This thesis studies the evolution of N-body Keplerian systems (N=500) using an algorithm developed by P.M. Visser that mathematically predicts when a close encounter occurs.

We present an implementation of the Wisdom-Holman integrator for simulating gravitational dynamics in planetary systems: systems with one dominant central mass and \(N\) orbiting bodies, such as the Solar System. The Wisdom-Holman integrator models the motion of non-central bodies as unperturbed Kepler orbits and integrates gravitational interactions between orbiting planets as weak perturbations. Two methods to advance a body along its orbits are investigated: one using coordinate transformations (Method A) and one based on the \(f\) and \(g\) functions (Method B). Method B is shown to be significantly faster than Method A, without a significant loss of accuracy, making it the preferred method for most simulations. Simulations of the Solar System using large time steps, including time steps exceeding the orbital period of some planets, are explored to determine whether the increase in computational speed justifies the loss in accuracy. Simulations with a fixed Sun and with a dynamic Sun are considered. Results indicate that for fixed Sun simulations, while accurate simulations require small time steps, larger steps still capture the qualitative behaviour of the system. The step size of simulations with a dynamic Sun is limited by \(\Dt\st{max} = T_\mercury / 6\). However, for small step sizes, dynamic Sun simulations accurately describe the Solar System and the restricted three-body problem in which resonance occurs. This is achieved without the use of Jacobian coordinates, which are commonly used in implementations of the Wisdom-Holman integrator. The fixed Sun and dynamic Sun simulations are shown to conserve energy, suggesting an accurate description of the system simulated.

MOdified Newtonian Dynamics (MOND) is an alternative explanation for the rotation-distance curves of galaxies that modifies Newton’s law for gravity at low gravitational accelerations, opposing dark matter.
Since its initial proposal, many interpolation functions for MOND have been proposed.
One potential interpolation function is the de Sitter interpolation, based on the properties of the de Sitter space. It is newly tested on globular clusters through simulation in this thesis. A simulation is needed because no analytical solution exits. Although the de Sitter interpolation function is quite old, limited research has gone into it so far.
In this thesis, the credibility of this function will be examined by comparing it to one of the most
popular functions: the standard interpolation. It will be tested on two globular clusters (NGC 6101 and NGC 5466), using the approximation of an isothermal sphere for initialization and an N-body particle mesh code to simulate the systems according to the gravitational law obtained through both of these functions. Afterwards, the resulting star density and velocity dispersion, both as function of distance to the centre of the cluster, will be compared to their observed counterparts.
The credibility of the results will be discussed in depth, comparing the system to the isothermal sphere, considering what changes variations in the parameters cause and assuring the system is in equilibrium after simulation.
The results show the de Sitter interpolation is on par with the standard interpolation, producing very similar results. Both match very well to the observed star count densities, and the velocity dispersion of NGC 6101 matches well to the observations too, whereas the velocity dispersion of NGC 5466 does not, in which case both functions match equally badly and predict a too high velocity dispersion.
In short, this thesis proves the usability of the de Sitter interpolation on globular clusters approximated as isothermal spheres, but further research is required to prove or disprove its superiority - in terms of its ability to match the observations - to the standard function.

The long-term simulation of planetary systems poses significant challenges due to the inherently chaotic and non-integrable nature of gravitational interactions in the N-body problem. This thesis examines the Wisdom–Holman symplectic integration scheme, a method specifically designed for nearly integrable systems. This scheme separates the dominant Keplerian motion from weaker perturbative forces, enabling stable integration over astronomical timescales. Emphasis is placed on understanding the practical limitations and capabilities of this method when using large time steps, particularly in the presence of mean-motion resonances and step-size resonances. Through extensive numerical experiments using the Rebound simulation package, the scaling behavior of integration errors is characterized, revealing a transition from secondorder to lower-order error regimes at large time steps. This shows that in certain systems, time steps significantly larger than the shortest orbital period can still yield acceptable accuracy. However, in systems with mean-motion resonance, strong sensitivity to step-size resonances is observed, requiring careful step-size selection. A comparison between Jacobi and Democratic Heliocentric coordinates shows that the former performs best when orbits are nested, while the latter is better suited to systems with crossing or unordered orbits. These findings provide practical guidelines for applying Wisdom–Holman integration effectively across a range of dynamical regimes.

