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Chapter 19 THE PRINCIPLE OF Least EFFECT Addition made after a lecture When I was at school, our physics teacher, named Bader, once called me in after class and said: “You look as if you are terribly tired of everything; listen to one interesting thing

The principle of least action is the most important among the family; he is one of key provisions modern physics.

The first formulation of the principle was given (P. Maupertuis (French)) in 1744. From here he derived the laws of reflection and refraction of light.

Principle of least action in classical mechanics

Let us first recall, using the example physical system with one, that, which we are talking about here, is, that is, a rule that associates a certain number with each function x(t). The action looks like: S[x] = \int \mathcal(L)(x(t),\dot(x)(t),t) dt, Where \mathcal(L)(x(t),\dot(x)(t),t) there are systems that depend on the trajectory (i.e. the coordinates, which in turn depends on time), its first in time, and can also explicitly depend on .

The action can be calculated for a completely arbitrary trajectory, no matter how “wild” and “unnatural” it may be. However, among the entire set possible trajectories there is only one path along which the body will actually go. The principle of least action precisely answers the question of how the body will actually move:

This means that if the Lagrangian of the system is given, then we can use it to establish exactly how the body will move.

Note that if the law of motion can in principle be found from the conditions of the problem, this automatically means that it is possible to construct a functional that takes an extreme value for true motion.

They obey it, and therefore this principle is one of the key provisions of modern physics. The equations of motion obtained with its help are called the Euler-Lagrange equations.

The first formulation of the principle was given by P. Maupertuis in the year, immediately pointing out its universal nature, considering it applicable to optics and mechanics. From this principle he derived the laws of reflection and refraction of light.

Story

Maupertuis came to this principle from the feeling that the perfection of the Universe requires a certain economy in nature and contradicts any useless expenditure of energy. Natural movement must be such as to make some quantity minimal. All he had to do was find this value, which he continued to do. It was the product of the duration (time) of movement within the system by twice the value, which we now call the kinetic energy of the system.

Euler (in "Réflexions sur quelques loix générales de la nature", 1748) adopts the principle of least amount of action, calling action "effort". His expression in statics corresponds to what we would now call potential energy, so his statement of the least action in statics is equivalent to the minimum condition potential energy for the equilibrium configuration.

In classical mechanics

The principle of least action serves as the fundamental and standard basis of the Lagrangian and Hamiltonian formulations of mechanics.

First let's look at the construction like this: Lagrangian mechanics. Using the example of a physical system with one degree of freedom, recall that an action is a functional with respect to (generalized) coordinates (in the case of one degree of freedom - one coordinate), that is, it is expressed through such that each conceivable version of the function is associated with a certain number - an action (in In this sense, we can say that action as a functional is a rule that allows for any given function calculate completely a certain number- also called action). The action looks like:

where is the Lagrangian of the system, depending on the generalized coordinate, its first derivative with respect to time, and also, possibly, explicitly on time. If the system has a greater number of degrees of freedom, then the Lagrangian depends on more generalized coordinates and their first time derivatives. Thus, the action is a scalar functional depending on the trajectory of the body.

The fact that the action is a scalar makes it easy to write it in any generalized coordinates, the main thing is that the position (configuration) of the system is unambiguously characterized by them (for example, instead of Cartesian coordinates, these can be polar coordinates, distances between points of the system, angles or their functions, etc. .d.).

The action can be calculated for a completely arbitrary trajectory, no matter how “wild” and “unnatural” it may be. However, in classical mechanics, among the entire set of possible trajectories, there is only one along which the body will actually go. The principle of stationary action precisely gives the answer to the question of how the body will actually move:

This means that if the Lagrangian of the system is given, then using the calculus of variations we can establish exactly how the body will move by first obtaining the equations of motion - the Euler-Lagrange equations, and then solving them. This allows not only to seriously generalize the formulation of mechanics, but also to choose the most convenient coordinates for each specific problem, not limited to Cartesian ones, which can be very useful for obtaining the simplest and most easily solved equations.

where is the Hamilton function of this system; - (generalized) coordinates, - conjugate (generalized) impulses, characterizing together in each this moment time, the dynamic state of the system and, each being a function of time, thus characterizing the evolution (movement) of the system. In this case, to obtain the equations of motion of the system in the form of Hamilton’s canonical equations, it is necessary to vary the action written in this way independently for all and .

It should be noted that if from the conditions of the problem it is possible in principle to find the law of motion, then this is automatically Not means that it is possible to construct a functional that takes stationary value with true movement. An example would be a joint movement electric charges and monopoles - magnetic charges- in an electromagnetic field. Their equations of motion cannot be derived from the principle of stationary action. Likewise, some Hamiltonian systems have equations of motion that cannot be derived from this principle.

