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Near the sample edge, or a sharp magnetic field step the drift of two-dimensional (2D) electrons in a magnetic field has th'e form of skipping/snake orbits. We show that families of skipping/snake orbits of electrons injected at one point inside a 2D metal generically exhibit caustics folds, cusps and cusp triplets, and, in one exceptional case, and extreme section of the butterfly bifurcation. Periodic appearance of singularities along the ±B-interface leads to the magneto-oscillations of non local conductance in multi-terminal electronic devices. We move onto propose a semi-classical theory that predicts two types .~ of oscillations in the flow of current injected from a point source near a ballistic p-n junction in graphene in a strong magnetic field. One originates from the classical effect of bunching of cyclotron orbits of electrons passing back and forth across the p-n interface, which displays a pronounced dependence on the commensurability between the cyclotron radii in the n- and p-regions. The other effect is caused by the interference of monochromatic electron waves in p-n junctions with equal carrier densities on the two sides and it consists in magneto-oscillations in the current transmission through the interface with periodicity similar to the Shubnikov-de Haas oscillations.

In recent history our understanding of the magnetic field and the generating motions of the molten iron in the Earths core have increased dramatically. The two major approaches to investigate fluid flow have been core surface flow modelling from magnetic data inversion and modelling of the dynamo itself. Core flow modelling involves downward continuation of the magnetic field to the core mantle boundary (CMB), then adopting the frozen flux approximation plus added assumptions e.g. tangential geostrophy to reduce non-uniqueness, to obtain fluid flow at the surface of the core that produces the observed secular variation (SV). The main check on the validity of these flows has been observed changes in length· of day. This study aims to test the fluid flow inversion more rigorously by using synthetic data of main field, SV and fluid flow from two self-consistent convection driven dipole dominated dynamos. The dynamo magnetic data are inverted and comparisons made with the true dynamo flow. The use of two large scale assumptions, the strong norm and the KE norm, has been tested. Forward models of advection, a neglected advection, and diffusion, from the dynamo data have been calculated to compare contributions to the secular variation. It is shown that within the dynamos the definition of the magnetic Reynolds number is flawed, relating to a failure of the frozen flux approximation. The effects of truncation of field and flow on the generated advection has been studied. It was found that both the failure of the frozen flux approximation and truncation had a large effect on the flow inversions. Another possible reason for non-recovery of some parts of the flow was found to be that much of the true and inverted flow was along contours of Br / cos 0, the null space caused by the geostrophic assumption. With reducing this non-uniqueness in mind, the validity of of a new assumption called helical flow was checked by studying the true properties of the dynamo flow. A new spectral helical flow constraint that can be applied separately to tangential geostrophy has been developed. With the caveat that the results have been found on dynamos with parameters very different to the Earth, cautious conclusions have been made on the best combinations of assumptions to use in Earth core flow models. It has been shown that, at the truncation of Earth models, when the new helical flow constraint is used with the KE norm and weak geostrophy more of the dynamo true flow has been recovered. The results have been applied to the Earth and validated by using changes in the length of day.

The search for new phases of quantum matter is at the heart of modern condensed matter physics, and frustrated magnets are one of the most promising classes of material in which to carry out this search. Each distinct phase can be thought of as its own 'universe', with its own set of unique, 'fundamental' excitations. A way of recognising unusual magnetic phases is by the experimental signature of their excitations, and measurement of the NMR T1 relaxation rate is a powerful method of probing these excitations. However, the dynamical information locked in NMR T1 measurements remains notoriously difficult to interpret. The difficulty arises from the fact that information about all possible low-energy spin excitations of the electrons, and their coupling to the nuclear moments, is folded into a single number, 1/T1 In this thesis we first develop a quantitative theory of T1, focusing on the specific example of the collinear antiferromagnet BaFe2As2. One of the most striking features of magnetism in BaFe2As2 is a strong dependence of 1/T1 on the orientation of the applied magnetic field, and we make convincing, quantitative fits to experimental data for different field orientations. This leads to the idea of 'angle-resolved' NMR. The quantum spin-nematic state - the magnetic analogue of the liquid crystal - is an unusual state of matter in which measurement of the T1 relaxation rate promises to be particularly revealing. Such a state has been proposed in the context of a number of magnetic insulators, including the quasi-two dimensional magnet iGa2S4, thin films of 3He, and the spin-chain system LiCuV04 in high magnetic field - but never yet observed in experiment. In all of these cases, the models studied predict an 'antiferroquadrupolar' order, in which spin fluctuations select perpendicular axes on neighbouring sites (or bonds) of the lattice. Progress in understanding these systems has been limited by the difficulty in performing calculations for any realistic microscopic model. With this in mind, we develop a phenomenological, field-theoretical description of anti- ferroquadrupolar spin-nematic order. The resulting action depends only on the symmetry of the order parameter, and so is applicable to a wide range of systems. Observation of the spin-nematic state is complicated by the fact that the order parameter does not break time-reversal symmetry, and is therefore 'invisible' to the tests commonly used to discern magnetic order. However, excitations of the spin-nematic state induce a fluctuating spin- dipole moment, and this can, in principle, be detected by dynamic probes of magnetism, including the NMR T1 relaxation rate. We make predictions for the 'fingerprint' of spin-nematic order in T1 measurements, and a particularly striking finding is the absence of a critical divergence at the onset of ordering. We also make predictions for the signature in inelastic neutron scattering experiments. These predictions could potentially lead to experimental verification of the long- elusive, spin-nematic state.

In this thesis, the results of time-resolved magneto-optical imaging experiments, micromagnetic simulations and analytical theory will be presented. These three approaches were used in order to understand how spin waves can be both excited and steered using magnetic non-uniformities (which gives rise to the graded magnonic index). The results presented in this thesis, as a whole, reveal that the graded magnonic index not only can be exploited either to excite or steer propagating spin waves in a deliberate manner, but is in fact a ubiquitous feature that needs to be taken into account when considering any dynamical phenomena in nano- and micro-magnetism.

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