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.
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. The natural movement must be such as to make a certain quantity minimum. 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 the least amount of action, calling action "effort". Its expression in statics corresponds to what we would now call potential energy, so that its statement of least action in statics is equivalent to the minimum potential energy condition for an equilibrium configuration.
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, let us 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 an action as a functional is a rule that allows for any given function to calculate a completely specific number - also called an 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 larger number of degrees of freedom, then the Lagrangian depends on a larger number of generalized coordinates and their first derivatives with respect to time. 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 a 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, which together characterize at each given moment of time the dynamic state of the system and, each being a function of time, thus characterizing the evolution (motion) 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 a stationary value during true motion. An example is the joint movement of 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.
Trivial examples help to evaluate the use of the operating principle through the Euler-Lagrange equations. Free particle (mass 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
in an orthogonal coordinate system.
In polar coordinates, the 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 at ). 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 a purely 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, moves simultaneously along all trajectories, but under normal conditions only trajectories close to stationary (that is, classical) contribute to the observed values. Since quantum mechanics transforms into classical mechanics in the limit of high energies, we can assume that this is quantum mechanical derivation of the classical principle of stationarity of action.
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 the configuration or phase space of these fields - using the just mentioned Lagrangian density.
More broadly, an action is understood as a functional that defines a mapping from a configuration space to a set of real numbers and, in general, it does not have to be an integral, because non-local actions are possible in principle, at least theoretically. Moreover, a configuration space is not necessarily a function space because it can have non-commutative geometry.
LEAST EFFECTIVE PRINCIPLE
One of the variational principles of mechanics, according to Krom, for a given class of mechanical movements compared with each other. system, the valid one is that for which physical. size, called action, has the smallest (more precisely, stationary) value. Usually N. d. p. is used in one of two forms.
a) N. d. p. in the form of Hamilton - Ostrogradsky establishes that among all kinematically possible movements of a system from one configuration to another (close to the first), accomplished in the same period of time, the valid one is the one for which the Hamiltonian action S will be the smallest. Math. the expression of the N. d.p. in this case has the form: dS = 0, where d is the symbol of incomplete (isochronous) variation (i.e., unlike complete variation, time does not vary in it).
b) N. d. p. in the form of Maupertuis - Lagrange establishes that among all kinematically possible movements of a system from one configuration to another close to it, performed while maintaining the same value of the total energy of the system, the one that is valid is the one for which - Therefore, the Lagrange action W will be the smallest. Math. the expression of the N. d.p. in this case has the form DW = 0, where D is the symbol of total variation (unlike the Hamilton-Ostrogradsky principle, here not only the coordinates and velocities vary, but also the time of movement of the system from one configuration to another) . N.d.p.v. In this case, it is valid only for conservative and, moreover, holonomic systems, while in the first case, the non-conservative principle is more general and, in particular, can be extended to non-conservative systems. N.D.P. are used to compile equations of mechanical motion. systems and to study the general principles of these movements. With an appropriate generalization of concepts, the NDP finds applications in the mechanics of a continuous medium, in electrodynamics, and quantum. mechanics, etc.
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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.
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. The natural movement must be such as to make a certain quantity minimum. 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 the least amount of action, calling action "effort". Its expression in statics corresponds to what we would now call potential energy, so that its statement of least action in statics is equivalent to the minimum potential energy condition for an equilibrium configuration.
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, let us 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 an action as a functional is a rule that allows for any given function to calculate a completely specific number - also called an 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 larger number of degrees of freedom, then the Lagrangian depends on a larger number of generalized coordinates and their first derivatives with respect to time. 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 a 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, which together characterize at each given moment of time the dynamic state of the system and, each being a function of time, thus characterizing the evolution (motion) 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 a stationary value during true motion. An example is the joint movement of 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.
Trivial examples help to evaluate the use of the operating principle through the Euler-Lagrange equations. Free particle (mass 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
in an orthogonal coordinate system.
In polar coordinates, the 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 at ). 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 a purely 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, moves simultaneously along all trajectories, but under normal conditions only trajectories close to stationary (that is, classical) contribute to the observed values. Since quantum mechanics transforms into classical mechanics in the limit of high energies, we can assume that this is quantum mechanical derivation of the classical principle of stationarity of action.
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 the configuration or phase space of these fields - using the just mentioned Lagrangian density.
More broadly, an action is understood as a functional that defines a mapping from a configuration space to a set of real numbers and, in general, it does not have to be an integral, because non-local actions are possible in principle, at least theoretically. Moreover, a configuration space is not necessarily a function space because it can have non-commutative geometry.
