On the symplectic integration of Hamiltonian systems

Resumo

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.

Ficha do documento

Tipo

Dissertação

Ano

2018

Instituição

Fundação Getulio Vargas

Fonte

Repositório da FGV

Idioma

Inglês

Acesso

Não informado

Identificador

oai:repositorio.fgv.br:10438/25673

Palavras-chave

Sistemas hamiltonianosVariedades simpléticasSymplectic integrationHamiltonian systemsMatemática