Examples

Trivial examples help to evaluate the use of the operating principle through the Euler-Lagrange equations. Free particle(weight m and speed v) in Euclidean space moves in a straight line. Using the Euler-Lagrange equations, this can be shown in polar coordinates as follows. In the absence of potential, the Lagrange function is simply equal to the kinetic energy

V orthogonal system coordinates

IN polar coordinates kinetic energy, and therefore the Lagrange function becomes

The radial and angular components of the equations become, respectively:

Solving these two equations

Here is a conditional notation for infinitely multiple functional integration over all trajectories x(t), and is Planck’s constant. Let us emphasize that, in principle, the action in the exponential appears (or can appear) itself when studying the evolution operator in quantum mechanics, but for systems that have an exact classical (non-quantum) analogue, it is exactly equal to the usual classical action.

Mathematical analysis of this expression in the classical limit - for sufficiently large , that is, for very fast oscillations of the imaginary exponential - shows that the overwhelming majority of all possible trajectories in this integral cancel each other in the limit (formally for ). For almost any path there is a path on which the phase shift will be exactly the opposite, and they will add up to zero contribution. Only those trajectories for which the action is close to the extreme value (for most systems - to the minimum) are not reduced. This is clean mathematical fact from the theory of functions of a complex variable; For example, the stationary phase method is based on it.

As a result, the particle in full agreement with the laws of quantum mechanics, it moves simultaneously along all trajectories, but under normal conditions only trajectories close to stationary (that is, classical) contribute to the observed values. Because the quantum mechanics transforms into classical in the limit of high energies, then we can assume that this is quantum mechanical derivation classical principle stationarity of action.

In quantum field theory

In quantum field theory, the principle of stationary action is also successfully applied. The Lagrangian density here includes the operators of the corresponding quantum fields. Although it is more correct here in essence (with the exception of the classical limit and partly quasi-classics) to speak not about the principle of stationarity of action, but about Feynman integration along trajectories in configuration or phase space these fields - using the Lagrangian density just mentioned.

Further generalizations

More broadly, an action is understood as a functional that specifies a mapping from the configuration space to a set real numbers and, in general, it does not have to be an integral, because nonlocal actions are in principle possible, at least in theory. Moreover, a configuration space is not necessarily a function space because it can have non-commutative geometry.

Notes

Literature

• Variational principles of mechanics. Sat. articles by classics of science. Edited by Polak L.S. M.: Fizmatgiz. 1959.
• Lanczos K. Variational principles of mechanics. - M.: Fizmatgiz. 1965.
• Berdichevsky V. L. Variational principles mechanics continuum. M.: Nauka, 1983. - 448 p.

The principle of least action, first formulated precisely by Jacobi, is similar to Hamilton's principle, but less general and more difficult to prove. This principle is applicable only to the case when the connections and force function do not depend on time and when, therefore, there is an integral of living force.

This integral has the form:

Hamilton's principle stated above states that the variation of the integral

is equal to zero upon the transition of actual motion to any other infinitely close motion that transfers the system from the same initial position to the same final position in the same period of time.

Jacobi's principle, on the contrary, expresses a property of motion that does not depend on time. Jacobi considers the integral

determining action. The principle he established states that the variation of this integral is zero when we compare the actual motion of the system with any other infinitely close motion that takes the system from the same initial position to the same final position. In this case, we do not pay attention to the time period spent, but we observe equation (1), i.e., the equation of manpower with the same value of the constant h as in actual movement.

This necessary condition for an extremum leads, generally speaking, to a minimum of integral (2), hence the name principle of least action. The minimum condition seems to be the most natural, since the value of T is essentially positive, and therefore integral (2) must necessarily have a minimum. The existence of a minimum can be strictly proven if only the time period is small enough. The proof of this position can be found in Darboux's famous course on surface theory. We, however, will not present it here and will limit ourselves to deriving the condition

432. Proof of the principle of least action.

At actual calculation we encounter one difficulty that is not present in the proof of Hamilton's theorem. The variable t no longer remains independent of variation; therefore variations of q i and q. are related to the variation of t by a complex relationship that follows from equation (1). The simplest way to get around this difficulty is to change the independent variable, choosing one whose values ​​fall between constant limits that do not depend on time. Let k be a new independent variable, the limits of which are assumed to be independent of t. When moving the system, the parameters and t will be functions of this variable

Let letters with primes q denote derivatives of parameters q with respect to time.

Since the connections are assumed to be independent of time, then Cartesian coordinates x, y, z are functions of q that do not contain time. Therefore, their derivatives will be linear homogeneous functions of q and 7 will be a homogeneous quadratic form of q, the coefficients of which are functions of q. We have

To distinguish the derivatives of q with respect to time, we denote, using parentheses, (q), the derivatives of q taken with respect to and put in accordance with this

then we will have

and integral (2), expressed through the new independent variable A, will take the form;

The derivative can be eliminated using the living force theorem. Indeed, the integral of manpower will be

Substituting this expression into the formula for, we reduce integral (2) to the form

The integral defining the action thus took its final form (3). There is an integrand function Square root from quadratic form from values

Let's show that differential equations extremals of integral (3) are exactly the Lagrange equations. Equations of extremals, based on general formulas calculus of variations will be:

Let's multiply the equations by 2 and perform partial differentiations, taking into account that it does not contain, then we get, if we do not write an index,

These are equations of extremals expressed through the independent variable Task is now to return to the independent variable

Since G is homogeneous function of the second degree in and is a homogeneous function of the first degree, then we have

On the other hand, the living force theorem can be applied to the factors of derivatives in the equations of extremals, which leads, as we saw above, to the substitution

As a result of all substitutions, the equations of extremals are reduced to the form

We have thus arrived at the Lagrange equations.

