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Under the aalaniz blog, I just posted step-by-step notes starting for ODEs and PDEs both linear and nonlinear, of the type you will see in grad physics programs. The notes connect Lagrangian, Hamiltonian and Poisson Bracket methods in classical physics to the various approaches to quantum physics and quantum field theory. You'll see in detail how deeply symmetry methods underlie physics.

The basis of the notes are Lie symmetry methods. You will see that differential equations, abstract algebra, topology all go hand-in-hand towards practical methods for differential equations and mathematical physics.

I got a 36 hour MS in pure math and a PhD in theoretical physics in particles and fields. I've spent ten years dotting i's and crossing t's on techniques I never felt I truly understood. I never felt like an honest PhD as long as math seemed ad hoc and full of tricks. I finally feel honest. The material in the notes should serve as the foundations for grad physics (possibly grad math), but it has been forgotten. Schools now teach each subject in isolation, which is not good.

Give the notes a look. I hope they are useful.

Cheers,

Alex

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# Step-by-step mathematical physics from classical to quantum physics

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