The principle of least action in analytical mechanics

3r3-31. 3r33814. Prehistory

3r3822.

3r3822.

3r3822.

3r3822.

The reason for this publication is ambiguous article on the principle of least action (HDPE) 3r3357. published on the resource a few days ago. It is ambiguous because its author in a popular form is trying to convey to the reader one of the fundamental principles of the mathematical description of nature, and this is partly possible for him. If it were not for one thing, lurking at the end of the publication. Under the spoiler is a full quote of this passage

3r3822.

The ball motion problem [/b]

3r3733.

## Not so simple

3r3822.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. 3r3822.

3r3822.

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. 3r3822.

3r3757. 3r3335. 3r33838. 3r3822.

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. 3r3822.

3r3822.

How to save the principle of least action, so that it is fair in such situations? We will talk about this next time. 3r3822.

3r3734. 3r3822.

3r3822.

3r33838. 3r33838. 3r3822.

So what, in my opinion, is the problem? 3r3822.

3r3822.

The problem is that the author, citing this example, made a number of fundamental errors. It is aggravated by the fact that the planned second part, according to the author, will be based on these errors. Guided by the principle of filling the resource with reliable information, I have to speak in more detail about my position on this issue, and the format of comments for this is too small. 3r3822.

3r3822.

This article will talk about how to build mechanics on the basis of the HDPE, and will try to explain to the reader that the problem posed by the author of the cited publication is missing. 3r3822.

3r3822.

3r33814. 1. Definition of Hamilton action. The principle of least action

3r3822.

3r3822.

Hamilton action refers to the functional 3r3-3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r380. 3r33737. 3r33737. 3r3822.

where

3r33714. 3r33737. 3r33714. 3r33737. 3r3391. 3r33737. 3r33737. 3r3822.

- Lagrange function, for some mechanical system, in which (omitting the arguments in the future)

*T*- kinetic energy of the system; P - its potential energy; 3r3784. q [/b] (t) is the vector of generalized coordinates of this system, which is a function of time. at the same time, it is assumed that the times t

_{ 1 }and t

_{ 2 }- fixed. 3r3822.

3r3822.

Why functional, but not function? Because a function, by definition, is a rule according to which one number from the domain (function argument) is assigned a different number from the domain. The functional differs in that the quality of its argument is not a number, but an entire function. In this case, it is the law of motion of the mechanical system

**q**(t), defined at least in the time interval between t

_{ 1 }and t

_{ 2 }. 3r3822.

3r3822.

The long-term (and this is putting it mildly!) Works of mechanical scientists (including the astounding Leonard Euler) allowed us to formulate 3r3822.

3r3822.

3r3784. The principle of least action: [/b] 3r3822.

3r3733. A mechanical system for which the Lagrange function is set to be

3r3128. 3r33737. , moving in such a way that the law of its movement

**q**(t) provides a minimum of the functionality 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r3138. 3r33737. 3r33737. 3r3822.

called the Hamilton action. 3r3822.

3r3734. 3r3822.

Already from the very definition of PND it follows the fact that this principle leads to the equations of motion only for a limited class of mechanical systems. For what? And let's see. 3r3822.

3r3822.

3r33814. 2. Limits of applicability of the principle of least action. Some definitions for the smallest

3r3822.

3r3822.

As follows from the definition, again, the Lagrange function, PND allows to obtain the equations of motion for mechanical systems, the force action on which is determined solely by the potential energy. In order to figure out which systems we are talking about, we will give a few definitions, which, to save the volume of the article, I place 3r3822 under the spoiler.

3r3822.

Work force on the move [/b]

Consider a point moving along the trajectory AB to which a force is applied 3r3373715. 3r33737. . The infinitesimal movement of a point along the path is determined by the vector

3r33180. 3r33737. directed tangentially to the trajectory. 3r3822.

3r3822.

Elementary work force

3r33737. on moving 3r3715. 3r33180. 3r33737. call a scalar value equal to

3r33714. 3r33737. 3r33714. 3r33737. 3r3188. 3r33737. 3r33737. 3r3822.

