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This report provides an introduction to geometric mechanics, which seeks to model the behavior of physical mechanical systems using differential geometric objects. In addition to its elegance as a method of representation, this formulation also admits the application of powerful analytical techniques from geometry as an aid to understanding these systems. In particular, it reveals the fundamental role that symplectic geometry plays in mechanics (something which is not at all obvious from the traditional Newtonian formulation), and in the case of systems exhibiting symmetry, leads to an elucidation of conservation and reduction laws which can be used to simplify the analysis of these systems. The contribution here is primarily one of exposition. Geometric mechanics was developed as an aid to understanding physics, and we have endeavored throughout to highlight the physical principles at work behind the mathematical formalism. In particular, we show quite explicitly the entire development of mechanics from first principles, beginning with Newton's laws of motion and culminating in the geometric reformulation of Lagrangian and Hamiltonian mechanics. Self-contained presentations of this entire range of material do not appear to be common in either the physics or the mathematics literature, but we feel very strongly that this is essential in order to understand how the more abstract mathematical developments that follow actually relate to the real world. We have also attempted to make many of the proofs contained herein more explicit than they appear in the standard references, both as an aid in understanding and simply to make them easier to follow, and several of them are original where we feel that their presentation in the literature was unacceptably opaque (this occurs primarily in the presentation of the geometric formulation of Lagrangian mechanics and the appendix on symplectic geometry). Finally, we point out that the fields of geometric mechanics and symplectic geometry are vast, and one could not hope to get more than a fragmentary glimpse of them in a single work, which necessiates some parsimony in the presentation of material. The subject matter covered herein was chosen because it is of particular interest from an applied or engineering perspective in addition to its mathematical appeal. / text

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.

Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found. We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

In studies of the ocean it has become conventional to retain only the component of the Coriolis force associated with the radial component of the Earth’s rotation vector, the so-called “traditional approximation”. We investigate the role of the “non-traditional” component of the Coriolis force, corresponding to the non-radial component of the rotation vector, in transporting abyssal waters across the equator. We first derive a non-traditional generalisation of the multi-layer shallow water equations, which describe the flow of multiple superposed layers of inviscid, incompressible fluid with constant densities over prescribed topography in a rotating frame. We derive these equations both by averaging the three-dimensional governing equations over each layer, and via Hamilton’s principle. The latter derivation guarantees that conservation laws for mass, momentum, energy and potential vorticity are preserved. Within geophysically realistic parameters, including the complete Coriolis force modifies the domain of hyperbolicity of the multi-layer equations by no more than 5%. By contrast, long linear plane waves exhibit dramatic structural changes due to reconnection of the surface and internal wave modes in the long-wave limit. We use our non-traditional shallow water equations as an idealised model of an abyssal current flowing beneath a less dense upper ocean. We focus on the Antarctic Bottom Water, which crosses the equator in the western Atlantic ocean, where the bathymetry forms an almost-westward channel. Cross-equatorial flow is strongly constrained by potential vorticity conservation, which requires fluid to acquire a large relative vorticity in order to move between hemispheres. Including the complete Coriolis force accounts for the fact that fluid crossing the equator in an eastward/westward channel experiences a smaller change in angular momentum, and therefore acquires less relative vorticity. Our analytical and numerical solutions for shallow water flow over idealised channel topography show that the non-traditional component of the Coriolis force facilitates cross-equatorial flow through an almost-westward channel.

