>>14423203It's because when you're looking at paths that require the least effort in a certain sense, you actually want to minimise the difference between the kinetic and potential energy.
People often give shit explanations for this and just go dude geodesics lmao but they don't explain what it means because they weren't ever taught why that minus sign is there. Here is why.
Imagine you throw a ball along some path. Intuitively speaking we expect them to travel along fairly smooth paths from our real world experience. That means things like zipping around are not true paths of motion, and neither are paths where the ball travels up and then comes back down a little bit to hit the target, or came down in a really weird way.
How this manifests in the expression T - V being minimised is clear when V = 0, because in that case to minimise the integral over T means that you travel at a constant speed over the definite period of time which you're considering. This is because if you didn't travel at a constant speed, for some paths you might be going faster or slower, but to get from A to B in a fixed time interval, all paths must have the same average speed. Since the square of the mean is always bounded above by the mean square, you can only minimise this for constant speeds when V = 0. So this trivial case is intuitively clear.
Now when V is non zero we know from experience that this should manifest as something like a force, and this means your mass should travel faster in some places. For example, your ball should travel upwards more quickly and then slow and fall.
Now things are slightly different, but only slightly, because we know that there will be a point where the potential energy will be greatest (or least). So again, from experience, we expect that the true path follows a path where it quickly goes where potential energy increases. Also, as potential energy increases, this reduces the kinetic energy. This last part is important, because you want T - V.