Then, the total work of the force on moving a point along the trajectory AB is a curvilinear integral

3r33714. 3r33737. 3r33714. 3r33737. 3r-33199. 3r33737. 3r33737. 3r3822.

3r33838. 3r33838. 3r3822.

The kinetic energy of the point is [/b]

3r3733. The kinetic energy of the point is

*T*call the work that must be done attached to a point of mass

*m 3r33737. forces, in order to transfer a point from motion to rest at a speed of 3r-3715. 3r33737. 3r3822.*

3r3734. 3r3822.

Calculate the kinetic energy according to this definition. Let a point start moving from a state of rest under the action of forces applied to it. On the segment of the trajectory AB, it acquires the speed of 3r33715. 3r33737. . Let us calculate the work done by the forces applied to the point, which, according to the principle of independence of the action of the forces, will replace the resultant 3r3715. 3r33737. 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

In accordance with the second law of Newton 3r3-3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

then

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

We compute the scalar product strictly standing under the integral sign, for which we differentiate in time the scalar product of the velocity vector itself into 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

On the other hand,

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

Differentiating this equality in time, we have

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

Comparing (1) and (2) we come to the conclusion that

3r33714. 3r33737. 3r33714. 3r33737. 3r3303. 3r33737. 3r33737. 3r3822.

Then, we calmly calculate the work, revealing the curvilinear integral over a definite one, taking as a limit the module of velocity of a point at the beginning and at the end of the trajectory 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

3r33838. 3r33838. 3r3822.

Conservative forces and the potential energy of the point [/b]

Consider a force acting on a point, such that the magnitude and direction of this force depends exclusively on the position of the point in the space 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

Let a point move in space along an arbitrary trajectory AB. Let us calculate what work the force will do in this case (3)

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

Since the projections of the force on the coordinate axes depend solely on these very coordinates, you can always find the function

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

such that 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

Then, the expression for the work is converted to the form

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

where

3r33383. 3r33737. - values of the function U (x, y, z) at points A and B, respectively. Thus, the work of the force under consideration does not depend on the trajectory of the point, but is determined only by the values of the function U at the beginning and at the end of the trajectory. This strength is called [i] conservative power of, and the corresponding function U (x, y, z) is a force function. Obviously,

3r3734. 3r3822.

Calculate the kinetic energy according to this definition. Let a point start moving from a state of rest under the action of forces applied to it. On the segment of the trajectory AB, it acquires the speed of 3r33715. 3r33737. . Let us calculate the work done by the forces applied to the point, which, according to the principle of independence of the action of the forces, will replace the resultant 3r3715. 3r33737. 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

In accordance with the second law of Newton 3r3-3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

then

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

We compute the scalar product strictly standing under the integral sign, for which we differentiate in time the scalar product of the velocity vector itself into 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

On the other hand,

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

Differentiating this equality in time, we have

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

Comparing (1) and (2) we come to the conclusion that

3r33714. 3r33737. 3r33714. 3r33737. 3r3303. 3r33737. 3r33737. 3r3822.

Then, we calmly calculate the work, revealing the curvilinear integral over a definite one, taking as a limit the module of velocity of a point at the beginning and at the end of the trajectory 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

3r33838. 3r33838. 3r3822.

Conservative forces and the potential energy of the point [/b]

Consider a force acting on a point, such that the magnitude and direction of this force depends exclusively on the position of the point in the space 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

Let a point move in space along an arbitrary trajectory AB. Let us calculate what work the force will do in this case (3)

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

Since the projections of the force on the coordinate axes depend solely on these very coordinates, you can always find the function

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

such that 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

Then, the expression for the work is converted to the form

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

where

3r33383. 3r33737. - values of the function U (x, y, z) at points A and B, respectively. Thus, the work of the force under consideration does not depend on the trajectory of the point, but is determined only by the values of the function U at the beginning and at the end of the trajectory. This strength is called [i] conservative power of

3r33333. 3r33737. , as well as the equality to zero of the work of a conservative force when moving along a closed trajectory. It is also said that the function U (x, y, z) in a space defines a force field. 3r3822.

