Consider oscillations of a weight m on a spring with stiffness coefficient k, which lies on a flat horizontal table, assuming that there is no friction of the weight on the table surface. If the weight is removed from the equilibrium position, it will oscillate about this position. We will describe these oscillations by a time-dependent function, assuming that it determines the deviation of the weight from its equilibrium position at time t.

In the horizontal direction, only one force acts on the weight - the elastic force of the spring, determined by the well-known Hooke's law

The deformation of the spring is a function of time, which is why it is also a variable.

From Newton's second law we have

because the acceleration is the second derivative of the displacement: .

Equation (9) can be rewritten in the form

where. This equation is called the harmonic oscillator equation.

Comment. In the mathematical literature, when writing a differential equation, one usually does not indicate the argument (t) near all functions that depend on it. This dependency is assumed by default. When using the mathematical package Maple in (10), it is necessary to indicate the explicit dependence of the function.

In contrast to the previous example of body motion under the action of a constant force, in our case the force changes over time, and equation (10) can no longer be solved using the usual integration procedure. Let's try to guess the solution of this equation, knowing that it describes some oscillatory process. As one of the possible solutions to equation (10), we can choose the following function:

Differentiating function (11), we have

Substituting expression (12) into equation (10), we make sure that it is satisfied identically for any value of t.

However, function (11) is not the only solution to the harmonic oscillator equation. For example, one can choose a function as another solution, which is also easy to check in a similar way. Moreover, one can check that any linear combination of these two randomly named solutions

with constant coefficients A and B is also a solution to the harmonic oscillator equation.

It can be proved that the two-constant solution (13) is the general solution of the harmonic oscillator equation (10). This means that formula (13) exhausts all possible solutions to this equation. In other words, the harmonic oscillator equation has no other particular solutions, except for those obtained from formula (13) by fixing arbitrary constants A and B.

Note that in physics it is most often necessary to look for just some particular solutions of individual ODEs or their systems. Let's consider this question in more detail.

It is possible to excite oscillations in the system of weight on a spring we are considering different ways. Let us set the following initial conditions

This means that at the initial moment of time, the weight was removed from the equilibrium position by a value a and freely released (ie, it starts its movement with zero initial speed). One can imagine many other ways of excitation, for example, a weight in the equilibrium position is “clicked” to give some initial speed, etc. [ general case, ].

We consider the initial conditions (14) as some additional conditions for separating from the general solution (13) some particular solution corresponding to our method of excitation of the weight oscillations.

Assuming t=0 in expression (13), we have, whence it follows that B=a. Thus, we have found one of the previously arbitrary constants in solution (13). Further, differentiating in formula (13), we have

Assuming t=0 in this expression and taking into account the second initial condition from (14), we obtain, hence it follows that A=0 and, thus, the initial particular solution has the form

It describes the oscillatory mode of the considered mechanical system, which is determined by the conditions of the initial excitation (14).

It is known from the school physics course that in formula (16) a is the amplitude of the oscillations (it sets the maximum deviation of the weight from its equilibrium position), is the cyclic frequency, and is the phase of the oscillations (the initial phase turns out to be equal to zero).

The harmonic oscillator equation (10) is an example of a linear ODE. This means that the unknown function and all its derivatives are included in each term of the equation to the first degree. Linear differential equations have an extremely important distinctive property: they satisfy the principle of superposition. This means that any linear combination of any two solutions of a linear ODE is also its solution.

In the example of the harmonic oscillator equation we are considering, an arbitrary linear combination of two particular solutions is not just some new solution, but a general solution to this equation (it exhausts all its possible solutions).

In general, this is not the case. For example, if we were dealing with a third-order linear differential equation (i.e., if the equation included a third derivative), then a linear combination of any two of its particular solutions would also be a solution to this equation, but would not represent him common decision.

