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University of Illinois – Urbana-Champaign
1.
Burkhardt, Paul.
Explicit, multi-map symplectic integrator for three-body classical trajectory studies in hyperspherical coordinates.
Degree: PhD, Chemistry, 2004, University of Illinois – Urbana-Champaign
URL: http://hdl.handle.net/2142/17345 
► Symplectic integrators are well known for preserving the phase space volume in Hamiltonian dynamics and are particularly suited for problems that require long integration times.…
(more)
▼ Symplectic integrators are well known for preserving the phase space volume in Hamiltonian dynamics and are particularly suited for problems that require long integration times. There is a general operator splitting method for developing explicit
symplectic integration algorithms to any arbitrary even order for separable Hamiltonians where the position and momentum coordinates are uncoupled. Explicit
symplectic integrators for general Hamiltonians are more difficult to obtain, but can be developed by a composition of
symplectic maps if the Hamiltonian can be split into exactly integrable parts. No general technique exists for splitting any Hamiltonian of general form. Many three body problems in classical mechanics can be effectively investigated in symmetrized, hyperspherical polar coordinates, but the Hamiltonian expressed in these coordinates is non-separable. In molecular dynamics, the hyperspherical coordinates facilitate the validation and visualization of potential energy surfaces and for quantum reactive scattering problems, the coordinates eliminate the need for adjusting the wavefunction between product and reactant channels. An explicit
symplectic integrator for hyperspherical coordinates has not yet been devised. This dissertation presents an explicit, multi-map symmetrized composition method
symplectic integrator for three-body Hamiltonians in symmetrized, hyperspherical polar coordinates, specifically for classical trajectory studies in the plane.
Advisors/Committee Members: Belford, R. Linn (Committee Chair), Martinez, Todd J. (committee member), McDonald, J. Douglas (committee member).
Subjects/Keywords: Chemistry; Symplectic integrator; Three-body classical trajectory; Hyperspherical coordinates; Classical mechanics; Hamiltonian dynamics
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APA (6th Edition):
Burkhardt, P. (2004). Explicit, multi-map symplectic integrator for three-body classical trajectory studies in hyperspherical coordinates. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/17345
Chicago Manual of Style (16th Edition):
Burkhardt, Paul. “Explicit, multi-map symplectic integrator for three-body classical trajectory studies in hyperspherical coordinates.” 2004. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed January 22, 2021.
http://hdl.handle.net/2142/17345.
MLA Handbook (7th Edition):
Burkhardt, Paul. “Explicit, multi-map symplectic integrator for three-body classical trajectory studies in hyperspherical coordinates.” 2004. Web. 22 Jan 2021.
Vancouver:
Burkhardt P. Explicit, multi-map symplectic integrator for three-body classical trajectory studies in hyperspherical coordinates. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2004. [cited 2021 Jan 22].
Available from: http://hdl.handle.net/2142/17345.
Council of Science Editors:
Burkhardt P. Explicit, multi-map symplectic integrator for three-body classical trajectory studies in hyperspherical coordinates. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2004. Available from: http://hdl.handle.net/2142/17345
University of Alberta
2.
Deng, Jian.
Uncertainty Quantification of Dynamical Systems and
Stochastic Symplectic Schemes.
Degree: PhD, Department of Mathematical and Statistical
Sciences, 2013, University of Alberta
URL: https://era.library.ualberta.ca/files/n583xv59r
► It has been known that for some physical problems, a small change in the system parameters or in the initial/boundary conditions could leas to a…
(more)
▼ It has been known that for some physical problems, a
small change in the system parameters or in the initial/boundary
conditions could leas to a significant change in the system
response. Hence, it is of importance to investigate the impact of
uncertainty on dynamical system in order to fully understand the
system behavior. In this thesis, numerical methods used to simulate
the effect of random/stochastic perturbation on dynamical systems
are studied. In the first part of this thesis, an aeroelastic
system model representing an oscillating airfoil in pitch and
plunge with random variations in the flow speed, the structural
stiffness terms and initial conditions are concerned. Two
approaches, stochastic normal form and stochastic collocation
method, are proposed to investigate the Hopf bifurcation and the
secondary bifurcation behavior, respectively. Stochastic normal
form allows us to study analytically the Hopf bifurcation scenario
and to predict the amplitude and frequency of the limit cycle
oscillation; while numerical simulations demonstrate the
effectiveness of stochastic collocation method for long term
computation and discontinuous problems. In the second part of this
work, we focus the construction of efficient and robust
computational schemes for stochastic system, and the stochastic
symplectic schemes for stochastic Hamiltonian system are developed.
A systematic procedure to construct symplectic numerical schemes
for stochastic Hamiltonian systems is presented. The approach is an
extension to the stochastic case of the methods based on generating
functions. The idea is also extended to the symplectic weak scheme
construction. Theoretical analysis of the convergence is reported
for strong/weak symplectic integrators. The numerical simulations
are carried out to confirm that the symplectic methods are
efficient computational tools for long-term behaviors. Moreover,
the coefficients of the generating function are invariant under
permutations for the stochastic Hamiltonian system preserving
Hamiltonian functions. As a consequence the high-order symplectic
weak and strong methods have simpler forms than the Taylor
expansion schemes with the same order.
