Numerical methods are usually necessary in solving Hamiltonian systems since there is often no closed-form solution. By utilizing a general property of Hamiltonians, namely the symplectic property, all of the qualities of the system may be preserved for indefinitely long integration times because all of the integral (Poincare) invariants are conserved. This allows for more reliable results and frequently leads to significantly shorter execution times as compared to conventional methods. The resonant triad Hamiltonian with one degree of freedom will be focused upon for most of the numerical tests because of its difficult nature and, moreover, analytical results exist whereby useful comparisons can be made.

Submitted by Diego Navarro Pozo (the.electric.me@gmail.com) on 2018-10-23T14:56:18Z No. of bitstreams: 1 dissert diego revisada + ficha + assinaturas.pdf: 953096 bytes, checksum: 005110857b3e2e871af759d632f8ef55 (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2018-10-23T15:26:47Z (GMT) No. of bitstreams: 1 dissert diego revisada + ficha + assinaturas.pdf: 953096 bytes, checksum: 005110857b3e2e871af759d632f8ef55 (MD5) / Made available in DSpace on 2018-10-29T18:11:10Z (GMT). No. of bitstreams: 1 dissert diego revisada + ficha + assinaturas.pdf: 953096 bytes, checksum: 005110857b3e2e871af759d632f8ef55 (MD5) Previous issue date: 2018-07-30 / Os sistemas Hamiltonianos formam uma das classes mais importantes de equações diferenciais. Além de constituírem o formalismo central da física clássica, sua aplicação se estende a uma grande variedade de outros campos de estudo. Esses sistemas possuem uma característica notória do ponto de vista da matemática, a saber, que a sua ação sobre seus estados iniciais preserva uma estrutura geométrica conhecida como simpleticidade. Este fato tem importantes consequências sobre as características qualitativas do comportamento do sistema, em especial no longo prazo. Neste trabalho, são estudados métodos numéricos para obter soluções aproximadas para sistemas Hamiltonianos (já que, via de regra, soluções exatas não podem ser encontradas) que preservem a estrutura simplética das equações originais. Para tal, é feita uma revisão da teoria clássica da integração numérica de equações diferenciais, bem como de temas mais recentes como os integradores exponenciais. Além de expor a literatura mais recente sobre integradores simpléticos do tipo Runge-Kutta Exponencial, o trabalho propõe um algoritmo para o cálculo computacionalmente eficientes de integrais envolvendo exponenciais de matrizes, que são centrais para a integração simplética estável de ordem alta. / Hamiltonian systems form one of the most important classes of differential equations describing the evolution of physical phenomena. They are the backbone of classical mechanics and their application covers many different areas such as molecular dynamics, hydrodynamics, celestial and statistical mechanics, just to mention a few of them. A noteworthy feature of Hamiltonian systems is that their flow possesses a geometric property -known as symplecticity- which has a major impact on the long-time behavior of the solution. Since in general closed-form solutions can be found only in few particular cases, the construction and analysis of numerical integrators -able to produce discrete approximations that are also symplecticity preserving- is crucial for studying these systems. In this work we present the key ideas about Hamiltonian systems and their theoretical properties. We also review the main numerical methods and techniques to design and analyze symplectic integrators. Special attention is given to the stability and dynamical properties of the methods, as well as their effectiveness for long-term simulations. Finally, we propose an algorithm to improve the computational implementation of the family of exponential-based symplectic integrators recently found in the literature.

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic integrators, and often, these techniques are developed for well-understood Hamiltonian systems such as Hill’s lunar equation. In this presentation, we explore how well symplectic techniques developed for well-understood systems (specifically, Hill’s Lunar equation) address discretization errors in MD systems which fail for one or more reasons.