In the course of differential equations, a theorem is proved that the general solution of an ODE of the Nth order (linear or non-linear) depends on N arbitrary constants. In the case of a nonlinear equation, these arbitrary constants can enter the general solution (in contrast to (13)), in a non-linear manner.

The superposition principle plays an extremely important role in the theory of ODEs, since it can be used to construct a general solution of a differential equation in the form of a superposition of its particular solutions. For example, for the case of linear ODEs with constant coefficients and their systems (the harmonic oscillator equation belongs precisely to this type of equations), a general solution method has been developed in the theory of differential equations. Its essence is as follows. We are looking for a particular solution in the form As a result of its substitution into the original equation, all time-dependent factors cancel and we arrive at some characteristic equation, which for the Nth order ODE is algebraic equation Nth degree. Solving it, we find, thereby, all possible particular solutions, an arbitrary linear combination of which gives the general solution of the original ODE. We will not dwell on this issue further, referring the reader to the appropriate textbooks on the theory of differential equations, where one can find further details, in particular, the case when the characteristic equation contains multiple roots.

If a linear ODE with variable coefficients is considered (its coefficients depend on time), then the superposition principle is also valid, but it is no longer possible to construct a general solution to this equation in an explicit form by any standard method. We will return to this issue later, discussing the phenomenon of parametric resonance and the Mathieu equation related to its study.

Perhaps the simplest mechanical system whose motion is described by a linear differential equation with constant coefficients is a mass on a spring. After a weight is hung from the spring, it will stretch a little to balance the force of gravity. Let us now follow the vertical deviations of the mass from the equilibrium position (Fig. 21.1). We denote upward deviations from the equilibrium position by and assume that we are dealing with a perfectly elastic spring. In this case, the forces opposing the stretch are directly proportional to the stretch. This means that the force is equal (the minus sign reminds us that the force opposes displacements). Thus, the acceleration multiplied by the mass should be equal to

For simplicity, let's assume that it happened (or we changed the system of units as necessary) that . We have to solve the equation

Fig. 21.1. A weight suspended on a spring. A simple example of a harmonic oscillator.

After that, we return to equation (21.2), in which and are contained explicitly.

We have already encountered equation (21.3) when we first began to study mechanics. We solved it numerically to find the motion. By numerical integration, we have found a curve that shows that if the particle is initially unbalanced, but at rest, then it returns to the equilibrium position. We did not follow the particle after it reached the equilibrium position, but it is clear that it will not stop there, but will oscillate (oscillate). With numerical integration, we found the time to return to the equilibrium point: . The duration of a complete cycle is four times longer: "sec". We found all this by numerical integration, because we didn’t know how to solve it better. But mathematicians have given us a certain function, which, if it is differentiated twice, goes into itself, multiplied by . (You can, of course, do the direct calculation of such functions, but this is much more difficult than just finding out the answer.)

This function is: . Let's differentiate it: , a . At the initial moment , , and the initial speed is equal to zero; these are exactly the assumptions that we made in numerical integration. Now, knowing that , we find the exact value of the time at which . Answer: , or 1.57108. We made a mistake earlier in the last sign, because the numerical integration was approximate, but the error is very small!

To move on, let's go back to the unit system, where time is measured in real seconds. What will be the solution in this case? Maybe we will take into account the constants and by multiplying by the corresponding factor ? Let's try. Let then and . To our chagrin, we did not succeed in solving equation (21.2), but again returned to (21.3). But we have discovered the most important property of linear differential equations: if we multiply the solution of the equation by a constant, then we again get the solution. It's mathematically clear why. If there is a solution to the equation, then after multiplying both parts of the equation by the derivatives, they will also be multiplied by and therefore satisfy the equation just as well as . Let's hear what the physicist has to say about this. If the weight stretches the spring twice as much as before, then the force will double, the acceleration will double, the acquired speed will be twice the previous speed, and in the same time the weight will cover twice the distance. But this is twice the distance - just the same distance that the weight needs to go to the equilibrium position. Thus, it takes the same amount of time to reach equilibrium and it does not depend on the initial bias. In other words, if the motion is described linear equation, then regardless of the "strength" it will develop in time in the same way.