This research presents a comprehensive re-evaluation and improvement of map projection distortion analysis, building upon the foundational framework of Goldberg and Gott. The fundamental challenge in cartography lies in representing the Earth’s spherical surface on a flat plane, called map projection, a process that inevitably introduces distortions in area, shape, or distance. This study aims to refine the quantification of these distortions by employing an improved distance metric that accurately measures projected geodesics, utilising high-precision numerical methods with over 10 million sample points to minimise error, and introducing an unbiased total distortion score that eliminates the arbitrary normalisation of previous methods. The results identify the Winkel Tripel projection as the best-performing overall map based on the improved, unbiased total distortion score.

Furthermore, this work critically investigates the viability of using the fractal dimension of coastlines as a novel distortion metric, leading to a shift in analysis towards evaluating distortions specifically along coastal geometries. However, theoretical computations using fractal dimension show that it is unsuitable as a global distortion metric due to its invariance under the bi-Lipschitz transformations that define many map projections. By establishing a normalisation approach based on coastline arc length, this study provides a robust and resolution-independent framework for tailored cartographic evaluation. The findings underscore the sensitivity of distortion scores to globe orientation and establish a foundational methodology for future property-specific distortion analyses, ensuring a more reliable and application-focused approach to selecting map projections than previously done. For the specific application of representing coastlines with minimal distortion, the Azimuthal Equidistant projection is found to be the most fitting.

Quaternions prove to be a useful tool when determining rotation groups of regualr polytopes and other object in three and four dimensions. In this thesis it will be studied how the unit quaternions relate to the special orthogonal groups in three and four dimension. Thereafter, this theory is applied to the normal subgroups of the binary octahedral groups as well as the cube and its dual the octahedron. The normal subgroups of the binary octahedral group form 4-dimensional objects, for example the 4-orthoplex or 16-cell and the hyperdiamond or 24-cell.

Symmetries that keep the origin in place, consist of orientation preserving and orientation
flipping isometries, called rotations and reflections respectively. The special orthogonal group in n dimensions, containing all real n × n orthogonal matrices with positive determinant, is precisely the group containing all rotations in n dimensions. The map Φ : H_1→SO(3) defined by q → Φ_q, where Φ_q (p) = qpq^−1 for p∈H is a rotation of ImH ≅ R^3, is a two-to-one surjective homomorphism. The map Ψ : H_1 × H_1 →SO(4), defined by q →Ψ_(q_1,q_2), where
Ψ_(q_1,q_2) (p) = q_1pq_2^−1 is a rotation of H ≅ R^4, is a two-to-one surjective homomorphism.

The n-cube is the generalization of square and cube in general dimensions. Using Φ the binary octahedral group 2O can be established, which describes all quaternions that describe rotational symmetries of the cube and its dual the octahedron. 2O contains 48 unit quaternions, which is twice as many elements as the number of rotational symmetries of the cube and octahedron, due to the two-to-one relation of Φ.

The binary octahedral group 2O describes an object in four dimensions. Its normal subgroups do as well, the most interesting are Q_8, which describes a 4-orthoplex or 16-cell and has the hypercube or 8-cell as dual, and 2T, which describes a hyperdiamond or 24-cell. The binary rotation groups of these polytopes and the object that 2O itself describes, are studied in particular. These binary rotation groups can all be constructed in the same manner.
Write N ∈ {Q_8,2T,2O}, then 2O_N := {(q_1,q_2) ∈ 2O × 2O : q_1N = q_2N} ⊂ H_1 × H_1 is equal to the entire binary rotation group of N. This means that for the rotation group of N, denote this as O_N ⊂ SO(4), it holds that O_N ≅ 2O_N/{±(1, 1)}, where the isomorphism is given by the restriction of Ψ : H_1 × H_1→SO(4) to 2O_N. For the remaining normal subgroups {1} and {±1}, this construction only gives a subgroup of the entire binary rotation group.

