I am sure this is a duplicate but I could not spot it exactly. And I am sure folks have covered this topic online here in great detail. I am referring to the Lagrangian here in the "Action" formula and some textbooks express it as S.

I can see why in the action formula the Lagrangian is subtracted but is it just a matter of convention to have it the kinetic energy minus the potential energy ? Or is there some fundamental reason I am missing?

I can see the principle work just fine by subtracting the kinetic energy from the potential energy instead of the reverse.

It's the balance that the functional is looking for so who starts on top when you subtract seems to me to make little difference since you are honing in on the actual path which is a balance that gives neither kinetic or potential energy the advantage.

Using a parabola as an example at the top there would be no motion but plenty of kinetic energy and at close to the bottom you have low potential energy buy there would not be enough kinetic energy to get to the other side and it would drop.


An overall sign of the action $S$ doesn't change the EL equations, and is therefore immaterial in classical physics.

One benefit of the standard convention $L=T-V$ (rather than the opposite) is that the Lagrangian becomes positive definite in the free case $V=0$.

See also this related Phys.SE post.


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