433. The case when there are no driving forces.

In case driving forces no, there is an equation for manpower and we have

The condition that the integral is a minimum is in this case is that the corresponding value -10 should be the smallest. Thus, when there are no driving forces, then among all the movements in which manpower keeps the same given value, the actual motion is that which takes the system from its initial position to its final position in the shortest time.

If the system is reduced to one point moving on a stationary surface, then the actual motion, among all movements on the surface that occur at the same speed, is the motion in which the point moves from its initial position to the final position in the shortest

time interval. In other words, a point describes on the surface the shortest line between its two positions, i.e. a geodesic line.

434. Note.

The principle of least action assumes that the system has several degrees of freedom, since if there were only one degree of freedom, then one equation would be sufficient to determine the motion. Since the movement can in this case be completely determined by the equation of living force, then the actual movement will be the only one that satisfies this equation, and therefore cannot be compared with any other movement.

The most general formulation of the law of motion mechanical systems is given by the so-called principle of least action (or Hamilton's principle). According to this principle, every mechanical system is characterized by a specific function.

or in short note, and the motion of the system satisfies the following condition.

Let the system occupy certain positions at moments of time, characterized by two sets of coordinate values ​​(1) and Then between these positions the system moves in such a way that the integral

had the least possible meaning. The function L is called the Lagrange function of this system, and the integral (2.1) is called the action.

The fact that the Lagrange function contains only q and q, but not higher derivatives, is an expression of the above statement that the mechanical state is completely determined by the specification of coordinates and velocities.

Let's move on to the derivation of differential equations, solving the problem on determining the minimum of the integral (2.1). To simplify the writing of formulas, let us first assume that the system has only one degree of freedom, so only one function must be defined

Let there be just that function for which S has a minimum. This means that S increases when replaced by any function of the form

where is a function that is small over the entire time interval from to (it is called a variation of the function since at all compared functions (2.2) must take the same values, then it should be:

The change in 5 when q is replaced by is given by the difference

The expansion of this difference into powers (in the integrand) begins with first-order terms. A necessary condition the minimality of S) is the vanishing of the set of these terms; it is called the first variation (or usually just variation) of the integral. Thus, the principle of least action can be written as

or, by varying:

Noting that we integrate the second term by parts and get:

But due to conditions (2.3), the first term in this expression disappears. What remains is the integral, which must be equal to zero for arbitrary values. This is only possible if the integrand identically vanishes. Thus we get the equation

In the presence of several degrees of freedom, the principle of least action must vary independently s various functions Obviously, we will then obtain s equations of the form

These are the required differential equations; in mechanics they are called Lagrange equations. If the Lagrange function of a given mechanical system is known, then equations (2.6) establish the connection between accelerations, velocities and coordinates, i.e. they represent the equations of motion of the system.

WITH mathematical point From view, equations (2.6) constitute a system of s second-order equations for s unknown functions. Common decision such a system contains arbitrary constants. To determine them and thereby full definition movement of a mechanical system requires knowledge initial conditions, characterizing the state of the system at a certain point in time, for example, knowledge initial values all coordinates and velocities.

Let the mechanical system consist of two parts A and B, each of which, being closed, would have as a Lagrange function, respectively, the functions ? Then, in the limit, when the parts are separated so far that the interaction between them can be neglected, the Lagrangian function of the entire system tends to the limit

This property of additivity of the Lagrange function expresses the fact that the equations of motion of each of the non-interacting parts cannot contain quantities related to other parts of the system.

It is obvious that multiplying the Lagrange function of a mechanical system by an arbitrary constant does not in itself affect the equations of motion.

From here, it would seem, a significant uncertainty could follow: the Lagrange functions of various isolated mechanical systems could be multiplied by any different constants. The property of additivity eliminates this uncertainty - it only allows for the simultaneous multiplication of the Lagrangian functions of all systems by the same constant, which simply comes down to the natural arbitrariness in the choice of units of measurement of this physical quantity; We will return to this issue in §4.

The following general remark needs to be made. Let's consider two functions that differ from each other by the total time derivative of any function of coordinates and time

Integrals (2.1) calculated using these two functions are related by the relation

i.e. differ from each other by an additional term that disappears when the action is varied, so that the condition coincides with the condition and the form of the equations of motion remains unchanged.

Thus, the Lagrange function is defined only up to the addition of the total derivative of any function of coordinates and time.