The future fusion power plants that are based on magnetic confinement will deal with plasmas that inevitably contain energetic (non-thermal) particles. These particles come, for instance, from fusion reactions or from external heating of the plasma. Ensembles of energetic ions can excite eigenmodes in the Alfvén frequency range to such an extent that the resulting wave fields redistribute the energetic ions, and potentially eject them from the plasma. The redistribution of ions may cause a substantial reduction of heating efficiency. Understanding the dynamics of such instabilities is necessary to optimise the operation of fusion experiments and of future fusion power plants. Two models have been developed to simulate the interaction between energetic ions and Alfvén eigenmodes. One is a bump-on-tail model, of which two versions have been developed: one fully nonlinear and one quasilinear. The quasilinear version has a lower dimensionality of particle phase space than the nonlinear one. Unlike previous similar studies, the bump-on-tail model contains a decorrelation of the wave-particle phase in order to model stochasticity of the system. When the characteristic time scale for macroscopic phase decorrelation is similar to or shorter than the time scale of nonlinear wave-particle dynamics, the nonlinear and the quasilinear descriptions quantitatively agree. A finite phase decorrelation changes the growth rate and the saturation amplitude of the wave mode in systems with an inverted energy distribution around the wave-particle resonance. Analytical expressions for the correction of the growth rate and the saturation amplitude have been derived, which agree well with numerical simulations. A relatively weak phase decorrelation also diminishes frequency chirping events of the eigenmode. The second model is called FOXTAIL, and it has a wider regime of validity than the bump-on-tail model. FOXTAIL is able to simulate systems with multiple eigenmodes, and it includes effects of different individual particle orbits relative to the wave fields. Simulations with FOXTAIL and the nonlinear bump-on-tail model have been compared in order to determine the regimes of validity of the bump-on-tail model quantitatively. Studies of two-mode scenarios confirmed the expected consequences of a fulfillment of the Chirikov criterion for resonance overlap. The influence of ICRH on the eigenmode-energetic ion system has also been studied, showing qualitatively similar effects as seen by the presence of phase decorrelation. Another model, describing the efficiency of fast wave current drive, has been developed in order to study the influence of passive components close to the antenna, in which currents can be induced by the antenna generated wave field. It was found that the directivity of the launched wave, averaged over model parameters, was lowered by the presence of passive components in general, except for low values of the single pass damping of the wave, where the directivity was slightly increased, but reversed in the toroidal direction. / De framtida fusionskraftverken baserade på magnetisk inneslutning kommer att hantera plasmor som oundvikligen innehåller energetiska (icke-termiska) partiklar. Dessa partiklar kommer exempelvis från fusionsreaktioner eller från externa uppvärmningsmekanismer av plasmat. Ensembler av energetiska joner kan excitera egenmoder i Alfvén-frekvensområdet i en sådan utsträckning att de resulterande vågfälten omfördelar de energetiska jonerna i rummet, och potentiellt slungar ut jonerna ur plasmat. Omfördelningen av joner kan orsaka en väsentligen minskad uppvärmningseffekt. Det är nödvändigt att förstå dynamiken hos denna typ av instabilitet för att kunna optimera verkningsgraden hos experiment och hos framtida fusionskraftverk. Två modeller har utvecklats för att simulera interaktionen mellan energetiska joner och Alfvén-egenmoder. Den första är en bump-on-tail-modell, av vilken två versioner har utvecklats: en fullt icke-linjär och en kvasi-linjär. I den kvasi-linjära versionen har partiklarnas fasrum en lägre dimensionalitet än i den icke-linjära versionen. Till skillnad från tidigare liknande studier innehåller denna bump-on-tail-modell en dekorrelation av våg-partikelfasen för att modellera stokasticitet hos systemet. När den karakteristiska tidsskalan för makroskopisk fasdekorrelation är ungefär samma som eller kortare än tidsskalan för icke-linjär våg-partikeldynamik så stämmer den icke-linjära och den kvasi-linjära beskrivningen överens kvantitativt. En ändlig fasdekorrelation förändrar vågmodens tillväxthastighet och satureringsamplitud i system med en inverterad energifördelning omkring våg-partikelresonansen. Analytiska uttryck för korrektionen av tillväxthastigheten och satureringsamplituden har härletts, vilka stämmer väl överens med numeriska simuleringar. En relativt svag fasdekorrelation försvagar även "frequency chirping events" (snabba frekvensskiftningar i korttids-Fourier-transformen av egenmodens amplitudutveckling) hos egenmoden. Den andra modellen, kallad FOXTAIL, har ett mycket bredare giltighetsområde än bump-on-tail-modellen. FOXTAIL kan simulera system med flera egenmoder, och den inkluderar effekter av olika enskilda partikelbanor relativt vågfälten. Simuleringar med FOXTAIL och med bump-on-tail-modellen har jämförts för att kvantitativt bestämma bump-on-tail-modellens giltighetsområde. Studier av scenarier med två egenmoder bekräftar de förväntade effekterna av när Chirikov-kriteriet för resonansöverlapp uppfylls. Även inflytandet av ICRH på dynamiken mellan egenmoder och energetiska joner har studerats, vilket har visat kvalitativt liknande effekter som har observerats i närvaron av fasdekorrelation. En annan modell, vilken beskriver effektiviteten hos "fast wave current drive" (strömdrivning med snabba magnetosoniska vågor), har utvecklats för att studera inflytandet av passiva komponenter nära antennen, i vilka strömmar kan induceras av vågfälten som genereras av antennen. Det visades att den utskickade vågens direktivitet, medelvärdesbildat över modellparametrar, generellt sett minskade vid närvaron av passiva komponenter, förutom vid låg "sinlge pass damping" (dämpning av vågen vid propagering genom hela plasmat), då direktiviteten istället ökade något, men bytte tecken i toroidal riktning. / <p>QC 20160927</p>