3r3822.

3r3733. Potential energy

3r33737. points in space with a given force field, call the work of conservative forces applied to it, which they do when moving a point to a position given by coordinates (x, y, z) in space, from some arbitrary position chosen as the starting point of the potential energy level . 3r3822.

3r3734. 3r3822.

3r3822.

On the previously considered point trajectory, we choose an arbitrary point O lying between points A and B. We assume that at the point the potential energy is zero. Then

3r33714. 3r33737. 3r33714. 3r33737. 3r33411. 3r33737. 3r33737. 3r3822.

- the potential energy of a point at position A, and

3r33714. 3r33737. 3r33714. 3r33737. 3r33434. 3r33737. 3r33737. 3r3822.

- the potential energy of a point in position B. Considering the above, we again calculate the work of potential forces on the displacement from point A to point B

3r33714. 3r33737. 3r33714. 3r33737. 3r33333. 3r33737. 3r33737. 3r3822.

Thus, the work of conservative forces is equal to the change in the potential energy of a point, taken with the opposite sign 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r3444. 3r33737. 3r33737. 3r3822.

and the choice of the level at which we consider the potential energy equal to zero does not affect the result at all. From this we can conclude that the reference level of potential energy can be chosen completely arbitrarily. 3r3822.

3r33838. 3r33838. 3r3822.

3r3822.

3r33814. 3. The concept of variations of generalized coordinates. Statement of the variational problem 3r3r1515. 3r3822.

3r3822.

So, we now consider a mechanical system moving under the action of potential forces, the position of which is uniquely given by the vector of generalized coordinates

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r3822.

where

*s*- the number of degrees of freedom of the system. 3r3822.

3r3822.

Valid, 3r3784. but so far unknown to us [/b] , the law of motion of this system is determined by the dependence of the generalized coordinates (4) on time. Consider one of the generalized coordinates

3r33737. assuming similar reasoning for all other coordinates. 3r3822.

3r3822.

3r3757. 3r33490. 3r33838. 3r3822.

3r33762. Figure 1. Real and roundabout movement of a mechanical system 3r33763. 3r3822.

3r3822.

The figure shows the dependence

3r301501. 3r33737. depicted by a red curve. Choose two arbitrary fixed points in time t

_{ 1 }and t

_{ 2 }, considering t

_{ 2 }> t

_{ 1 }. The position of the system 3r33737. 3r33512. 3r33737. We agree to call the initial position of the system, and

3r33515. 3r33737. - the final position of the system. 3r3822.

3r3822.

However, I once again insist that the following text be read carefully! 3r3784. Despite the fact that we are given the initial and final position of the system, neither the first position nor the second is known to us in advance! [/b] As well as the unknown law of the system! These provisions are considered precisely as the initial and final position, regardless of specific values. 3r3822.

3r3822.

Further, we believe that from the initial position to the final system can come in different ways, that is, the dependence 3r3373715. 3r33528. 3r33737. can be any kinematically possible. The actual movement of the system will exist in a single variant (beautycurve, the remaining kinematically possible variants will be called

*in a roundabout*3r33737. 3r33333. 3r33737. (blue curve in the picture). The difference between real and roundabout traffic is

3r33714. 3r33737. 3r33714. 3r33737. 3r33541. 3r33737. 3r33737. 3r3822.

we will call

*isochronous variations of generalized coordinates*3r3822.

3r3822.

In this context, variations (5) should be understood as infinitesimal functions expressing the deviation of a roundabout motion from the real one. The small “delta” for the designation was not chosen randomly and underlines the fundamental difference between the variation and the differential of the function. The differential is the main linear part of the function increment, caused by the argument increment. In the case of variation, the change in the value of the function is

**with a constant value of the argument caused by a change in the type of the function itself!**We do not vary the argument, in the role of which time plays, therefore the variation is called isochronous. We vary the rule according to which each value of time corresponds to a certain value of generalized coordinates! 3r3822.

3r3822.