Decoherence plays an important role in the evolution of quantum systems. In this thesis, we theoretically investigate the decoherence dynamics of two-dimensional spin- 1/2 structures coupled to an electron bath, with a focus on the influence of exchange and dipolar coupling. Using a combination of analytical derivations and numerical simulations, we explore the impact of interatomic distance, lattice symmetry, and external magnetic fields on quantum coherence. The study extends previous one-dimensional investigations to two-dimensional square, triangular, and hexagonal Ti-spin structures on an MgO substrate, where spin interactions can be manipulated at the atomic scale using scanning tunneling microscopy (STM). We analyse the eigenenergies and eigenstates of these systems and employ the Lindblad master equation to model decoherence processes. Our results show a competition between the Heisenberg exchange and the dipole-dipole coupling, with special attention to the intermediate regime. Additionally, we characterise spin relaxation and flipping dynamics, highlighting their dependence on structural variations. These insights contribute to the broader understanding of spin coherence in engineered quantum systems and provide a foundation for future experimental and theoretical studies in quantum nanoscience.

Novel types of superconductivity is a very active area of research. Heterostructures are a promising and recent avenue for finding these. This thesis explores a theoretical model of a ferroelectric heterostructure to investigate whether dipole-mediated electron interactions can induce superconductivity. Two models were analyzed: a static case where the dipole aligns instantaneously with the net electric field and a dynamic case where the dipole has inertia and oscillates. The static model showed that under small lattice constants, dipole mediated interactions can overcome Coulomb repulsion between electrons, though such conditions are challenging to achieve. The dynamic model, analyzed using Floquet theory, revealed resonance effects that amplify electron motion. While definitive superconductivity was not established, experimental validation through pressure-tuned lattice constants and resonance conditions could validate the model’s predictions. Future research could extend this framework to quantum mechanical dipoles and more complex materials, enhancing the understanding of superconductivity in engineered heterostructures.

The rotational velocities of stars at the edges of galaxies are higher than predicted by Newtonian dynamics. One theory that is capable of explaining this anomaly is called MOdified Newtonian Dynamics
(MOND). To further test this MOND theory, the wide binary test has been suggested. In the wide binary test, the stability of wide binary systems is tested. The MOND theory deviates significantly from the
Newtonian theory when accelerations are smaller than a0 = 1.2·10−10 m/s2 , which is present in wide binary systems. The MOND theory is non-linear, which introduces a non-trivial effect of the external field
(EF) of the Milky Way on the wide binary orbit. To perform the wide binary test, an iterative particle mesh N-body code was used with initial conditions taken from the Gaia mission database. Additionally, the EF was modeled in three ways, where the Newton potential adjustment model was found to be the most realistic. The results include nine simulations, split up into three models and three different regimes. The models include the Newton model, the MOND without EF model, and the MOND with EF model. The regimes consist of the Newtonian regime (ainternal > a0), the transition regime (ainternal ≈ a0), and the deep MOND regime (ainternal < a0). In the Newton model, only the Newtonian regime gave stable orbits. The MOND without EF model gave stable orbits for all regimes, and the MOND with EF model gave stable orbits only for the Newtonian regime. The MOND without EF gave stable wide binary orbits, whereas the other models did not. If additional measurements detect these wide binary systems, then the MOND without EF model gains credibility. The orbits of the Newton model and the MOND with EF model were similar, with only a small difference. This is because the strength of the external field is 1.6 · a0, which makes the external field dominant over the internal accelerations, making the simulation Newtonian-like. To determine which model is correct, additional long-term measurements of Gaia are required. Further research recommendations include improving the performance and accuracy of the code. This is done by better implementation of the fast Fourier transform, translating the code into a faster programming language than Python, using adaptive time steps, and reducing self-interaction. The most important extension to the model is to introduce the galactic tidal effect.

The next generation of telescopes will be able to directly measure the light curves of exoplanets. This thesis demonstrates the potential of using light curve analysis to study exoplanetary ring systems, with the goal of retrieving their albedo map and optical depth. This can improve our understanding of the composition, formation, and evolution of such systems. We construct a mathematical model that describes how light interacts with ring particles through absorption, transmission, and scattering. Applying this model to data of Saturn’s rings from high-resolution images by NASA's Cassini space probe, albedo maps for the rings are constructed and the optical depth is found to be mostly consistent with existing literature. We then derive an analytic formula for the light curve of an exoplanetary ring system. This formula is converted into a numerical linear transformation that links the albedo map of a ring system to its light curve. Using the Moore-Penrose pseudoinverse and singular value decomposition this transformation is inverted, enabling the retrieval of the albedo map from the light curve. The method is tested using synthetic data of model ring systems while varying the axial tilt, observer direction, and signal-to-noise ratio from artificial Gaussian white noise. The results show that the method is generally effective at recovering the albedo map and optical depth for signal-to-noise ratios down to 10³.

