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The field equations come from differentiating the lagrangian. See https://physics.stackexchange.com/questions/3005/derivation-... for example. Youll need to read a bit on calculus of variations. Usually in university this is gently introduced through the lagrangian formalism of classical mechanics

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Ah thank you!! I've been searching for a straightforward derivation like that for a while. Maxwell's equations are familiar ground for me, so that is very helpful.

Well so this is for classical fields, where the standard model is a quantum field theory. So it will effectively follow the Schrodinger equation with the corresponding Hamiltonian from this Lagrangian. There are a lot of complications though, so I don't believe you will be able to just plug the Standard Model Hamiltonian in like this. Renormalization complicates things. You can do numerical simulations for certain parts of the Standard Model, specifically quantum chromodynamics (QCD), using lattice QCD, but that's not something I have experience with, so I can't speak to how easy that is to implement.

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