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 the actual motion to any other infinitely close motion, which 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
In the 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, by assumption, do not depend on time, the 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). The integrand function is the square root of the quadratic form of the quantities
Let us show that the differential equations of the extremals of the integral (3) are exactly the Lagrange equations. The equations of extremals, based on the general formulas of the 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 in terms of the independent variable. The task now is to return to the independent variable
Since Γ is a homogeneous function of the second degree of and is a homogeneous function of the first degree, 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.
In the case when there are no driving forces, there is an equation for living force and we have
The condition for the integral to be a minimum is in this case that the corresponding value of -10 must be the smallest. Thus, when there are no driving forces, then among all the movements in which the living force maintains the same given value, the actual movement is that which transfers 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 movement, among all movements on the surface, performed at the same speed, is the movement in which the point moves from its initial position to the final position in the shortest
period of time. In other words, a point describes on the surface the shortest line between its two positions, i.e., a geodesic line.
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.
In we briefly examined one of the most remarkable physical principles - the principle of least action, and stopped at an example that seemed to contradict it. In this article we will look at this principle in a little more detail and see what happens in this example.
This time we'll need a little more math. However, I will again try to present the main part of the article at an elementary level. I will highlight slightly more strict and complex points in color; they can be skipped without compromising the basic understanding of the article.
To accurately describe its movement, as a rule, initial conditions are specified. For example, it is specified that at the initial moment of time the ball was at a point with coordinate and had a speed . Having set the initial conditions in this form, we unambiguously determine the further movement of the ball - it will move at a constant speed, and its position at the moment of time will be equal to the initial position plus the speed multiplied by the elapsed time: . This method of setting initial conditions is very natural and intuitively familiar. We have specified all the necessary information about the motion of the ball at the initial moment of time, and then its motion is determined by Newton's laws.
However, this is not the only way to specify the movement of the ball. Another alternative way is to set the position of the ball at two different times and . Those. ask that:
1) at the moment of time the ball was at a point (with coordinate);
2) at the moment of time the ball was at the point (with coordinate ).
The expression “was at point” does not mean that the ball was at rest at point. At the moment of time he could fly through the point. This means that its position at the moment of time coincided with the point. The same applies to the point.
These two conditions also uniquely determine the motion of the ball. Its movement is easy to calculate. To satisfy both conditions, the speed of the ball must obviously be . The position of the ball at the moment of time will again be equal to the initial position plus the speed multiplied by the elapsed time:
Note that in the conditions of the problem we did not need to set the initial speed. It was uniquely determined from conditions 1) and 2).
Setting conditions in the second way looks unusual. It may be unclear why it might be necessary to ask them in this form at all. However, in the principle of least action, it is the conditions in the form of 1) and 2) that are used, and not in the form of specifying the initial position and initial speed.
For example, we can move the ball with the same speed equal to (green trajectory). Or we can keep it at point half the time, and then move it to the point at double speed (blue trajectory). You can first move it in the opposite direction, and then move it to (brown trajectory). You can move it back and forth (red path). In general, you can move it however you like, as long as conditions 1) and 2) are met.
For each such trajectory we can associate a number. In our example, i.e. in the absence of any forces acting on the ball, this number is equal to the total accumulated kinetic energy during the entire time of its movement in the time interval between and and is called action.
In this case, the word “accumulated” kinetic energy does not convey the meaning very accurately. In reality, kinetic energy is not accumulated anywhere; the accumulation is used only to calculate the action for the trajectory. In mathematics there is a very good concept for such accumulation - the integral:As an example, let's take a ball of mass 1 kg, set some boundary conditions and calculate the action for two different trajectories. Let the point be at a distance of 1 meter from the point, and the time is 1 second away from time. Those. we must move the ball, which at the initial moment of time was at point , in one second to a distance of 1 m along the axis.The action is usually indicated by the letter . The symbol means kinetic energy. This integral means that the action is equal to the accumulated kinetic energy of the ball over the time interval from to.
In the first example (green trajectory) we moved the ball uniformly, i.e. with the same speed, which obviously should be equal to: m/s. The kinetic energy of the ball at each moment of time is equal to: = 1/2 J. In one second, 1/2 J of kinetic energy will accumulate. Those. action for such a trajectory is equal to: J s.
Now let's not immediately move the ball from point to point, but hold it at point for half a second, and then, over the remaining time, evenly move it to point. In the first half a second, the ball is at rest and its kinetic energy is zero. Therefore, the contribution to the action of this part of the trajectory is also zero. The second half a second we move the ball at double speed: m/s. The kinetic energy will be equal to = 2 J. The contribution of this period of time to the action will be equal to 2 J times half a second, i.e. 1 J s. Therefore, the total action for such a trajectory is equal to J s.
