The most fundamental principle in physics is called 'The Principle of Least Action'. It also happens to be a good principle to live by during these trying times
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We do though, although it hasn't been made very pedagogical. It's basically because we want to make this exact differential: dL = p dv + F dx, and also have the convention that F = - dU/dx https://physics.stackexchange.com/questions/86008/motivation-for-form-of-lagrangian … and https://web2.ph.utexas.edu/~mwguthrie/t.lagrangian.pdf …
i wouldn't call this an a priori reason though. it just seems to say that it's because it returns the right EOM, but what the 'right' EOM are is an empirical fact
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I actually think it is a kinda of a priori reason in a way because I think this really is implicitly defining what we mean by "force".
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hmm i think i disagree with you on this one about this being a purely mathematical statement.
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