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 …
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F = ± dU/dx would cause L = T ± U. Is convention and math. What forces are needed to explain any given observation is physics.
fair point , i do recall that you can derive this minus sign from the classical limit of relativity and it's essentially just the metric signature/2.. however i could also recall there being something else to worry about, that i can't recall at the moment
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Wait for real
the minus sign there has always been “magic” to me in mechanics -
this minus sign is turning out to be far more subtle and interesting than i had anticipated
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right i wasn't saying fundamental, i was saying that it's empirical .. but i think if we take m<0 instead then everything works out
End of conversation
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