The mistake did us good - we learned that by multiplying the solution by a constant, we get the solution of the previous equation. After some trial and error, you may come to the conclusion that instead of manipulating with, you need to change the time scale. In other words, equation (21.2) must have a solution of the form

(Here - not the angular velocity of a rotating body at all, but we will not have enough of all the alphabets if each value is denoted by a special letter.) We have provided the index 0 here, because we have many more omegas to meet: remember what corresponds to the natural movement of the oscillator. An attempt to use (21.4) as a solution is more successful because and . We finally solved the equation we wanted to solve. This equation coincides with (21.2) if .

Now we need to understand the physical meaning. We know that the cosine "repeats" after the angle changes to . Therefore it will be periodic motion; a full cycle of this movement corresponds to a change in the "angle" by . The quantity is often referred to as the phase of motion. To change to , you need to change to (full swing period); of course, is found from the equation . This means that you need to calculate for one cycle, and everything will be repeated if you increase by ; in this case we will increase the phase by . In this way,

. (21.5)

This means that the heavier the weight, the slower the spring will oscillate back and forth. Inertia in this case will be greater, and if the force does not change, then it will take more time to accelerate and decelerate the load. If you take a stiffer spring, then the movement should be faster; and indeed, the period decreases with increasing spring constant.

Note now that the period of oscillation of the mass on the spring does not depend on how the oscillations begin. For the spring, it seems to be indifferent how much we stretch it. The equation of motion (21.2) determines the period of oscillation, but says nothing about the amplitude of the oscillation. Of course, the oscillation amplitude can be determined, and we will now deal with it, but for this it is necessary to set the initial conditions.

The point is that we have not yet found the most general solution of equation (21.2). There are several types of solutions. The solution corresponds to the case when at the initial moment the spring is stretched and its velocity is equal to zero. You can make the spring move in another way, for example, seize the moment when the balanced spring is at rest, and hit the weight sharply; this will mean that at the moment some speed is reported to the spring. Such a movement will correspond to another solution (21.2) - the cosine must be replaced by a sine. Let's throw one more stone into the cosine: if - solution, then, entering the room where the spring swings, at the moment (let's call it ""), when the weight passes through the equilibrium position, we will be forced to replace this solution with another one. Therefore, there cannot be a general solution; the general solution must allow, so to speak, the displacement of the origin of time. Such a property has, for example, the solution , where is some constant. Further, one can decompose called the angular frequency; is the number of radians by which the phase changes in 1 second. It is determined by a differential equation. Other quantities are not determined by the equation, but depend on initial conditions. The constant serves as a measure of the maximum deviation of the load and is called the oscillation amplitude. The constant is sometimes called the oscillation phase, but there may be misunderstandings here, because others call the phase and say that the phase depends on time. We can say that - this is a phase shift compared to some, taken as zero. Let's not argue about words. Different correspond to movements with different phases. This is true, but whether to call it a phase or not is another question.

Discoveries in the quantum field and other areas. At the same time, new devices and devices are invented, through which it is possible to conduct various studies and explain the phenomena of the microworld. One of these mechanisms is the harmonic oscillator, the principle of which was known even by representatives of ancient civilizations.

The device and its types

A harmonic oscillator is a mechanical system in motion, which is described by a differential with coefficients of constant value. Most simple examples such devices - a load on a spring, a pendulum, acoustic systems, the movement of molecular particles, etc.

Conventionally, the following types of this device can be distinguished:

Device Application

This device is used in various fields, mainly to study the nature of oscillatory systems. The quantum harmonic oscillator is used to study the behavior of photon elements. The results of experiments can be used in various fields. So, physicists from the American Institute found that beryllium atoms, located at fairly large distances from each other, can interact at the quantum level. At the same time, the behavior of these particles is similar to bodies (metal balls) in the macrocosm, moving in a forward-return order, similar to a harmonic oscillator. Beryllium ions, despite being physically long distances, exchanged the smallest units of energy (quanta). This discovery allows to significantly advance IT-technologies, and also provides a new solution in the production of computer equipment and electronics.