Subjects/Keywords: stochastic symplectic integrator; Uncertainty Quantification; Stochastic differential equations
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APA (6th Edition):
Deng, J. (2013). Uncertainty Quantification of Dynamical Systems and
Stochastic Symplectic Schemes. (Doctoral Dissertation). University of Alberta. Retrieved from https://era.library.ualberta.ca/files/n583xv59r
Chicago Manual of Style (16th Edition):
Deng, Jian. “Uncertainty Quantification of Dynamical Systems and
Stochastic Symplectic Schemes.” 2013. Doctoral Dissertation, University of Alberta. Accessed January 22, 2021.
https://era.library.ualberta.ca/files/n583xv59r.
MLA Handbook (7th Edition):
Deng, Jian. “Uncertainty Quantification of Dynamical Systems and
Stochastic Symplectic Schemes.” 2013. Web. 22 Jan 2021.
Vancouver:
Deng J. Uncertainty Quantification of Dynamical Systems and
Stochastic Symplectic Schemes. [Internet] [Doctoral dissertation]. University of Alberta; 2013. [cited 2021 Jan 22].
Available from: https://era.library.ualberta.ca/files/n583xv59r.
Council of Science Editors:
Deng J. Uncertainty Quantification of Dynamical Systems and
Stochastic Symplectic Schemes. [Doctoral Dissertation]. University of Alberta; 2013. Available from: https://era.library.ualberta.ca/files/n583xv59r
Delft University of Technology
3.
Kleinschneider, A.M. (author).
Modelling the orbital-tidal evolution of the Galilean moon Io.
Degree: 2016, Delft University of Technology
URL: http://resolver.tudelft.nl/uuid:573de551-1ee3-4fbb-bb40-6a1fb21d4b61
► Io, the innermost Galilean moon of Jupiter, is the most volcanically active body in the Solar System. Its volcanism is driven by tidal foces, which…
(more)
▼ Io, the innermost Galilean moon of Jupiter, is the most volcanically active body in the Solar System. Its volcanism is driven by tidal foces, which are in turn sustained by the Laplace resonance between Io, Europa, and Ganymede. Tides have significant impact on a body's characteristics. The liquid ocean underneath Europa's icy surface is sustained by tidal heating. Similarly, tidally-heated exomoons may be able to support life well outside the habitable zone. Tidal forces influence the orbital evolution of a body. Consequently, the orbital evolution also affects the tidal evolution. Thus, it is important to understand how tides drive the Jupiter system and how sustainable the tidal heating that Io experiences is. Additionally, the interior structure of both Jupiter and Io are not very well understood nor constrained, but govern the orbital-tidal evolution via the Love number k2 and the quality factor Q. The orbital motion and tidal evolution of the Jupiter system was modelled. For this, existing TUDAT functions, for example for third-body gravitational acceleration, were combined with new tidal acceleration models. To enable long-term stability and fast computation of the integration a 4 th -order symplectic integrator with Wisdom-Holman split was applied. This type of energy-conserving integrator has limited applicability in the case of small dissipative forces, but requires fewer force evaluation than, for example, a Runge-Kutta integrator of the same order. A variety of reasonable values for k2 and Q have been tested, as well as extreme cases. It was found that the tides raised on Jupiter by Io have a negligible effect on the evolution of the system. On the other hand, the tides raised on Io by Jupiter profoundly impact the evolution of the inner moons. Over the course of five thousand years, Io migrates inwards by several thousand kilometre. In doing so, Europa is brought into a closer orbit as well, to retain the resonance. Similarly, the eccentricity of both Io and Europa decreases, which in turn reduces the dissipated tidal energy and -heating. With Io being in spin-orbit resonance, tidal forces due to tides on Io are not readily applied. Multiple variations of analytical expressions of tides on Io have been evaluated. The results of this thesis provide a qualtiative assessment of the evolution of the Jupiter system and of the sustainability of Io's strong volcanism. The insight gathered on the modelling of tides and their effects on spin-orbit-resonant bodies in particular will benefit future work on the evolution of moons in the Solar System as well as exosystems.
Aerospace Engineering
Space Engineering
MSc Spaceflight
Advisors/Committee Members: Vermeersen, L.L.A. (mentor).
Subjects/Keywords: Io; Jupiter; tides; orbital evolution; orbital stability symplectic integrator; Love number; quality factor
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APA ·
Chicago ·
MLA ·
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CSE |
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to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Kleinschneider, A. M. (. (2016). Modelling the orbital-tidal evolution of the Galilean moon Io. (Masters Thesis). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:573de551-1ee3-4fbb-bb40-6a1fb21d4b61
Chicago Manual of Style (16th Edition):
Kleinschneider, A M (author). “Modelling the orbital-tidal evolution of the Galilean moon Io.” 2016. Masters Thesis, Delft University of Technology. Accessed January 22, 2021.
http://resolver.tudelft.nl/uuid:573de551-1ee3-4fbb-bb40-6a1fb21d4b61.
