Full Record

Author | Liu, Jian-Long |

Title | Preservation of Periodicity in Variational Integrators |

URL | https://doi.org/10.31979/etd.c4z9-y9yp https://scholarworks.sjsu.edu/etd_theses/4596 |

Publication Date | 2015 |

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 |

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…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…