We have defined the infinity boxes in general dimensions, which are mathematical objects that allow rays of light to continuously reflect. When the collision points of these rays are are positioned where they would appear to the viewer (along the inititial ray), then the mirror pattern will be revealed. These mirror patterns can be used to create mirror-tilings of two- and three-dimensional Euclidian space. We have answered the question whether a curved infinity box can create patterns that are nearly identical to exact tilings of Euclidian space. To obtain the answer, we set up the fundamental theory of rays and infinity boxes, and computed the radii of the curved mirror faces that exhibit certain properties. These properties are the alignmnent of internal angles, and the alignment of image points in an infinity box. We have simulated infinity boxes in two- and three-dimensions to visualise the patterns from these infinity boxes curved with these radii. Furthermore, we connected the theory of infinity boxes the theory of dynamical billiards by stating that infinity boxes visualise solutions of dynamical billiards. Lastly, we analysed the connection between the dodecahedron infinity box, and concluded that curving the mirrors of the dodecahedron infinity box results in a slight approximation of the interior of a stereographically projected 120-Cell. All the numerical methods used to visualise infinity boxes have been thoroughly explained, and the computer program that we created called MiRai, is available over the internet.

This paper aims to improve the Particle Mesh (PM) method for Newtonian Dynamics (ND) and MOdified Newtonian Dynamics (MOND). The PM method involves discretizing the mass density on a grid using Gaussian smoothing, which is used to calculate the gravitational potential using Fast Fourier Transforms (FFTs). A major downside of Gaussian smoothing is that the mass density of two particles overlaps whenever they are in close proximity, which leads to incorrect orbital dynamics. Hence, the linearity of ND is used to correct the accelerations of particles whenever this overlap in mass density occurs. These short-range corrections involve the use of analytical expressions to eliminate the inaccurate influence of the PM method, correcting the interactions between particle pairs to direct particle-particle interactions. These corrections can introduce instabilities whenever two particles are too close in proximity and the time steps are too large. This issue is mitigated by using a softening parameter in the corrections. In MOND, the linearity in the accelerations is removed, which means that these short-range corrections cannot always be used. They may only be used in the limit of high accelerations, for which MOND reduces to ND. The short-range corrections are tested for circular and elliptical two-body systems, showing a significant reduction in error of their expected orbital trajectories. To test the limitations of the PM method under non-linear accelerations in MOND where no corrections may be applied, a wide binary system is used to determine the minimum distance between particles that still yields somewhat accurate. This also demonstrates that without the use of short-range corrections large grids are required, highlighting a major inefficiency of PM methods. Lastly, the corrections are tested in both ND and MOND using the Plummer and Miyamoto-Nagai models, which represent a sphere and a disk, respectively. The Plummer model showed significant improvements in stability in both ND and MOND. However, the Miyamoto-Nagai model shows less optimistic results in MOND due to the difficulty in placing the particles at a minimum separating particles whenever the accelerations are non-lineair. Overall, careful use of these short-range corrections allows for significant improvements in the PM method for both ND and MOND. Further research into optimizing the computational performance of FFTs and short-range corrections using tree codes is still available. Additionally, a time-adaptive PM can be implemented to prevent possible instabilities due to large time steps when using the corrections.