In fact, we vary the law of motion, according to which the system moves from the initial state to the final state. The initial and final states are determined by the actual law of motion, but I emphasize once again - we don’t know their specific values and can be any kinematically possible, we just assume that they exist and the system is guaranteed to move from one position to another! In the initial and final positions of the system, we do not vary the law of motion; therefore, the variations of the generalized coordinates in the initial and final positions are zero 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33535. 3r33737. 3r33737. 3r3822.

Based on the principle of least action, the actual movement of the system should be such as to deliver a minimum of action functionality. Variation of coordinates causes a change in the action functional. A necessary condition for the achievement of an extremal value by a functional is that its variation equals

to zero.

3r33714. 3r33737. 3r33714. 3r33737. 3r33575. 3r33737. 3r33737. 3r3822.

3r3822.

3r33814. 4. The solution of the variational problem. Lagrange equations of the 2nd kind of 3r3-3815. 3r3822.

We solve the variational problem posed by us, for which we calculate the total variation of the action functional and equate it to zero 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r? 3592. 3r33737. 3r33737. 3r3822.

We will drive everything under one integral, and since for the variations all operations on infinitely small quantities are valid, we transform this crocodile into the form

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r3822.

Based on the definition of generalized speed

3r33714. 3r33737. 3r33714. 3r33737. 3r31414. 3r33737. 3r33737. 3r3822.

Then expression (8) is converted to the form

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r3822.

The second term is integrated in parts

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r3822.

Based on condition (7), we have

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

then, we get the equation

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r3822.

With arbitrary limits of integration, the equality to zero of a certain integral is ensured by the equality to zero of the integrand of

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r3822.

Taking into account the fact that the variations of the generalized coordinates are independent, (11) is valid only if all coefficients are zero for variations, that is, 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r38080. 3r33737. 3r33737. 3r3822.

Nobody stops us from multiplying each of the equations by (-1) and getting a more familiar record 3r3822.

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r3822.

3r3822.

3r3784. Equations (12) are the solution to the problem [/b] . And here at this moment the attention is again - the solution of a variational problem on the principle of least action, this is

**Not a function 3r3785. delivering a minimum to Hamilton action,**

3r3822.

And that's

3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r33737. 3r3822.

where is C 3r3804. 1 [/sub] , , C 3r3804. 2s [/sub] - arbitrary integration constants. 3r3822.

3r3822.

Therefore,

3r3733. PND is a fundamental principle that allows one to obtain the equations of motion for a system for which the Lagrange function

is defined.

3r3734. 3r3822.

3r3822.

Dot! In the problems of analytical mechanics, the above calculations no longer need to be done, it is enough to use their result (12). The function that satisfies equation (12) is the law of motion of the system that satisfies the MHP. 3r3822.

3r3822.

3r33814. 5. The challenge with the ball and the wall

3r3822.

3r3822.

Now let us return to the problem with which it all began - on the one-dimensional motion of a ball near an absolutely elastic wall. Of course, for this problem one can obtain differential equations of motion. 3r3784. Since these are differential equations of motion, then any, I emphasize this, any solution to them delivers a minimum to the action functional, which means the HDPE is executed!The general solution of the equations of motion of a ball can be represented in the form of the so-called

**and a system of differential equations, solving which such a function can be found**. In this case, this is the second-kind Lagrange differential equation written through the Lagrange function, that is, in the formulation for conservative mechanical systems. 3r3822.3r3822.

And that's

**on this principle of least action ends**, and the theory of ordinary differential equations begins, which, in particular, states that a solution of equation (12) is a vector function of the form3r33714. 3r33737. 3r33714. 3r33737. 3r33737. 3r33737. 3r33737. 3r3822.

where is C 3r3804. 1 [/sub] , , C 3r3804. 2s [/sub] - arbitrary integration constants. 3r3822.

3r3822.

Therefore,

3r3733. PND is a fundamental principle that allows one to obtain the equations of motion for a system for which the Lagrange function

is defined.

3r3734. 3r3822.

3r3822.

