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Classical Mechanics and Field Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
We are now ready to frame Liouville’s theorem in a different manner. The first thing we must do is to define what is meant by the phase space volume. After all, our previous discussion on Liouville’s theorem essentially regarded all phase space variables yr as coordinates in a Euclidean space when equating an incompressible flow with the divergence of the flow field. If the phase space is 2N-dimensional (to show that a symplectic manifold must be of even dimension is left as Problem 10.54), then we can define the phase space volume formη as () η=ωN,
A geometric look at MHD and the Braginsky dynamo
Published in Geophysical & Astrophysical Fluid Dynamics, 2021
Andrew D. Gilbert, Jacques Vanneste
Although all this machinery works beautifully and is well established in the literature, together with applications, we argue that it is only by stepping back and considering some of these fluid dynamical systems in a more abstract setting that it is clear why these methods work, what is the origin of various transformation and conservation laws, and where choices can and cannot be made. For example working in Euclidean space with Cartesian coordinates and metric , it is too easy to switch between vector fields such as the velocity and 1-form fields such as the momentum , which have the same components (up to the factor of density ρ), even though they are very different objects in terms of their properties under transport. Working on a general manifold with a metric g and an induced volume form μ forces one to establish what type of object one is dealing with at the outset, and with this, theory can become easier and clearer, drawing on well-established results in differential geometry. The advantage of the use of differential geometry and particularly Lie derivatives has been stressed by several authors in MHD, including Tur and Yanovsky (1993), Webb (1994), Marsden and Shkoller (2001), Holm (2002b), Roberts and Soward (2006a, 2006b), Arter (2013a, 2013b) and Soward and Roberts (2014). Even working in , the use of a formulation based on a general metric g can avoid complications switching from Cartesian coordinates to, say, cylindrical polars, and is well-nigh essential if non-orthogonal coordinates are employed, for example when a coordinate labels surfaces of constant buoyancy in a geophysical fluid dynamical setting (Gilbert and Vanneste 2020b).