Maps, which have been in used for centuries, are constructed by map projections. A map projection is a transformation of the latitudes and longitudes positions on the globe to the x and y coordinates on a flat map. There are probably over a hundred of different map projections, some more practical than others. A perfect map is impossible to create, so it is essential to determine the next best alternative. The different maps can be classified in five different categories: Conformal, Equal-area, compromise, equidistant and true-direction. To evaluate this, various distortions are considered. The six distortions defined in this thesis are: area (A), isotropy (I), flexion (F), skewness (S), distance (D), and boundary cut (B). These distortions together form a comprehensive set of all distortions that occur due to map projections. Since these distortions are quite difficult to compute directly, they are numerically approximated. To achieve this, the data points of the coordinates need to be sampled, which can be done through various methods. This thesis examines systematic generation and random sampling of points. Additionally, it investigates how many points are needed to achieve a sufficiently homogeneous covering of the globe and a stable solution for the distortion, this is achieved from around 20.000 point on the globe. These methods are then applied to different map projections, including the Mercator, Equirectangular, Winkel Tripel, Gott-Wagner, and Azimuthal Equidistant and the Azimuthal Equidistant split into two-hemispheres.. Among these map projections, the Gott-Wagner projection has the lowest total distortion value and is therefore considered the best overall map projection. However, each distortion has its own advantages and disadvantages, so it is not possible to definitively declare one map projection as the best in all scenarios. Selecting the best map projection depends on minimising the distorting specific to its needs.

This bachelor's thesis explores the realm of map projections, aiming to establish a systematic approach for evaluating and ranking them based on angular and area distortions. Beginning with a definition of map projections, the study progresses to quantify the distortions that occur when projecting a spherical surface onto a flat one. Using these distortions, various maps are assessed; for each, figures of the map are displayed and a short summary of the map's features and the angular and area distortion are given. These distortions are also visualised in figures. To rank the map projections, distortion numbers are defined and refined throughout the study. The findings reveal a ranked list of map projections, providing a structured basis for selecting appropriate projections in cartographic endeavours. Through mathematical analysis, this thesis contributes to the understanding and evaluation of map projections, offering valuable insights for cartographers and researchers in the field.

Ever since their discovery, the Trojans have raised many questions among astrophysicists. In 1989 it was found that there were more L4 than L5 Trojans and the current L4:L5 ratio value is estimated to be 1.6. However, this asymmetry could not be explained by Jupiter’s current orbit and must therefore have arisen during the early development of the Solar System. It was suggested that an outward migration as described by the Nice model could cause the asymmetry. To investigate this, we extended a recent research by modelling an outward migration of Jupiter on the actual Trojans and their symmetric copies. In this manner, we had a broad initial Trojan distribution, which allowed us to investigate several effects of initial values like the Trojan inclinations and maximum angular deviations from the Lagrange point. The simulations used a recently found midpoint Yoshida integrator, which allowed fourth order symplectic simulations for our non-separable Hamiltonian problem. From the simulations it followed that the ratio could only be explained if the Trojans initially had large angular deviations from the Lagrange point, which is possible if they later lost energy due to for example mutual collisions. Also the small initial inclinations as predicted for the early Trojans would in general have risen due to the migration, but it could not explain the the inclinations up to 40 degrees for today Trojans, implying that the increase in inclination must have been caused by other reasons. We found that not only the total migration distance and duration are important, but also the function that describes the evolution of Jupiter’s semi-major axis and the eccentricity of Jupiter’s orbit. For further research we suggest to use data sets on those quantities from simulations with the Nice model, to investigate these related problems together. We believe that our model and code is of great advantage for such a research, because of its optimisations and it allows to implement the data on the distance and eccentricity directly into the model without having to alter the equations of motion.

There has been interesting research on the superconducting diode recently. A possible structure for this is the Superconductor/Ferroelectric/Superconductor (S/FE/S) Josephson junction. Our goal is to simulate a S/FE/S junction with a magnetic field and see if this leads to a superconducting diode effect. First this junction and its properties are studied through a numerical simulation. Then a magnetic field is applied and we look at how this affects the junction. For our simulation we use the Velocity Verlet method. We start by simulating a normal RCSJ. The results agree with what we expect from the theory, however there is small but noticeable error which we attribute to the numerical method. Next a S/FE/S junction is simulated. The polarisation of the ferroelectric only affects the RCSJ when it oscillates at a frequency which is close to the plasma frequency. If we apply a magnetic field we see small gaps for the retrapping current however these are close to the numerical error thus they are to small too be detected by our simulation.

Humans are efficient at moving due to their exceptional mastery of bipedal locomotion. Several models have been made that attempt to model the motion of the centre of mass with a spring-mass system with various degree of success. For example, a two dimensional model tracks the height of the centre of mass well and a three dimensional model explains the normal force exerted on the ground.