The aim of the presented thesis is to create a compact publication which deals with properties, solution and examination of behavior of dynamical systems as models of mechanical structures. The opening portion of the theoretical part leads the reader through the subject of description of dynamical systems, offers solution methods and investigates solution stability. As the introduction proceeds, possible forms of structure loading, damping and response are presented. Following chapters discuss extensively the possible approaches to system behavior observation and identification of nonlinear and chaotic phenomena. The attention is also paid to displaying methods and color spaces as these are essential for the examination of complex and sensitive systems. The theoretical part of the thesis ends with an introduction to fractal geometry. As the theoretical background is laid down, the thesis proceeds with an application of the knowledge and shows the approach to numerical simulation and study of models of real structures. First, the reader is introduced to the single pendulum model, as the simplest model to exhibit chaotic behavior. The following double pendulum model shows the obstacles of observing systems with more state variables. The models of free rod and cantilever serve as examples of real structure models with many degrees of freedom. These models show even more that a definite or at least sufficiently relevant monitoring of behavior of such deterministic systems is a challenging task which requires sophisticated approach.

In the present work, we introduce a never before studied soliton equation called the intermediate mixed Manakov (IMM) equation. Through a pole ansatz, we prove that the equation has N-soliton solutions with pole parameters governed by the hyperbolic Calogero-Moser system. We also show that there are spatially periodic N-soliton solutions with poles obeying elliptic Calogero-Moser dynamics. A Lax pair is given in the form of a Riemann-Hilbert problem on a cylinder. A similar Lax pair is shown to imply a novel spin generalization of the intermediate nonlinear Schrödinger equation. Some conservation laws for the IMM are proven. We demonstrate that the IMM can be written as a Hamiltonian system, with one of these conserved quantities as the Hamiltonian. Finally, a physical interpretation is given by showing that the IMM can be rewritten to describe a system of two nonlocally coupled fluids, with nonlinear self-interactions. / Vi presenterar en aldrig tidigare studerad solitonekvation som vi döper till ‘the intermediate mixed Manakov equation’ (ungefär ‘den mellanliggande kopplade Manakovekvationen’. Kortform: IMM). Genom en polansats bevisar vi att ekvationen har N-solitonlösningar där polparametrarna utgör ett hyperboliskt Calogero-Mosersystem. Vi visar också att det finns rumsligt periodiska N-solitonlösningar vars poler följer elliptisk Calogero-Moserdynamik. Ett Laxpar ges i form av ett Riemann-Hilbertproblem på en cylinder. Vi demonstrerar att ett liknande Laxpar leder till en ny spinngeneralisering av den s.k. INLS-ekvationen. Några bevarandelagar för IMM bevisas. Vi visar att IMM-ekvationen kan skrivas som ett Hamiltonskt system, där Hamiltonianen är en av våra tidigare bevarade storheter. Till sist ger vi en fysikalisk tolkning av vår ekvation genom att demonstrera hur den beskriver ett system av ickelokalt interagerande vätskor, med ickelinjära självinteraktioner.

Experiments for the development of fusion power that are based on magnetic confinement deal with plasmas that inevitably contain energetic (non-thermal) particles. These particles come e.g. from fusion reactions or from external heating of the plasma. Ensembles of energetic ions can excite plasma waves in the Alfvén frequency range to such an extent that the resulting wave fields redistribute the energetic ions, and potentially eject them from the plasma. The redistribution of ions may cause a substantial reduction heating efficiency, and it may damage the inner walls and other components of the vessel. Understanding the dynamics of such instabilities is necessary to optimise the operation of fusion experiments and of future fusion power plants. A Monte Carlo model that describes the nonlinear wave-particle dynamics in a toroidal plasma has been developed to study the excitation of the abovementioned instabilities. A decorrelation of the wave-particle phase is added in order to model stochasticity of the system (e.g. due to collisions between particles). Based on the nonlinear description with added phase decorrelation, a quasilinear version of the model has been developed, where the phase decorrelation has been replaced by a quasilinear diffusion coefficient in particle energy. When the characteristic time scale for macroscopic phase decorrelation becomes similar to or shorter than the time scales of nonlinear wave-particle dynamics, the two descriptions quantitatively agree on a macroscopic level. The quasilinear model is typically less computationally demanding than the nonlinear model, since it has a lower dimensionality of phase space. In the presented studies, several effects on the macroscopic wave-particle dynamics by the presence of phase decorrelation have been theoretically and numerically analysed, e.g. effects on the growth and saturation of the wave amplitude, and on the so called frequency chirping events with associated hole-clump pair formation in particle phase space. Several effects coming from structures of the energy distribution of particles around the wave-particle resonance has also been studied. / <p>QC 20150330</p>