The principle of least action. Part 1
When I first learned about this principle, I had a feeling of some kind of mysticism. It seems that nature mysteriously sorts through all possible paths of movement of the system and selects the best of them.
Today I want to talk a little bit about one of the most remarkable physical principles - the principle of least action.
Since the time of Galileo, it was known that bodies, on which no forces act, move along straight lines, that is, along the shortest path. Light lines propagate along straight lines.
When reflected, the light also moves in such a way as to get from one point to another in the shortest way. In the picture, the shortest path will be the green path at which the angle of incidence equals the angle of reflection. Any other path, such as red, will be longer.
It is easy to prove, simply reflecting the path of the rays on the opposite side of the mirror. In the picture they are shown dotted.
It can be seen that the green ACB path becomes a direct ACB ’. And the red path turns into a broken ADB ’line, which, of course, is longer than the green one.
In 166? Pierre Fermat suggested that the speed of light in a dense substance, for example, in glass, is less than in air. Prior to this, there was the generally accepted version of Descartes, according to which the speed of light in a substance must be greater than in air in order to get the correct law of refraction. For Fermat, the assumption that light can move in a denser environment faster than in a rarefied seemed unnatural. Therefore, he suggested that everything is exactly the opposite and proved an amazing thing - with this assumption, the light is refracted so as to reach the destination in the shortest possible time.
The figure again shows in green the path that the light beam actually moves. The path marked in red is the shortest, but not the fastest, because the light has to travel a greater way in the glass, and in it its speed is less. The fastest is precisely the real path of the light beam.
All these facts suggested that nature acts in some rational way, that light and bodies move in the most optimal way, expending as little effort as possible. But what kind of effort, and how to count them remained a mystery.
In 174? Maupertuis introduced the concept of "action" and formulated the principle that the true trajectory of a particle differs from any other in that the action for it is minimal. However, Maupertuis himself could not give a clear definition of what this action is equal to. The rigorous mathematical formulation of the principle of least action was already developed by other mathematicians - Euler, Lagrange, and was finally given by William Hamilton: 3r-3286.
In mathematical language, the principle of least action is formulated rather briefly, but the meaning of the notation used may not be clear to all readers. I want to try to explain this principle more clearly and in simple words.
So imagine that you are sitting in a car at
and at time
You have been given a simple task: by the time
You need to drive to the point
Fuel for the car is expensive and, of course, you want to spend it as little as possible. Your car is made with the latest super-technologies and can accelerate or brake as quickly as you wish. However, it is arranged so that the faster it goes, the more it consumes fuel. Moreover, fuel consumption is proportional to the square of the speed. If you drive twice as fast, then during the same period of time you consume 4 times more fuel. In addition to speed, the weight of the car also affects fuel consumption. The heavier our car, the more fuel it consumes. Our car's fuel consumption at any one time is
i.e. exactly equal to the kinetic energy of the car.
So how do you need to go to get to point 3r33252.
at exactly the right time and use less fuel? It is clear that you need to go in a straight line. When you increase the travel distance of the fuel consumed exactly no less. And then you can choose different tactics. For example, you can quickly arrive at point
just sit in advance and wait until the time comes
. The driving speed, and hence the fuel consumption at each time point, will turn out to be great, but then the driving time will be reduced. Perhaps, the total fuel consumption will not be so great. Or you can drive evenly, at the same speed, so that, without haste, just arrive at time point
. Or part of the road to drive fast, and some slower. How better to go?
It turns out that the most optimal, most economical way to drive is to drive at a constant speed, such as to end up at point 3r-3252.
at exactly the appointed time
. In any other case, more fuel will be consumed. You can check for yourself with a few examples. The reason is that fuel consumption increases in proportion to the square of the speed. Therefore, with an increase in speed, fuel consumption increases faster than driving time decreases, and overall fuel consumption also increases.
So, we found out that if a car consumes fuel at any one time in proportion to its kinetic energy, then the most economical way to get from point 3r33252. 3r3195.
to point 3r33252.
by the precisely designated time it is to travel evenly and rectilinearly, just as the body moves in the absence of forces acting on it. Any other method of driving will lead to greater overall fuel consumption.