The harmonic oscillator is used in the evaluation of musical works. This method is called spectroscopic examination. At the same time, it was found that the most stable system is a composition of four musicians (a quartet). And modern works are mostly anharmonic.

HARMONIC OSCILLATIONS

Lecture 1

VASCULATION

VASCULATION. WAVES. OPTICS

Oscillation is one of the most common processes in nature and technology. Fluctuations are processes that repeat over time. High-rise buildings and high-voltage wires oscillate under the influence of the wind, the pendulum of a wound clock and a car on springs during movement, the level of the river during the year and the temperature of the human body during illness. Sound is fluctuations in air pressure, radio waves are periodic changes in the strength of the electrical and magnetic field, light is also electromagnetic oscillations. Earthquakes - ground vibrations, ebbs and flows - changes in the levels of the seas and oceans caused by the attraction of the moon, etc.

Oscillations are mechanical, electromagnetic, chemical, thermodynamic, etc. Despite such a variety, all oscillations are described by the same differential equations.

The first scientists to study vibrations were Galileo Galilei and Christian Huygens. Galileo established the independence of the period of oscillations from the amplitude. Huygens invented the pendulum clock.

Any system that, when slightly out of balance, oscillates steadily is called a harmonic oscillator. In classical physics, such systems are a mathematical pendulum within small deflection angles, a load within small oscillation amplitudes, an electrical circuit consisting of linear capacitance and inductance elements.

A harmonic oscillator can be considered linear if the displacement from the equilibrium position is directly proportional to the perturbing force. The oscillation frequency of a harmonic oscillator does not depend on the amplitude. For the oscillator, the principle of superposition is fulfilled - if several perturbing forces act, then the effect of their total action can be obtained as a result of adding the effects from active forces separately.

Harmonic oscillations are described by the equation (Fig. 1.1.1)

(1.1.1)

where X- displacement of the oscillating value from the equilibrium position, BUT– amplitude of oscillations equal to the value of the maximum displacement, - phase of oscillations, which determines the displacement at the time , - initial phase, which determines the magnitude of the displacement at the initial moment of time, - cyclic frequency of oscillations.

The time of one complete oscillation is called the period, where is the number of oscillations completed during the time.

The oscillation frequency determines the number of oscillations per unit time, it is related to the cyclic frequency by the ratio, then the period.

The speed of an oscillating material point

acceleration

Thus, the speed and acceleration of the harmonic oscillator also change according to the harmonic law with amplitudes and respectively. In this case, the speed is ahead of the phase displacement by , and acceleration - by (Fig. 1.1.2).

From a comparison of the equations of motion of a harmonic oscillator (1.1.1) and (1.1.2) it follows that , or

it differential equation second order is called the harmonic oscillator equation. His solution contains two constants a and , which are determined by setting the initial conditions

.

If a periodically repeating process is described by equations that do not coincide with (1.1.1), it is called anharmonic. A system that performs anharmonic oscillations is called an anharmonic oscillator.

1.1.2 . Free oscillations of systems with one degree of freedom. complex form representations of harmonic vibrations

In nature, small oscillations that a system makes near its equilibrium position are very common. If a system taken out of equilibrium is left to itself, that is, external forces do not act on it, then such a system will perform free undamped oscillations. Consider a system with one degree of freedom.

