geometria diferencial
Òrbites
Differential topology
Differential geometry
Orbits
description [eng] We have divided this thesis in four parts: a) PART I: Exact symplectic geometry (introduction of the problems). This part contains the basic tools of symplectic geometry and outlines the four subjects that we have study along the thesis: the determination problem, the interpolation problem, the variational problem and the breakdown problem. b) PART II: On the standard symplectic manifold (analytical part). We recall the necessary tools to work on R(d) x R(d). That is we perform a coordinate treatment of the results. First of all we relate different kinds of generating functions to the primitive function and later we solve formally the determination problem. Then we introduce different variational principles: for fixed points, periodic orbits and orbital segments. Their invariance under certain kind of transformations of phase space is proved, and we interpret physically such results. Finally we give the basic properties of invariant exact Lagrangian graphs obtaining at last that if our graph is minimizing then its orbits are minimizing. c) PART III: On the cotangent bundle (geometrical part). The first three chapters are similar to the three previous ones with the difference that we do an intrinsic treatment of the results by considering any cotangent bundle. The fourth chapter in this part deals with the solution of the interpolation problem given in analytic set up. d) PART IV: Converse KAM theory (numerical part). The last part deals with the applications to converse Kolmogorv-Arnold-Moser (KAM) theory. First of all we give a small list of different examples that we shall study later. Then we generalize converse KAM theory and we related it to the Lipschitz theory by Birkhoff and Herman. Then we perform our variational Greene method and apply it to different examples. Also we study numerically the Aubry-Mather sets in higher dimensions. After this we apply our methods to the rotational standard map that is a symplectic skew product. Then we give some ideas about the geometrical obstructions for existence of invariant tori showing them with a simple example. We also find some known Birkhoff normal forms using our methods. Finally we explain briefly how our theory can be used for arbitrary Lagrangian foliations.
publishDate 1998
dc.date.none.fl_str_mv 1998
dc.type.none.fl_str_mv info:eu-repo/semantics/doctoralThesis
info:eu-repo/semantics/publishedVersion
format doctoralThesis
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/42094
http://www.tdx.cat/TDX-1030109-115035
http://hdl.handle.net/10803/2116
url https://hdl.handle.net/2445/42094
http://www.tdx.cat/TDX-1030109-115035
http://hdl.handle.net/10803/2116
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.rights.none.fl_str_mv (c) Haro Provinciale, 1998
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Haro Provinciale, 1998
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Universitat de Barcelona
publisher.none.fl_str_mv Universitat de Barcelona
dc.source.none.fl_str_mv Tesis Doctorals - Departament - Matemàtica Aplicada i Anàlisi
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869403455229001728
spelling The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolationHaro, ÀlexTopologia diferencialGeometria diferencialÒrbitesDifferential topologyDifferential geometryOrbits[eng] We have divided this thesis in four parts: a) PART I: Exact symplectic geometry (introduction of the problems). This part contains the basic tools of symplectic geometry and outlines the four subjects that we have study along the thesis: the determination problem, the interpolation problem, the variational problem and the breakdown problem. b) PART II: On the standard symplectic manifold (analytical part). We recall the necessary tools to work on R(d) x R(d). That is we perform a coordinate treatment of the results. First of all we relate different kinds of generating functions to the primitive function and later we solve formally the determination problem. Then we introduce different variational principles: for fixed points, periodic orbits and orbital segments. Their invariance under certain kind of transformations of phase space is proved, and we interpret physically such results. Finally we give the basic properties of invariant exact Lagrangian graphs obtaining at last that if our graph is minimizing then its orbits are minimizing. c) PART III: On the cotangent bundle (geometrical part). The first three chapters are similar to the three previous ones with the difference that we do an intrinsic treatment of the results by considering any cotangent bundle. The fourth chapter in this part deals with the solution of the interpolation problem given in analytic set up. d) PART IV: Converse KAM theory (numerical part). The last part deals with the applications to converse Kolmogorv-Arnold-Moser (KAM) theory. First of all we give a small list of different examples that we shall study later. Then we generalize converse KAM theory and we related it to the Lipschitz theory by Birkhoff and Herman. Then we perform our variational Greene method and apply it to different examples. Also we study numerically the Aubry-Mather sets in higher dimensions. After this we apply our methods to the rotational standard map that is a symplectic skew product. Then we give some ideas about the geometrical obstructions for existence of invariant tori showing them with a simple example. We also find some known Birkhoff normal forms using our methods. Finally we explain briefly how our theory can be used for arbitrary Lagrangian foliations.[cat] La present memòria es troba dividida en quatre parts ben diferenciades. La primera conté les eines bàsiques de la geometria simplèctica i planteja els quatre problemes que tractarem al llarg de la memòria: el problema de determinació, el problema d'interpolació, el problema variacional i el problema del trencament de tors invariants. La segona part tracta sobre la varietat simpléctica estàndard, i vindria a ser la part analítica. Aquí hem treballat a R(d) x R(d), és a dir hem fet un tractament coordenat dels resultats. Primer relacionem les funcions generatrius amb la funció primitiva i després resolem formalment el problema de determinación. Tot seguit tractem diferents principis variacionals per als punts fixos per a les òrbites periòdiques i per als segments orbitals. La seva invariància respecte a certs tipus de transformacions de l'espai de fase és demostrada donant una interpretació física. Finalment donem les propietats bàsiques dels grafs Lagrangians invariants, especialment aquella que diu que les òrbites sobre un graf minimitzant són minimitzants. La tercera part abraça el tema del fibrat cotangent, la part geométrica de l'obra. Els tres primers capítols segueixen més o menys la línia dels tres precedents amb la diferéncia fonamental que aquí considerem qualsevol fibrat cotangent. Fem llavors un tractament intrínsec. El quart capítol d'aquesta part està dedicat a resoldre el problema d'interpolació en el cas analític. La quarta i darrera part (que vindria a ser la secció numèrica de la tesi), tracta de les aplicacions a la teoria Kolmogorv, Arnold i Moser (KAM) inversa o del trencament dels tors invariants. Primer donem una llista d'exemples que utilitzarem més endavant. Després generalitzem la teoria KAM inversa i la relacionem amb la teoria Lipschitziana de Birkhoff i Herman. Llavors implementem el nostre criteri de Greene variacional i l'apliquem a diferents exemples. També estudiem els equivalents dels conjunts d'Aubry-Mather en dimensió alta (bé = 4). Després apliquem aquesta metodologia a l'aplicació estàndard rotacional (3D), indicant abans la teoria necessària. Llavors donem algunes idees de com generalitzar els criteris obstruccionals a dimensions altes hi ho mostrem amb un petit exemple. Finalment retrobem algunes formes normals de Birkhoff utilitzant la nostra metodologia basada en la funcióprimitiva i expliquem una mica com es podria considerar la nostra teoria tenint en compte foliacions Lagrangianes arbitràries.Universitat de BarcelonaSimó, CarlesUniversitat de Barcelona. Departament de Matemàtica Aplicada i Anàlisi1998info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/42094http://www.tdx.cat/TDX-1030109-115035http://hdl.handle.net/10803/2116Tesis Doctorals - Departament - Matemàtica Aplicada i Anàlisireponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglés(c) Haro Provinciale, 1998info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/420942026-05-27T06:46:51Z
score 15,301603

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