Dot! In the problems of analytical mechanics, the above calculations no longer need to be done, it is enough to use their result (12). The function that satisfies equation (12) is the law of motion of the system that satisfies the MHP. 3r3822.

3r3822.

3r33814. 5. The challenge with the ball and the wall

3r3822.

3r3822.

Now let us return to the problem with which it all began - on the one-dimensional motion of a ball near an absolutely elastic wall. Of course, for this problem one can obtain differential equations of motion. 3r3784. Since these are differential equations of motion, then any, I emphasize this, any solution to them delivers a minimum to the action functional, which means the HDPE is executed!

*phase portrait*considered mechanical system. Here is the phase portrait 3r3822.

3r3822.

3r3757. 3r3758. 3r33838. 3r3822.

3r33762. Figure 2. Phase portrait of the system in the problem with the ball 3r33737. 3r3822.

3r3822.

The coordinate of the ball is plotted on the horizontal axis, and the projection of speed on the x-axis is plotted on the vertical axis. It may seem strange, but this drawing reflects all possible phase trajectories of the ball, with any initial, or if you so wish, boundary conditions. In fact, there are infinitely many parallel lines on the graph; the drawing shows some of them and the direction of movement along the phase trajectory. 3r3822.

3r3822.

This is a general solution of the ball motion equation. Each of these phase trajectories provides a minimum of action functionality, which directly follows from the calculations performed above. 3r3822.

3r3822.

What does the author of the task? He says: the ball is at rest, and for the time interval from t

_{ A }to t

_{ B }action is zero. If the ball is pushed to the wall, then for the same period of time the action will be longer, since the ball has a nonzero and constant kinetic energy. But why the ball moves to the wall, because at rest the action will be less? So PND is experiencing problems and does not work! But we will definitely solve it in the next article. 3r3822.

3r3822.

What the author says is nonsense. Why? 3r3784. Yes, because it compares the actions on different branches of the same real phase trajectory! [/b] Meanwhile, when using PND, the effect on the real trajectory and on the set of roundabout trajectories is compared. That is, the action on the real trajectory is compared with the action on those trajectories that are not in nature, and never will be! 3r3822.

3r3822.

Unclear? I will explain it even more clearly. Consider the state of rest. It is described by a branch of the phase portrait that coincides with the abscissa axis. Coordinate does not change over time. This is a real movement. And what kind of movement will be roundabout. Any other kinematically possible. For example, small oscillations of a ball near the rest position considered by us. The task allows the ball to oscillate along the x axis? Admits, then such a movement is kinematically possible and can be considered as one of the roundabout

3r3822.

Why is the ball still resting? Yes, because the action is at rest, calculated on a fixed time interval from t

_{ A }to t

_{ B }, will be less action, with small fluctuations in the same period of time. It means that nature prefers rest to oscillations and any other “stirring” of the ball. In full accordance with the PND. 3r3822.

3r3822.

Suppose we pushed the ball toward the wall. Suppose we pushed it as the author wants, at a speed selected from the boundary conditions, so that at the moment of time t

_{ B }the ball was in the same position from which it started. The ball, at a constant speed, reaches the wall, bounces elastically and returns to its initial position at the moment of time t

_{ B }, again with constant speed. Ok, this is a real move. What movement will be one of the roundabouts? For example, if the ball moves to the wall and away from the wall at a rate varying with time. Such a movement is possible kinematically? Maybe. Why doesn't the speed of the ball change? Yes, because the effect on such a phase trajectory will have a minimum value, in comparison with any other option, where the speed depends on time. 3r3822.

3r3822.

That's all. Nothing such magic happens here. HDPE works without any problems. 3r3822.

3r3822.

3r33814. Conclusions and wishes

3r3822.

3r3822.

PND is a fundamental law of nature. In particular, laws of mechanics, for example, differential equations of motion (12), follow from it. PND tells us that nature is arranged in such a way that the equation of motion of a conservative mechanical system looks exactly like expression (12) and in no other way. More from him and is not required. 3r3822.

3r3822.

No need to invent problems where there are none. 3r33838.

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3r33838.

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