A stable equilibrium corresponds to a position of the system in which its potential energy has a minimum ( q is the generalized coordinate of the system). The deviation of the system from the equilibrium position leads to the emergence of a force that tends to bring the system back. We denote the value of the generalized coordinate corresponding to the equilibrium position, then the deviation from the equilibrium position

We will count the potential energy from the minimum value . Let's take the resulting function, expand it in a Maclaurin series and leave the first term of the expansion, we have: o

VASCULATION. WAVES. OPTICS

VASCULATION

Lecture 1

HARMONIC OSCILLATIONS

Ideal harmonic oscillator. The equation ideal oscillator and his decision. Amplitude, frequency and phase of oscillations

Oscillation is one of the most common processes in nature and technology. Fluctuations are processes that repeat over time. High-rise buildings and high-voltage wires oscillate under the influence of the wind, the pendulum of a wound clock and a car on springs during movement, the level of the river during the year and the temperature of the human body during illness. Sound is fluctuations in air pressure, radio waves are periodic changes in the strength of the electric and magnetic fields, light is also electromagnetic vibrations. Earthquakes - ground vibrations, ebbs and flows - changes in the levels of the seas and oceans caused by the attraction of the moon, etc.

Oscillations are mechanical, electromagnetic, chemical, thermodynamic, etc. Despite such a variety, all oscillations are described by the same differential equations.

The first scientists to study vibrations were Galileo Galilei and Christian Huygens. Galileo established the independence of the period of oscillations from the amplitude. Huygens invented the pendulum clock.

Any system that, when slightly out of balance, oscillates steadily is called a harmonic oscillator. In classical physics, such systems are a mathematical pendulum within small deflection angles, a load within small oscillation amplitudes, an electrical circuit consisting of linear capacitance and inductance elements.

A harmonic oscillator can be considered linear if the displacement from the equilibrium position is directly proportional to the perturbing force. The oscillation frequency of a harmonic oscillator does not depend on the amplitude. For the oscillator, the principle of superposition is fulfilled - if several perturbing forces act, then the effect of their total action can be obtained as a result of adding the effects of the acting forces separately.

Harmonic oscillations are described by the equation (Fig. 1.1.1)

(1.1.1)

where X- displacement of the oscillating value from the equilibrium position, BUT– amplitude of oscillations equal to the value of the maximum displacement, - phase of oscillations, which determines the displacement at the time , - initial phase, which determines the magnitude of the displacement at the initial moment of time, - cyclic frequency of oscillations.

The time of one complete oscillation is called the period, where is the number of oscillations completed during the time.

The oscillation frequency determines the number of oscillations per unit time, it is related to the cyclic frequency by the ratio, then the period.

The speed of an oscillating material point

acceleration

Thus, the speed and acceleration of the harmonic oscillator also change according to the harmonic law with amplitudes and respectively. In this case, the speed is ahead of the phase displacement by , and acceleration - by (Fig. 1.1.2).

From a comparison of the equations of motion of a harmonic oscillator (1.1.1) and (1.1.2) it follows that , or

This second-order differential equation is called the harmonic oscillator equation. His solution contains two constants a and , which are determined by setting the initial conditions

.

If a periodically repeating process is described by equations that do not coincide with (1.1.1), it is called anharmonic. A system that performs anharmonic oscillations is called an anharmonic oscillator.

1.1.2 . Free oscillations of systems with one degree of freedom. Complex form of representation of harmonic oscillations

In nature, small oscillations that a system makes near its equilibrium position are very common. If a system taken out of equilibrium is left to itself, that is, external forces do not act on it, then such a system will perform free undamped oscillations. Consider a system with one degree of freedom.

A stable equilibrium corresponds to a position of the system in which its potential energy has a minimum ( q is the generalized coordinate of the system). The deviation of the system from the equilibrium position leads to the emergence of a force that tends to bring the system back. We denote the value of the generalized coordinate corresponding to the equilibrium position, then the deviation from the equilibrium position

We will count the potential energy from the minimum value . Let's take the resulting function, expand it in a Maclaurin series and leave the first term of the expansion, we have: o

,

where . Then, taking into account the introduced notation:

, (1.1.4)

Taking into account the expression (1.1.4) for the force acting on the system, we obtain:

According to Newton's second law, the equation of motion of the system has the form:

Expression (1.1.5) coincides with the equation (1.1.3) of free harmonic oscillations, provided that

and has two independent solutions: and , so the general solution is:

,

From formula (1.1.6) it follows that the frequency is determined only by the intrinsic properties of the mechanical system and does not depend on the amplitude and on the initial conditions of motion.

