This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical systems are described by the Hamiltonian equations of a function E on the tangent bundle TM endowed with a symplectic form ω_σ, where E is the kinetic energy. Our main goal is to prove existence results for a) periodic orbits, and b) Poincare sections for motions on a fixed energy level Σ_m := {E = m^2/2} ⊂ T M . We tackle this problem by studying the contact geometry of the level set Σ_m . This will allow us to a) count periodic orbits using algebraic invariants such as the Symplectic Cohomology SH of the sublevels ({E ≤ m^2/2}, ω_σ ); b) find Poincare sections starting from pseudo-holomorphic foliations, using the techniques developed by Hofer, Wysocki and Zehnder in 1998. In Chapter 3 we give a proof of the invariance of SH under deformation in an abstract setting, suitable for the applications. In Chapter 4 we present some new results on the energy values of contact type. First, we give explicit examples of exact magnetic systems on T^2 which are of contact type at the strict critical value. Then, we analyse the case of non-exact systems on M different from T^2 and prove that, for large m and for small m with symplectic σ, Σ_m is of contact type. Finally, we compute SH in all cases where Σ_m is convex. On the other hand, we are also interested in non-exact examples where the contact property fails. While for surfaces of genus at least two, there is always a level not of contact type for topological reasons, this is not true anymore for S^2 . In Chapter 5, after developing the theory of magnetic flows on surfaces of revolution, we exhibit the first example on S^2 of an energy level not of contact type. We also give a numerical algorithm to check the contact property when the level has positive magnetic curvature. In Chapter 7 we restrict the attention to low energy levels on S^2 with a symplectic σ and we show that these levels are of dynamically convex contact type. Hence, we prove that, in the non-degenerate case, there exists a Poincare section of disc-type and at least an elliptic periodic orbit. In the general case, we show that there are either 2 or infinitely many periodic orbits on Σ_m and that we can divide the periodic orbits in two distinguished classes, short and long, depending on their period. Then, we look at the case of surfaces of revolution, where we give a sufficient condition for the existence of infinitely many periodic orbits. Finally, we discuss a generalisation of dynamical convexity introduced recently by Abreu and Macarini, which applies also to surfaces with genus at least two.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-07-08T13:38:37Z No. of bitstreams: 1 paulocesarignaciodasilvafilho.pdf: 478866 bytes, checksum: 883dfa24e474221cdd52a8dc34720114 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T16:32:06Z (GMT) No. of bitstreams: 1 paulocesarignaciodasilvafilho.pdf: 478866 bytes, checksum: 883dfa24e474221cdd52a8dc34720114 (MD5) / Made available in DSpace on 2016-07-13T16:32:06Z (GMT). No. of bitstreams: 1 paulocesarignaciodasilvafilho.pdf: 478866 bytes, checksum: 883dfa24e474221cdd52a8dc34720114 (MD5) Previous issue date: 2012-02-24 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Esta dissertação é dedicada ao estudo de Aplicações de Campos de Jacobi aos Sistemas Dinâmicos, seguindo alguns trabalhos desenvolvidas por [6] que utilizam tais campos para caracterizar fluxos geodésicos do tipo Anosov. Em seguida foram desenvolvidas alguns conceitos envolvendo Fluxo Magnético com o trabalho de Gabriel P. Paternain e Keith Burns [2] e por último foram desenvolvidos aplicações de tais campos para a dinâmica do Bilhar [14]. / This dissertation treat the study of Aplications of Jacobi Fields in the Dinamycal System, following some works by [6], that use these fields to characterizae geodesic flows of Anosov type. Then such apllications have been developed some concepts concerning Magnetic Flows with the work of Gabriel P. Paternain e Keith Burns [2] and were finally developed for the dynamic Billiards [14].

