We consider systems of Euler-Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lie's infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples.

This thesis is mainly concerned with problems in the areas of the Calculus of Variations and Partial Differential Equations (PDEs). The properties of the functional to minimize with respect to the given topology play an important role in the existence of minimizers of integral problems. We will introduce the important concepts of quasiconvexity and polyconvexity. Inspired by finite element methods from Numerical Analysis, we introduce a perturbed problem which has some surprising uniqueness properties.

Made available in DSpace on 2016-08-31T13:33:46Z (GMT). No. of bitstreams: 1 TayaraCPB_Dissert.pdf: 1984521 bytes, checksum: 5a46c1781124a49b404a083b87b969bd (MD5) Previous issue date: 2015-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / With advances in technology and the popularization of the use of Unmanned Aerial Vehicles (UAV's) so does the need to use more robust and more effective control techniques. Among the various types of unmanned aerial vehicles, this paper will focus on quadrotor model, which has a mechanical structure in the form of x, whose ends have an engine and propeller assembly, where the rotation of this group is responsible for the lift and the movements developed by quadrotor. This feeling, aiming to apply drivers that provide stability to the dynamic system. This study aims to conduct mathematical modeling using the Euler-Lagrange. With this, it is proposed a PID controller (Proportional Integral Derivative) to maintain stable the three orientation angles and height to a desired value. The development of the proposed controller will be validated via simulation confirming the application feasibility of the technique presented stability / Com o avanço tecnológico e a popularização do uso dos Veículos Aéreos Não Tripulados (VANT s) cresce também a necessidade do uso de técnicas de controle mais robustas e mais eficazes. Dentre os mais diversos tipos de veículos aéreos não tripulados, este trabalho irá focar no modelo do quadrotor, que possui uma estrutura mecânica em forma de cruz, cujas extremidades têm um conjunto de motor e hélice, onde a rotação desse conjunto é responsável pela força de sustentação e pelos movimentos desenvolvidos pelo mesmo. Objetivando aplicar controladores que proporcione estabilidade ao sistema dinâmico deste veiculo aéreo. O presente trabalho tem como objetivo realizar a modelagem matemática deste sistema usando as equações de Euler-Lagrange. Tendo isso, é proposto um controlador PID (Proporcional Integral Derivativo) para manter os três ângulos de orientação estáveis e a altura em um valor desejado. O desenvolvimento do controlador proposto será validado via simulação confirmando a viabilidade da aplicação da técnica de estabilidade apresentada

Infimalfolgen nichtkonvexer Variationsprobleme haben aufgrund feiner Oszillationen häufig keinen starken Grenzwert in Sobolevräumen. Diese Oszillationen haben eine physikalische Bedeutung; Finite-Element-Approximationen können sie jedoch im Allgemeinen nicht auflösen. Relaxationsmethoden ersetzen die nichtkonvexe Energie durch ihre (semi)konvexe Hülle. Das entstehende makroskopische Modell ist degeneriert: es ist nicht strikt konvex und hat eventuell mehrere Minimalstellen. Die fehlende Kontrolle der primalen Variablen führt zu Schwierigkeiten bei der a priori und a posteriori Fehlerschätzung, wie der Zuverlässigkeits- Effizienz-Lücke und fehlender starker Konvergenz. Zur Überwindung dieser Schwierigkeiten erweitern Stabilisierungstechniken die relaxierte Energie um einen diskreten, positiv definiten Term. Bartels et al. (IFB, 2004) wenden Stabilisierung auf zweidimensionale Probleme an und beweisen dabei starke Konvergenz der Gradienten. Dieses Ergebnis ist auf glatte Lösungen und quasi-uniforme Netze beschränkt, was adaptive Netzverfeinerungen ausschließt. Die vorliegende Arbeit behandelt einen modifizierten Stabilisierungsterm und beweist auf unstrukturierten Netzen sowohl Konvergenz der Spannungstensoren, als auch starke Konvergenz der Gradienten für glatte Lösungen. Ferner wird der sogenannte Fluss-Fehlerschätzer hergeleitet und dessen Zuverlässigkeit und Effizienz gezeigt. Für Interface-Probleme mit stückweise glatter Lösung wird eine Verfeinerung des Fehlerschätzers entwickelt, die den Fehler der primalen Variablen und ihres Gradienten beschränkt und so starke Konvergenz der Gradienten sichert. Der verfeinerte Fehlerschätzer konvergiert schneller als der Fluss- Fehlerschätzer, und verringert so die Zuverlässigkeits-Effizienz-Lücke. Numerische Experimente mit fünf Benchmark-Tests der Mikrostruktursimulation und Topologieoptimierung ergänzen und bestätigen die theoretischen Ergebnisse. / Infimising sequences of nonconvex variational problems often do not converge strongly in Sobolev spaces due to fine oscillations. These oscillations are physically meaningful; finite element approximations, however, fail to resolve them in general. Relaxation methods replace the nonconvex energy with its (semi)convex hull. This leads to a macroscopic model which is degenerate in the sense that it is not strictly convex and possibly admits multiple minimisers. The lack of control on the primal variable leads to difficulties in the a priori and a posteriori finite element error analysis, such as the reliability-efficiency gap and no strong convergence. To overcome these difficulties, stabilisation techniques add a discrete positive definite term to the relaxed energy. Bartels et al. (IFB, 2004) apply stabilisation to two-dimensional problems and thereby prove strong convergence of gradients. This result is restricted to smooth solutions and quasi-uniform meshes, which prohibit adaptive mesh refinements. This thesis concerns a modified stabilisation term and proves convergence of the stress and, for smooth solutions, strong convergence of gradients, even on unstructured meshes. Furthermore, the thesis derives the so-called flux error estimator and proves its reliability and efficiency. For interface problems with piecewise smooth solutions, a refined version of this error estimator is developed, which provides control of the error of the primal variable and its gradient and thus yields strong convergence of gradients. The refined error estimator converges faster than the flux error estimator and therefore narrows the reliability-efficiency gap. Numerical experiments with five benchmark examples from computational microstructure and topology optimisation complement and confirm the theoretical results.