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I'm a senior undergraduate student studying differential geometry. I have experience with smooth manifolds, some elementary theory of Lie groups, and a little multi-linear algebra. I understand this is all very "par for the course" for a first exposure to any serious amount of differential geometry.

It's come to my attention that a lot of the material I've encountered has interesting applications to mathematical physics, a discipline I know very little about. Since physics is interesting to me, I would like to dabble into these applications a little bit. So the function of this post is two-fold.

First, what kind of background material is necessary to really understand the applications of geometry to physics? The last real course in physics I had was at the Newtonian level. I've done some reading past that in fluids, and a little bit of electromagnetism, but that was just light reading.

Of course, with the background material covered, the next question is for a solid reference in physics that I could use. I find electricity and magnetism interesting and fluid dynamics interesting. I'm not so interested in mechanics, but if it's necessary I learn about it in more detail first, that's fine by me. I understand that field theories of any type eventually run up against quantum mechanics, which is also an area of thought I know relatively little about from a rigorous perspective.

I suppose I prefer a reference which is not too long, because I'll only have about 8 weeks between semesters to study from. If it isn't possible to get a good understanding in that time, then I can always continue with the material in the summer time. I also prefer an approach which is more geometric than analytic, if possible.

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In order to understand the applications of geometry to physics it is first of all necessary have a solid knowledge of geometry and physics! This means that you have to know quite well the subject and the tools you're planning to use. What does it mean in practice, it dependes on the level of formalism required. For example, Newtonian mechanics is satisfactory written in the language of vectorial analysis. At this level, more geometry is pretty unuseful. Lagrangian formulation of classical mechanics requires more geometry to be fully understood. Here, the main objects are a (differentiable) manifold associated to the system, its tangent space and a function defined on the tangent space satisfying Euler-Lagrange equations.

Even more geometry it's mandatory for the Hamiltonian formalism. This is in a sense a more symmetric reformulation of Lagrangian one. Suppose for the moment that we aim to describe autonomous systems. Then we need a symplectic manifold. Equations of motion can be found requiring the Lie derivative of the symplectic form identically vanish along the vector field associated to their integral curves. In the case the system is nonautonomous, symplectic geometry is not enough and contact structure theory come up into discussion.

This discussion is by no means complete. Its goal is to show that even for describing the same things one can use very different tools depending on what you are looking for. There are problems more suitably treated in Newtonian formalism, others in Lagrangian formalism and many others in Hamiltonian one.

For e.m. the situation is analagous. The simplest treatment can be done in terms of vector fields and functions defined on the physical space but a more appropriated one is in terms of manifolds (endowed with a metric) and differential forms defined on it.

I think that you might greatly appreciate the followings books:

  • B. Schutz, Geometrical methods of mathematical physics (especially Chapter 5 and references therein; reasonably short),
  • W. Thirring, Course of mathematical physics. (more difficult.)

Chapter 5 of Schutz's book deals with the following topics, rewritten in geometrical language: Thermodynamics, Hamiltonian mechanics, Electromagnetism, Fluid mechanics, Cosmology.

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  • $\begingroup$ I see. My buddy recommended me this text, which seems to go through some basic Lagrangian and Hamiltonian mechanics using calculus of variations. Would this be an okay starting point? I will look into the texts you posted above as well. amazon.co.uk/Variational-Principles-Mechanics-Dover-Physics/dp/… $\endgroup$ – Alfred Yerger Sep 2 '15 at 21:44
  • $\begingroup$ Looks pretty good, but I'd say, if you're familiar with differential geometry, that classical Arnold's text "Mathematical methods of classical mechanics" is better suited to your interests. Another classic book, that takes the geometrical point of view, is Abraham-Marsden-Ratiou "Foundation of mechanics" (other books by same authors are great as well) but it at very high(est) level and your time is limited. $\endgroup$ – Federico Sep 2 '15 at 22:01

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