〈 Berger | Dillon 〉

It is a consequence of the convention F = - dU/dx. That minus sign is what caries through to make the minus sign in L = T-U.

right but the *relative* minus sign, no matter where it goes, is physical. That's the empirical fact i'm talking about. The fact that nature wants to "go downhill", as it were.

That is our convention of what "down" means.

yes, but it's the fact that nature chooses one thing over the other no matter what we call it, mathematically represented as a minus sign

i.e. define "down" as the thing nature chooses , call it plus or minus or what you want it doesn't matter. But we need to distinguish between them somehow mathematically. But this i argue is physical content.

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

Wait for real the minus sign there has always been “magic” to me in mechanics