Similarly, any other trajectory with the boundary conditions 1) and 2) given by us corresponds to a certain number equal to the action for this trajectory. Among all such trajectories, there is a trajectory that has the least action. It can be proven that this trajectory is the green trajectory, i.e. uniform movement of the ball. For any other trajectory, no matter how tricky it is, the action will be more than 1/2.
In mathematics, such a comparison for each function of a certain number is called a functional. Quite often in physics and mathematics problems similar to ours arise, i.e. to find a function for which the value of a certain functional is minimal. For example, one of the problems that had great historical significance for the development of mathematics is the problem of the bachistochrone. Those. finding the curve along which the ball rolls the fastest. Again, each curve can be represented by a function h(x), and each function can be associated with a number, in this case the time of rolling the ball. Again, the problem comes down to finding a function for which the value of the functional is minimal. The branch of mathematics that deals with such problems is called the calculus of variations.
The first trajectory is obtained from the laws of physics and corresponds to the real trajectory of a free ball, on which no forces act and for which boundary conditions are specified in the form 1) and 2).
The second trajectory is obtained from the mathematical problem of finding a trajectory with given boundary conditions 1) and 2), for which the action is minimal.
The principle of least action states that these two trajectories must coincide. In other words, if it is known that the ball moved in such a way that the boundary conditions 1) and 2) were satisfied, then it necessarily moved along a trajectory for which the action is minimal compared to any other trajectory with the same boundary conditions.
One might consider this a mere coincidence. There are many problems in which uniform trajectories and straight lines appear. However, the principle of least action turns out to be a very general principle, valid in other situations, for example, for the motion of a ball in a uniform gravitational field. To do this, you just need to replace the kinetic energy with the difference between kinetic and potential energy. This difference is called the Lagrangian or Lagrangian function and the action now becomes equal to the total accumulated Lagrangian. In fact, the Lagrange function contains all the necessary information about the dynamic properties of the system.
If we launch a ball in a uniform gravitational field in such a way that it passes a point at an instant of time and arrives at a point at an instant of time, then, according to Newton’s laws, it will fly along a parabola. It is this parabola that will coincide with the trajectories for which the action will be minimal.
Thus, for a body moving in a potential field, for example, in the gravitational field of the Earth, the Lagrange function is equal to: . Kinetic energy depends on the speed of the body, and potential energy depends on its position, i.e. coordinates In analytical mechanics, the entire set of coordinates that determine the position of the system is usually denoted by one letter. For a ball moving freely in a gravitational field, means coordinates , and .To indicate the rate of change of any quantity, in physics very often they simply put a dot over this quantity. For example, it denotes the rate of change of coordinate, or, in other words, the speed of the body in the direction. Using these conventions, the speed of our ball in analytical mechanics is denoted as . Those. stands for velocity components.
Since the Lagrange function depends on speed and coordinates, and can also explicitly depend on time (explicitly depends on time means that the value is different at different times, for the same speeds and positions of the ball), then the action in general is written as
Strictly speaking, potential energy can be taken to be equal not to zero, but to any number, since the difference in potential energy at different points in space is important. However, changing the potential energy value does not affect the search for a trajectory with minimal action. It’s just that for all trajectories the action value will change to the same number, and the trajectory with the minimum action will remain the trajectory with the minimum action. For convenience, for our ball we will choose the potential energy equal to zero.Another possible physical trajectory with the same boundary conditions would be a trajectory in which the ball first flies to the right, passing the point at time . Then he collides with the spring, compresses it, the spring, straightening, pushes the ball back, and it again flies past the point. You can select the speed of the ball such that it bounces off the wall and passes the point exactly at the moment . The action on such a trajectory will be basically equal to the accumulated kinetic energy during the flight between the point and the wall and back. There will be some period of time when the ball will compress the spring and its potential energy will increase, and during this period of time the potential energy will make a negative contribution to the action. But such a period of time will not be very long and will not significantly reduce the effect.
The figure shows both physically possible trajectories of the ball's motion. The green trajectory corresponds to a ball at rest, while the blue trajectory corresponds to a ball bouncing off a spring wall.
However, only one of them has a minimal effect, namely the first! The second trajectory has more action. It turns out that in this problem there are two physically possible trajectories and only one with minimal action. Those. In this case, the principle of least action does not work.
On the graph I marked four special points in green. What do these points have in common? Let's imagine that the graph of a function is a real slide along which a ball can roll. The four designated points are special in that if you place the ball exactly at this point, it will not roll away anywhere. At all other points, for example, point E, he will not be able to stand still and will begin to slide down. Such points are called stationary. Finding such points is a useful task, since any maximum or minimum of a function, if it does not have sharp breaks, must necessarily be a stationary point.