In this report a three dimensional model is constructed to describe the motion of the centre of mass, with the purpose to determine the spring constant and the rest length in a simple prosthetic leg. The research question is: can a three dimensional spring-mass model accurately track the centre of mass and can this be used to determine the optimal properties of the spring of a simple prosthetic leg?

The spring mass model consists of two springs connected to the ground and the centre of mass. The springs do not exert a force on the centre of mass if they are extended. The model incorporates steps by moving the ground connection points instantaneously over fixed points on a rail. The model can be used in two distinct ways. The first generates a periodic trajectory by finding the optimal initial positions y₀ and z₀ (PPFA) and the second finds the optimal spring parameters k and u to fit a data set (DFA). The optimal conditions are found by discretizing the parameter space and using an iterative process. The space around the best parameters becomes the parameter space for the next iteration.

The PPFA shows that the initial lateral displacement is nearly constant with respect to the spring constant when the rest length is chosen such that the centre of mass is at constant height if the model is stationary. The PPFA also shows that there are two distinct y-trajectories possible for walking. Running, however, has only one trajectory. Several versions of the model are fit to the data: only discretised step widths, discretised step widths and a x₀-offset, and distinct legs. There is not much difference between these models. The x₀-offset is only Δs = -0.02m confirming that the position of the feet does align with the extreme values of the y- and z-position. The model fits the general characteristics well but fine details in the motion are lost. Future research should investigate how cumbersome these deviations of the data are perceived by people using a simple prosthetic.

The rotational velocities of stars in galaxies indicate that either Newtonian gravity breaks down on this scale, or that there is extra mass in the universe that has not been observed. The former is known as Modified Newtonian Dynamics (MOND), whereas the latter option is known as the dark matter paradigm. In this thesis a version of MOND is considered where the field equation for gravity is modified and becomes nonlinear. To simulate MOND, new N-body codes are needed due to the field equation not being linear anymore and in this thesis a new particle mesh method is developed. This new particle mesh method follows the same steps as the normal particle mesh method, except at the step where the acceleration field on the grid is calculated. This step is replaced by an iterative method to find the MONDian acceleration field on the grid. The code was then tested in the deep MOND regime using four cases of which analytical solutions of the MOND Poisson equation are known. These cases are: linear motion, the two-body problem, a ring system and an isothermal sphere. Using these test cases, numerous conclusions were drawn. From the linear motion test it was found that in the code bodies can interact with themselves, resulting in non-physical accelerations. The two-body test showed that the magnitude of the relative error in the force as calculated by the particle mesh code is roughly on the same order as the same error when calculating the Newtonian force using a particle mesh code. Furthermore, all tests showed that energy is conserved quite well, even though momentum need not be conserved due to the self interactions, and that generally the conservation of angular momentum is violated by the code. The tests also showed that the analytical formulae that were derived for these systems gave largely the same predictions as the algorithm, hence verifying both the formulae as the algorithm. The code could be sped up by writing it in C, the accuracy can be improved by reducing self-interactions and it can be extended by adding a method to handle close encounters. Some applications of the code are given, including: simulating the solar system, wide binary stars, tidal dwarf galaxies and galaxy clusters.

This research investigates the impact of gravitational scatterings caused by close encounters between particles in an N-body Kepler system, addressing three main questions: (1) the influence of scatterings on system evolution, (2) the correspondence between simulated and expected average times between scatterings, and (3) the effect of increasing different parameters individually on the average scattering time. Simulations demonstrate an average scattering angle of 15.2 degrees for particles involved in the top 10 percent of scatterings. This would indicate a non-negligible impact of gravitational scatterings, especially for systems with heavier bodies. The results indicate that the simulated average time between scatterings is higher than the expected average, necessitating further research for accurate estimation. Moreover, the time between scatterings decreases over time, before reaching a stationary state after roughly 300 scatterings. On this domain, the correlation coefficient between the scattering time and the scattering counter was found to be  -0.08. By varying the test domains for different parameters, a new expression for the expected time between two scatterings is proposed based on simulation data. A clear connection was found between the scattering time and the number of particles, the maximum orbital radius and the maximum inclination angle. The study acknowledges limitations, including the non-stationary initialization state and linear approximations to most computations, suggesting avenues for future improvement. Overall, this research aims to find the role of gravitational scatterings in Kepler systems and underscores the need to consider these interactions, which are now often considered to be negligible.