Full Record

New Search | Similar Records

Title Preservation of Periodicity in Variational Integrators
Publication Date
Degree MS
Discipline/Department Mathematics
Degree Level masters
University/Publisher San Jose State University
Abstract Classical numerical integrators do not preserve symplecticity, a structure inherent in Hamiltonian systems. Thus, the trajectories they produce cannot be expected to possess the same qualitative behavior observed in the original system. Pooling recent results from O'Neale and West, we explore a particular class of numerical integrators, the variational integrator, that preserves one aspect of the range of behavior present in Hamiltonian systems, the periodicity of trajectories. We first establish the prerequisites and some key concepts from Hamiltonian systems, particularly symplecticity and action-angle coordinates. Through perturbation theory and its complications manifested in small divisor problems, we motivate the necessity for KAM theory. With O'Neale's KAM-type theorem, we observe the preservation of periodicity by symplectic one-step methods. Lastly, we show that the variational integrator introduced by West possesses the defining characteristics of symplectic one-step methods, and therefore also preserves periodicity of the original trajectories.
Subjects/Keywords hamiltonian system; kam theory; periodicity; perturbation theory; symplectic integrator; variational integrator
Country of Publication us
Record ID oai:scholarworks.sjsu.edu:etd_theses-8143
Repository sjsu
Date Indexed 2020-10-15

Sample Search Hits | Sample Images | Cited Works

…40 5.3 A return map produced by the variational integrator . . . . . . . . . . . . . 44 5.4 Return maps of the pendulum produced by the variational integrator . . . . 47 5.5 Phase space of the pendulum produced by the variational integrator

…produced by the numerical integrator is periodic or not. While the answer to this question 2 does not establish all of the sufficient conditions for the periodicity of the original trajectory, it gives us some basis for what to expect in the remaining…

…are introduced as we define Hamiltonian systems and will evolve in parallel to solidify our understanding of the theory, concluding with the results of applying the variational integrator to them. We first explore most of the background required for…

…periodicity of trajectories. Lastly, we delve into an intuitive way presented by West ([Wes04]), in which the variational integrator, an integrator of the type described by O’Neale, can be easily derived by utilizing the idea of…

…first topic of discussion is how they are defined. Rather than advancing along the historical progression of developing Hamiltonian systems from Lagrangian systems, we start with the definition of a symplectic manifold. From there, we arrive naturally at…

…from Hamiltonian systems. 2.1 Symplecticity Symplecticity is a concept central to Hamiltonian systems. We thus begin by introducing some basic ideas. Definition 2.1.1. A symplectic form on an even-dimensional manifold is a skew-symmetric…

…nondegenerate, bilinear differential form. An even-dimensional manifold M equipped with a symplectic form ω, or (M, ω), is called a symplectic manifold. On a symplectic manifold, the coordinates are usually written as (q, p), where q is…

…called the generalized position and p is the generalized momentum, both of which are n-dimensional. 5 Example 2.1.2. A simple example of a symplectic form on R2n is the standard symplectic form ω0 (u, v) = u, Jv for all u, v ∈ R2n , with…