Principle of least Action

Fermat's Principle

When the light moves from point A to point B, it chooses the path that requires the least time. It seems that the light “appreciates” its time.


The most fundamental principle in modern physics is the principle of least action. All objects (stone, light, electron …), moving, obey this principle. On the one hand, the light obeys the principle of least action, but, on the other hand, it moves so as to spend a minimum of time on the path traveled. It is unlikely that the movement of light is governed by two different principles. Therefore, the idea arises that the principle of least time and the principle of least action are one and the same principle.

Let’s check this.

In empty space (there are no different fields) the ball moves from point A to point B in a straight line. It moves to spend the minimum time on the traversed path. Just like light! How will the ball move if there are forces? Everyone knows that in the field of Earth gravity the ball moves along a parabola. That is, when moving from point A to point B (for simplicity these points lie on the same horizontal line) the ball moves along a trajectory that is slightly convex upward and passes through point C:

Рис. 1

If the ball saved its time (as well as the light), then it would turn slightly down, would move through point D. Because below the speed of the ball is higher: at point D, its speed is greater than at points A and B. Moving along parabola, curved down, the ball would have come to point B faster than if it were moving along a straight segment AB. Instead, the ball, moving from A to B, turns up, where its speed goes down. As a result, it moves along a longer path and at a lower speed. Obviously, the ball does not want to obey the principle of least time. Why?

Maybe the matter is that the ball is a solid body, and the light is a wave? The electron also has wave properties and it moves like a ball. In the field of Earth gravity, it moves along a convex upward parabola. In a homogeneous electric field, it moves in a similar way.

Why does the same principle of least action lead to the principle of the least time in the case of light, but in the case of a ball, stone or electron no?

What do you think of it?

We will find an answer to this question in the following articles.

Vasily Yanchilin

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