Problems with physics self-study: completeness

[TomCurious]

Hello, I am looking for some advice on self-studying physics.

For a long time, I have essentially followed Ian Stewart's approach to learning - namely, flick through a textbook until you find something interesting, and read as much as you need to in order to understand it. In such a way, I have acquired a store of physics knowledge and tools. While I may not be 'mathematically mature,' I nonetheless have a good command of mathematics, at the level of 'mathematical methods for scientistis and engineers,' with some more PDE's, complex analysis, abstract algebra, and riemannian geometry.

My main worry is that I am not developing a complete and thorough understanding of the physics I am studying. Indeed, I recently talked to a professor, who pointed out that I had an incomplete understanding of the equivalence principle (in the mathematical sense of the local vanishing of the christoffel symbols)

Perhaps this would be better clarified with some examples (note, again, that this is self study: I am not enrolled in any advanced undergraduate programs at the moment):

1. Classical Mechanics: I understand Functional Analysis, Hamiltonian Theory, Nambu mechanics, Hamilton Jacobi Equation, classical field theory, Central force and two body problem,
missing: The majority of Rigid Body Motion, Representations of Lie Groups, chaotic motion, perturbation theory

2. Quantum Mechanics: I understand Wave and Matrix Mechanics, Propagators and Path Integrals (evaluated via steepest descent), relation to projective geometry, density operator formalism, entanglement
missing: time-independant perturbation theory, rigorous treatment of bound states, WKB method,

By understand, I mean I can solve problems in a standard textbook on the subject, and am able to explain the material to someone who is unfamiliar with it.

I am simply in love with all this material and would like to learn it as fast as possible. Do you have any advice on how to clear my understanding (specifically, in high energy physics, gravitation, atomic, particle, and nuclear physics, as well as fluid dynamics and geophysics)

Thanks a bunch!

 

[R.Mesri]

My advice would be to start by finding a good textbook in the area you are interested in, and to work through it in detail. Make sure that you understand all of the concepts and equations, and take notes as you go. As you progress, try to find worked examples or videos online that break down the solutions to more difficult problems. You can also look for online lectures, tutorials, or courses that cover the topics you are interested in. This will help you test your understanding and practice problem solving. Additionally, you could try reading papers in the field and trying to replicate their results. This can help you learn the techniques used to solve problems, and can help you gain insight into the research methods used in your field. Finally, don’t forget to ask questions when you are stuck!