Nous étudions l'effet thermo-électrique et les phénomènes qui en résultent, forces et les courants thermoélectriques (TEC) sous l'action d'un champ magnétique externe imposé lors de la solidification d'alliages métalliques. Nous avons utilisé des simulations numériques, des observations directes et des examens de laboratoire. L'interaction entre les courants thermo-électriques et le champ magnétique externe lors de la solidification se produit des forces électromagnétiques et donc un écoulement du métal liquide. Le résultat est nommé effet magnétique thermoélectrique (TEME). Les formulations de TEC, les forces et les équations gouvernant les écoulements TEM sont donnés. Afin de mieux prouver l'existence de la TEME, des expériences par méthode d'imagerie à rayons X menées au synchrtron ont été utilisées pour observer in-situ et en temps réel l'action directe des forces et les mouvements TEM pendant la solidification directionnelle des alliages Al-Cu. Nous avons montré la cohérence raisonnable entre les calculs analytiques et des simulations numériques qui ont exécuté avec les mêmes conditions de traitement. En outre, la capacité des écoulements thermo-électriques à influer sur la microstructure lors de la solidification directionnelle sont expérimentalement évaluées dans les autres cas en réalité. La solidification directionnelle d'une seule phase de formation des alliages Al-Cu sous divers champs magnétiques montre que les écoulements TEM sont capables de modifier la forme de l'interface liquide-solide conduisant à des morphologies différentes. L'effet le plus intense se produit dans différents champs magnétiques pour différentes morphologies, en effet, le champ magnétique élevé est nécessaire pour la morphologie a une plus petite longueur typique. Ceci est en accord avec le comportement des vitesses de TEM qui varient avec les champs magnétiques imposés ainsi que les différentes échelles de longueur typique. Cette variation est confirmée par des simulations numériques 3D. Nous montrons que les dendrites primaires et à l'avant de la phase eutectique, peuvent être modifiés par les mouvements TEM et les forces de TEM dans le solide pour améliorer la croissance de la phase de Al2Cu facettes primaire pendant la solidification des Al-40wt%Cu hypereutectiques. Le mécanisme de renforcement de la croissance de la phase facettes Al2Cu est confirmé par la transmission électronique observation au microscope, et la raison de la formation de la structure de croissance de couple de Al-26wt% Cu alliages est vérifiée par le test de l'analyse thermique différentielle. Ainsi, nous pouvons affirmer que le champ magnétique élevé facilite la formation de la structure de la croissance de couple pour hypoeutectiques alliages Al-Cu, et favorise la croissance de la phase Al2Cu primaire pour hypereutectiques Al-Cu alliages. / We have investigated the thermoelectric magnetic (TEM) forces and flows resulting from the interaction between the internal thermoelectric currents (TEC) and the imposed external magnetic field during solidification. Numerical simulations, direct observations and experimental examinations were undertaken. As the natural phenomenon, TEC was discovered almost 200 years ago, therefore, our introduction begins from then on. It is shown that the interaction between TEC and external magnetic field during solidification in the cont put forth new interesting phenomena in the context of a rising field named Electromagnetic Processing of Materials. After that, it is discussed how the TEC appear and the TEM effect (TEME, referring to both TEM forces and flows) behaves at the liquid-solid interface in directional solidification under external magnetic field. Meanwhile, formulations of TEC, TEM forces and flows are given, and numerical simulations of TEME are performed to visually display the TEM forces and flows. In order to further prove the existence of TEME, in situ synchrotron X-ray imaging method was used to observe the direct resultant of TEM forces and flows during directionally solidifying the Al-Cu alloys. The observations show reasonable consistency with the analytical calculations and numerical simulations performed with the same process conditions. Except confirmation the existence of TEME, its abilities to affect the microstructure during directional solidification are experimentally investigated in the more realistic cases. The single phase forming Al-Cu alloys are directionally solidified under various magnetic fields, which shows that TEM flows are capable to modify the shape of liquid-solid interface, and the most intensive affect occurs under different magnetic fields for different interface morphologies. Indeed, the smaller the typical length of the morphology is the higher the magnetic field is needed. This agrees with the estimating regulation of the velocity of TEM flows changing with magnetic fields for different typical length scales, and is confirmed by 3D numerical simulations. Directional solidification of multiphase forming Al-Cu alloys under various magnetic fields shows that the mushy zone length (distance between the front of primary dendrites and eutectic phases) varies with the magnetic fields, which can be attributed to the redistribution of rejected solutes by TEM flows. In addition, apparent enhanced growth of the primary faceted Al2Cu phase is founded when Al-40wt%Cu alloys are solidified under sufficient high magnetic fields, this should be ascribed to the TEM forces acting on the solid because strains are able to lead the formation of defects and thus benefit to the growth of faceted phase. This is confirmed by comparison of the dislocations in samples solidified without and with a 10T magnetic field via transmission electron microscopy observation. In another aspect, an almost entire couple growth structure is achieved when Al-26wt%Cu alloys are directionally solidified under a 4T magnetic field, which can be explained by the effect of high magnetic field on changing the nucleation temperature and growth velocity of each phase. Moreover, the differential thermal analysis test on the nucleation temperature of both α-Al and eutectic phases verified this explanation. Therefore, we conclude that high magnetic field facilitates the formation of couple growth structure for hypoeutectic Al-Cu alloys, reversely, enhances the growth of primary dendrite for hypereutectic Al-Cu alloys.