In the field of gravity
Now let's improve our car a little. Let's attach jet engines to it so that it can fly freely in any direction. In general, the design remained the same, so the fuel consumption again remained strictly proportional to the kinetic energy of the car. If the task is now given to fly out of the point 3r33252. 3r3195.
and fly to point
, the most economical way, as before, of course, will fly evenly and straightforwardly, to be at the point 3r-3252. 3r3164.
at exactly the appointed time
. This again corresponds to the free movement of the body in three-dimensional space.
However, in the latest car model installed an unusual machine. This unit can produce fuel from virtually nothing. But the design is such that the higher the car is, the more fuel the vehicle produces at each time point. Fuel production is directly proportional to the height of
on which the car is currently located. Also, the heavier the car, the more powerful the machine is installed on it and the more fuel it generates, and the output is directly proportional to the mass of the car 3r33252. 3r3182.
. The device turned out such that the production of fuel is exactly equal to r3r3252. 3r3185.
is the acceleration of gravity), i.e. the potential energy of the car.
The fuel consumption at each time point is equal to the kinetic energy minus the potential energy of the car (minus the potential energy, because the installed device produces fuel, and does not waste it). Now our task is the most economical movement of the car between the points 3r-3252. 3r3195.
getting harder. Rectilinear uniform motion is not the most effective in this case. It turns out that it is more optimal to gain some height, to linger there for a while, having developed more fuel, and then go down to the point 3r33252.
. With the right flight path, total fuel generation due to climb will override additional fuel costs to increase the path length and increase speed. If you carefully calculate, then the most economical way for a car would be to fly along a parabola, exactly along such a trajectory and with exactly the speed with which a stone would fly in the earth's gravity field.
Here it is worth making an explanation. Of course, it is possible from the point
Throw a stone in many different ways so that it gets to the point
. But you need to throw it so that it takes off from the point
, hit the mark
exactly at time
. This movement will be the most economical for our car.
Lagrange function and the principle of least action 3r32r6868.
Now we can transfer this analogy to real physical bodies. The analogue of the intensity of fuel consumption for the bodies is called the Lagrange function or Lagrangian (in honor of Lagrange) and is denoted by the letter 3r-3252.
. Lagrangian shows how much "fuel" the body consumes at a given time. For a body moving in a potential field, Lagrangian is equal to its kinetic energy minus potential energy.
The analogue of the total amount of fuel consumed for the entire period of movement, i.e. the value of Lagrangian accumulated over the entire time of movement is called “action”.
The principle of least action is that the body moves in such a way that the action (which depends on the trajectory of movement) is minimal. It should not be forgotten that the initial and final conditions are specified, i.e. where the body is at time point
and at time
In this case, the body does not necessarily have to move in a uniform field of aggression, which we considered for our car. You can consider completely different situations. The body can oscillate on an elastic band, swing on a pendulum or fly around the sun, in all these cases it moves so as to minimize “total fuel consumption” i.e. act.
If the system consists of several bodies, then Lagrangian of such a system will be equal to the total kinetic energy of all bodies minus the total potential energy of all bodies. And again, all bodies will move in a coordinated manner so that the action of the entire system during such a movement is minimal.
Not so simple
In fact, I was a little deceived, saying that the bodies always move in such a way as to minimize the action. Although in many cases this is true, you can come up with situations in which the action is clearly not minimal.
For example, take the ball and place it in the empty space. At some distance from him put the elastic wall. Suppose we want the ball to be in the same place after some time. Under such specified conditions, the ball can move in two different ways. First, he can just stay in place. Secondly, you can push it towards the wall. The ball will fly to the wall, bounce off of it and come back. It is clear that you can push it so fast that it returns at exactly the right time.
Both variants of the ball movement are possible, but the action in the second case will be more, because all this time the ball will move with non-zero kinetic energy.
How to save the principle of least action, so that it is fair in such situations? We will talk about this next time.
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