Can we visually see the principle of least action?

Visualizing the principle of least action is generally not possible.

But there is one particular case that lends itself quite well.

It is taken from, I don’t remember which part, of one of Feynman’s books of physics.

Let’s take, for example, a ball that falls from a certain height.

Given the Lagrangian L = K — U (where K is the kinetic energy, U the potential energy, and their difference the quantity called action) it can be imagined that the physical trajectory of the ball is given by a compromise between maximizing the energy potential and minimizing kinetic energy.

The ball must therefore spend most of its time close to where it started to maximize potential energy, which is greater when it is far from the ground.

On the other hand, it has to spend the rest of the time accelerating and thus acquiring kinetic energy. But if the ball spends too much time on top it has to accelerate a lot to get to the ground in the short time remaining, and doing so would increase its kinetic energy too much.

The best compromise is reached with a gradual acceleration. The ball therefore falls with a uniformly accelerated motion.

The chart is taken from the book I mentioned before, with the only difference being that it refers to the case where the ball is first thrown up from the ground and then falls back down. It is quantitatively different from the example I have selected, but it qualitatively and conceptually conveys the idea.