I've seen many courses labelled as 'mathematical physics', but I'm interested in knowing about the opposite: 'physical mathematics'.
I've noticed that some areas of mathematics which I found extremely abstract and not interesting were directly inspired by or popularized by physics: functional analysis, differential geometry, symplectic geometry and contact structures, partial and ordinary differential equations.
I've noticed, however, that students often dislike interrupting a mathematical course to introduce some motivating physics. So my question is,
What topics could be included into a 3-credit hour, one semester graduate course dedicated to describing physical systems and the math which results from them? Specifically, is there a set of topics that would form a somewhat cohesive whole while allowing enough time to get to interesting math?
One specific example I've thought of is doing the first few chapters of Dirac's Principles of Quantum Mechanics to teach functional analysis from a physics viewpoint and combining this with the first three chapters of Wald's General Relativity to teach differential geometry from a physics viewpoint. However, these topics are very different, and their is probably enough material in those combined chapters to teach several courses. What topics would be interconnected but not contain too much material?