Numerical methods for solving linearly damped Hamiltonian ordinary differential equations are analyzed and compared. The methods are constructed from the well-known Störmer-Verlet and implicit midpoint methods. The structure preservation properties of each method are shown analytically and numerically. Each method is shown to preserve a symplectic form up to a constant and are therefore conformal symplectic integrators, with each method shown to accurately preserve the rate of momentum dissipation. An analytical linear stability analysis is completed for each method, establishing thresholds between the value of the damping coefficient and the step-size that ensure stability. The methods are all second order and the preservation of the rate of energy dissipation is compared to that of a third order Runge-Kutta method that does not preserve conformal properties. Numerical experiments will include the damped harmonic oscillator and the damped nonlinear pendulum.

<p> Both the theories of differential geometry and of Lie groups and their algebras have been invaluable to the physicist. In the theory of fibre bundles and in the symplectic formulation of mechanics, these fields coalesce to provide a rich structure that enables her/him to obtain a more unified overview of the modern theories in physics. In this short work, we introduce this structure and examine its consequences in general relativity and quantum mechanics.</p> / Thesis / Master of Science (MSc)

Differential equations are frequently used for modeling systems in the physical sciences, biology, and other important real-world disciplines. Oftentimes, however, these equations cannot be solved exactly, so suitable computer algorithms are necessary to provide an approximated solution. While these computational simulations fail to exactly represent all behaviors of the true solution, they can be constructed to exactly, or very closely, reproduce certain properties which are key to the physical or scientific applications of a problem. This paper explores a computational method specifically constructed for modeling the behavior of systems with linear damping, or a reduction of energy, introduced in them. The method was designed to be conformal symplectic, and closely reproduce dissipation of physical properties such as linear and angular momentum, mass, and energy, caused by the damping. The algorithm was constructed in such a way that it maintains low computational cost to implement. Additionally, the method demonstrates favorable accuracy and stability properties in simulation. The method can also handle more complex scenarios, such as systems with forcing terms, and nonlinear systems. In these cases, it has been shown to hold advantages over other commonly used methods in particular circumstances.

Quantum systems with finite Hilbert space are considered. Position and mo- mentum states and their relation through a Fourier transform, displacement in the position-momentum phase-space, and symplectic transformations are introduced and their properties are studied. Symplectic Sp(2l;Zp) trans- formations in l-partite finite system are explicit constructed. The general method is applied to bi-partite and tri-partite systems. The effect of these transformations on the correlations is discussed. Entanglement calculations between the subsystems in a bi-partite system and a tri-partite system are presented. The effect of measurements is also studied.

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Much effort in recent times has been devoted to the study of low energy transport in multibody gravitational systems. Despite continuing advancements in computational abilities, such studies can often be demanding or time consuming in the three-body and four-body settings. In this work, the Hamiltonian describing the planar circular restricted three-body problem is rewritten for systems having small mass parameters, resulting in a 2D symplectic twist map describing the evolution of a particle's Keplerian motion following successive close approaches with the secondary. This map, like the true dynamics, admits resonances and other invariant structures in its phase space to be analyzed. Particularly, the map contains rotational invariant circles reminiscent of McGehee's invariant tori blocking transport in the true phase space, adding a new quantitative description to existing chaotic zone estimates about the secondary. Used in a patched three-body setting, the map also serves as a tool for investigating transfer trajectories connecting loose captures about one secondary to the other without any propulsion systems. Any identified initial conditions resulting in such a transfer could then serve as initial guesses to be iterated upon in the continuous system. In this work, the projection of the McGehee torus within the interior realm is identified and quantified, and a transfer from Earth to Venus is exemplified. / Master of Science / The transport of a particle between celestial bodies, such as planets and moons, is an important phenomenon in astrodynamics. There are multiple ways to mediate this objective; commonly, the motion can be influenced directly via propulsion systems or, more exotically, by utilizing the passive dynamics admitted by the system (such as gravitational assists). Gravitational assists are traditionally modelled using two-body dynamics. That is, a space- craft or particle performs a flyby within that body's sphere of influence where momentum is exchanged in the process. Doing so provides accurate and reliable results, but the design space effecting the desired outcome is limited when considering the space of all possibilities. Utilizing three-body dynamics, however, provides a significant improvement in the fidelity and variety of trajectories over the two-body approach, and thus a broader space through which to search. Through a series of approximations from the three-body problem, a discrete map describing the evolution of nearly Keplerian orbits through successive close encounters with the body is formed. These encounters occur outside of the body's sphere of influence and are thus uniquely formed from three-body dynamics. The map enables computation of a trajectory's fate (in terms of transit) over numerical integration and also provides a boundary for which transit is no longer possible. Both of these features are explored to develop an algorithm able to rapidly supply guesses of initial conditions for a transfer in higher fidelity models and further develop the existing literature on the chaotic zone surrounding the body.

The odd symplectic Grassmannian IG := IG(k, 2n + 1) parametrizes k dimensional subspaces of C^2n+1 which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IG is itself a smooth Schubert variety in the submaximal isotropic Grassmannian IG(k, 2n + 2). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case k = 2, and it gives an algorithm to calculate any quantum multiplication in the equivariant quantum cohomology ring. / Ph. D. / The thesis studies a problem in the general area of Combinatorial Algebraic Geometry. The goal of Algebraic Geometry is to study solutions to systems to polynomial equations. Such systems are ubiquitous in scientific research. We study a problem in enumerative geometry on a space called the odd symplectic Grassmannian. The problem seeks to find the number of curves which are incident to certain subspaces of the given Grassmannian. Due to subtle geometric considerations, the count is sometimes virtual, meaning that some curves need to be counted negatively. The rigorous context of such questions is that of Gromov-Witten theory, a subject with roots in physics. Our space affords a large number of symmetries, and the given counting problems translate into significant amount of combinatorial manipulations. The main result in the dissertation is a combinatorial algorithm to perform the virtual curve counting in the odd-symplectic Grassmannian.

In this thesis I show that indecomposable projective and tilting modules in hypertoric category O are obtained by applying a variant of the geometric Jacquet functor of Emerton, Nadler, and Vilonen to certain Gel'fand-Kapranov-Zelevinsky hypergeometric systems. This proves the abelian case of a conjecture of Bullimore, Gaiotto, Dimofte, and Hilburn on the behavior of generic Dirichlet boundary conditions in 3d N=4 SUSY gauge theories.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21537号 / 理博第4444号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 小野 薫, 教授 向井 茂, 教授 望月 拓郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

The 4-genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4-genus and related invariants of homology classes is the Thom conjecture, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsv´ath-Szab´o, which say that closed symplectic surfaces minimize genus. In this thesis, we prove a relative version of the symplectic capping theorem. More precisely, suppose (X, ω) is a symplectic 4-manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. We show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, Σ) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in S3 . We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in S3 . Using this result, we give a criterion for quasipostive fibered knots to be strongly quasipositive. Symplectic convexity disc bundles is a useful tool in constructing symplectic fillings of contact manifolds. We show the symplectic convexity of the unit disc bundle in a Hermitian holomorphic line bundle over a Riemann surface.