MLA Handbook (7th Edition):
Kleinschneider, A M (author). “Modelling the orbital-tidal evolution of the Galilean moon Io.” 2016. Web. 22 Jan 2021.
Vancouver:
Kleinschneider AM(. Modelling the orbital-tidal evolution of the Galilean moon Io. [Internet] [Masters thesis]. Delft University of Technology; 2016. [cited 2021 Jan 22].
Available from: http://resolver.tudelft.nl/uuid:573de551-1ee3-4fbb-bb40-6a1fb21d4b61.
Council of Science Editors:
Kleinschneider AM(. Modelling the orbital-tidal evolution of the Galilean moon Io. [Masters Thesis]. Delft University of Technology; 2016. Available from: http://resolver.tudelft.nl/uuid:573de551-1ee3-4fbb-bb40-6a1fb21d4b61
4.
Shen, Xuefeng.
Geometric Integrators for Stiff Systems, Lie Groups and Control Systems.
Degree: Mathematics, 2019, University of California – San Diego
URL: http://www.escholarship.org/uc/item/9g2730gd
► The main idea of a geometric integrator is to adopt a geometric viewpoint of the problem and to construct integrators that preserve the geometric properties…
(more)
▼ The main idea of a geometric integrator is to adopt a geometric viewpoint of the problem and to construct integrators that preserve the geometric properties of the continuous dynamical system. For classical mechanics, both the Lagrangian and the Hamiltonian formulations can be described using the language of geometry. Due to the rich conservation properties of mechanics, it is natural to study the construction of numerical integrators that preserve some geometric properties, such as the symplectic structure, energy, and momentum maps. Such geometric structure-preserving numerical integrators exhibit nice properties compared to traditional numerical methods. This is especially true in galaxy simulations and molecular dynamics, where long time simulations are required to answer the corresponding scientific questions. Variational integrators have attracted interest in the geometric integration community as it discretizes Hamilton's principle, as opposed to the corresponding differential equation, to obtain a numerical integrator that is automatically symplectic, and which exhibits a discrete Noether's theorem. Besides classical mechanics, such an approach has also been applied to other fields, such as optimal control~, partial differential equations~, stochastic differential equations~, and so on. In this thesis, we consider generalizations of geometric integrators that are adapted to three special settings. One is the case of stiff systems of the form, \dot{q} = Aq + f(q), where the coefficient matrix A has a large spectral radius that is responsible for the stiffness of the system, while the nonlinear term f(q) is relatively smooth. Traditionally, exponential integrators have been used to address the issue of stiffness. In Chapter~\ref{exp}, we consider a special semilinear problem with A=JD, f(q)=J∇ V(q), where JT = -J, DT=D, and JD=DJ. Then, the system is described by \dot{q} = J(Dq+∇ V(q)), which naturally arises from the discretization of Hamiltonian partial differential equations. It is a constant Poisson system with Poisson structure Jij\frac{\partial}{\partial xi}\otimes \frac{\partial}{\partial xj}, and Hamiltonian H(q) = \frac{1}{2}qTDq + V(q). Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs are presented. These geometric exponential integrators exhibit better long time stability properties as compared to non-geometric integrators, and are computationally more efficient than traditional symplectic integrators and energy-preserving methods based on the discrete gradient method. The other generalization is to Lie groups. When configuration manifold is a Lie group, we would like to utilize the group structure rather than simply regard it as embedded submanifold. This is particularly useful when…
Subjects/Keywords: Mathematics; geometric reduction; kalman filter; lie group; stiff system; symplectic integrator; variational integrator
…3.3 Lie group variational integrator… …3.3.2 Variational integrator on the Lagrangian side… …3.3.3 Variational integrator on the Hamiltonian side… …61
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High-Order Symplectic Lie Group Methods on SO(n… …93
Hamiltonian variational integrator on the rotation group SO(n)…
Record Details
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Record Details
Similar Records
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Shen, X. (2019). Geometric Integrators for Stiff Systems, Lie Groups and Control Systems. (Thesis). University of California – San Diego. Retrieved from http://www.escholarship.org/uc/item/9g2730gd
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Shen, Xuefeng. “Geometric Integrators for Stiff Systems, Lie Groups and Control Systems.” 2019. Thesis, University of California – San Diego. Accessed January 22, 2021.
http://www.escholarship.org/uc/item/9g2730gd.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Shen, Xuefeng. “Geometric Integrators for Stiff Systems, Lie Groups and Control Systems.” 2019. Web. 22 Jan 2021.
Vancouver:
Shen X. Geometric Integrators for Stiff Systems, Lie Groups and Control Systems. [Internet] [Thesis]. University of California – San Diego; 2019. [cited 2021 Jan 22].
Available from: http://www.escholarship.org/uc/item/9g2730gd.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Shen X. Geometric Integrators for Stiff Systems, Lie Groups and Control Systems. [Thesis]. University of California – San Diego; 2019. Available from: http://www.escholarship.org/uc/item/9g2730gd
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
. 
Open Access Theses and Dissertations