If we more accurately classify these points, then point A is the absolute minimum of the function, i.e. its value is less than any other function value. Point B is neither a maximum nor a minimum and is called a saddle point. Point C is called a local maximum, i.e. the value in it is greater than at neighboring points of the function. And point D is a local minimum, i.e. the value in it is less than at neighboring points of the function.
The search for such points is carried out by a branch of mathematics called mathematical analysis. Otherwise, it is sometimes called infinitesimal analysis, since it can work with infinitesimal quantities. From the point of view of mathematical analysis, stationary points have one special property, thanks to which they are found. To understand what this property is, we need to understand what the function looks like at very small distances from these points. To do this, we will take a microscope and look through it at our points. The figure shows what the function looks like in the vicinity of various points at different magnifications.
It can be seen that at very high magnification (i.e. for very small deviations x) the stationary points look exactly the same and are very different from the non-stationary point. It is easy to understand what this difference is - the graph of a function at a stationary point becomes a strictly horizontal line when increased, and at a non-stationary point it becomes an inclined line. That is why a ball installed at a stationary point will not roll down.
The horizontality of a function at a stationary point can be expressed differently: the function at a stationary point practically does not change with a very small change in its argument, even compared to the change in argument itself. The function at a non-stationary point with a small change changes in proportion to the change. And the greater the slope of the function, the more the function changes when . In fact, as the function increases, it becomes more and more like a tangent to the graph at the point in question.
In strict mathematical language, the expression “a function practically does not change at a point with a very small change” means that the ratio of a change in a function and a change in its argument tends to 0 as it tends to 0:$$display$$\lim_(∆x \to 0) \frac (∆y(x_0))(∆x) = \lim_(x \to 0) \frac (y(x_0+∆x)-y(x_0) )(∆x) = 0$$display$$
For a non-stationary point, this ratio tends to a non-zero number, which is equal to the tangent of the slope of the function at this point. This same number is called the derivative of the function at a given point. The derivative of a function shows how quickly the function changes around a given point with a small change in its argument. Thus, stationary points are points at which the derivative of the function is equal to 0.
Again, in mathematical language, “slightly different” has the following precise meaning. Let us assume that we have a given functional for functions with the required boundary conditions 1) and 2), i.e. And . Let us assume that the trajectory is stationary.We can take any other function such that it takes zero values at the ends, i.e. = = 0. Let's also take a variable, which we will make smaller and smaller. From these two functions and the variable, we can compose a third function, which will also satisfy the boundary conditions and. As it decreases, the trajectory corresponding to the function will become increasingly closer to the trajectory.
Moreover, for stationary trajectories at small values of the functional for the trajectories will differ very little from the value of the functional for even in comparison with . Those.
$$display$$\lim_(ε \to 0) \frac (S(x"(t))-S(x(t)))ε=\lim_(ε \to 0) \frac (S(x( t)+εg(t))-S(x(t)))ε = 0$$display$$
Moreover, this should be true for any trajectory satisfying the boundary conditions = = 0.A change in the functional with a small change in the function (more precisely, the linear part of the change in the functional, proportional to the change in the function) is called a variation of the functional and is denoted by . The name “calculus of variations” comes from the term “variation”.
For stationary trajectories, variation of the functional.
A method for finding stationary functions (not only for the principle of least action, but also for many other problems) was found by two mathematicians - Euler and Lagrange. It turns out that a stationary function, whose functional is expressed by an integral similar to the action integral, must satisfy a certain equation, which is now called the Euler-Lagrange equation.
It turns out that real bodies may not necessarily move along trajectories with the least action. They can move along a wider set of special trajectories, namely stationary trajectories. Those. the real trajectory of the body will always be stationary. Therefore, the principle of least action is more correctly called the principle of stationary action. However, according to established tradition, it is often called the principle of least action, implying not only the minimality, but also the stationarity of trajectories.
Now we can write down the principle of stationary action in mathematical language, as it is usually written in textbooks: .If we return to the example with a ball and an elastic wall, then the explanation of this situation now becomes very simple. Under given boundary conditions that the ball must end up at a point at both time and time, there are two stationary trajectories. And the ball can actually move along any of these trajectories. To explicitly select one of the trajectories, you can impose an additional condition on the motion of the ball. For example, say that the ball should bounce off the wall. Then the trajectory will be determined unambiguously.Here these are generalized coordinates, i.e. a set of variables that uniquely define the position of the system.
- rate of change of generalized coordinates.
- Lagrange function, which depends on the generalized coordinates, their velocities and, possibly, time.
- an action that depends on the specific trajectory of the system (i.e. on ).The real trajectories of the system are stationary, i.e. for them a variation of the action.
Some remarkable consequences follow from the principle of least (more precisely stationary) action, which we will discuss in the next part.