The dependence of the coordinate of the oscillating system on time can be determined as the real part of the complex expression , where A=Xe-iα is a complex amplitude, its modulus coincides with the usual amplitude, and its argument coincides with the initial phase.

1.1.3 . Examples of oscillatory motions of various physical nature

Fluctuations of the load on the spring

Consider the oscillations of a load on a spring, provided that the spring is not deformed beyond the limits of elasticity. We will show that such a load will perform harmonic oscillations relative to the equilibrium position (Fig. 1.1.3). Indeed, according to Hooke's law, a compressed or stretched spring creates a harmonic force:

where - coefficient of spring stiffness, is the coordinate of the equilibrium position, X is the coordinate of the load (material point) at the moment of time , is the displacement from the equilibrium position.

Let us place the origin of the coordinate in the equilibrium position of the system. In this case .

If the spring is stretched by X, then release at time t=0, then the equation of motion of the load according to Newton's second law will take the form -kx=ma, or , and

(1.1.6)

This equation coincides in form with the equation of motion (1.1.3) of a system performing harmonic oscillations, we will look for its solution in the form:

. (1.1.7)

We substitute (1.17) into (1.1.6), we have: that is, expression (1.1.7) is a solution to equation (1.1.6) provided that

If at the initial moment of time the position of the load was arbitrary, then the equation of motion will take the form:

.

Let's consider how the energy of the load changes, making harmonic oscillations in the absence of external forces (Fig. 1.14). If at the time t=0 send offset to cargo x=A, then its total energy will become equal to the potential energy of the deformed spring, kinetic energy equals zero (point 1).

Force acting on the load F= -kx, seeking to return it to the equilibrium position, so the load moves with acceleration and increases its speed, and, consequently, its kinetic energy. This force reduces the displacement of the load X, the potential energy of the load decreases, turning into kinetic. The "load - spring" system is closed, so its total energy is conserved, that is:

. (1.1.8)

At the moment of time, the load is in equilibrium (point 2), its potential energy is zero, and its kinetic energy is maximum. We find the maximum speed of the load from the law of conservation of energy (1.1.8):

Due to the stock of kinetic energy, the load does work against the elastic force – and passes through the equilibrium position. Kinetic energy gradually turns into potential. When the load has a maximum negative displacement - BUT, kinetic energy wk=0, the load stops and starts moving to the equilibrium position under the action of an elastic force F= -kx. Further movement is similar.

Pendulums

Under the pendulum understand solid which, under the influence of gravity, oscillates around a fixed point or axis. There are physical and mathematical pendulums.

A mathematical pendulum is an idealized system consisting of a weightless inextensible thread on which a mass concentrated at one material point is suspended.

A mathematical pendulum, for example, is a ball on a long thin thread.

The deviation of the pendulum from the equilibrium position is characterized by the angle φ , which forms a thread with a vertical (Fig. 1.15). When the pendulum deviates from the equilibrium position, a moment of external forces (gravity) arises: , where m- weight, - pendulum length

This moment tends to return the pendulum to the equilibrium position (similar to the quasi-elastic force) and is directed opposite to the displacement φ , so there is a minus sign in the formula.

The equation for the dynamics of rotational motion for a pendulum has the form: Iε=,

.

We will consider the case of small fluctuations, therefore sin φ ≈φ, denote ,

we have: , or , and finally

This is the equation of harmonic oscillations, its solution:

.

The oscillation frequency of a mathematical pendulum is determined only by its length and the acceleration of gravity, and does not depend on the mass of the pendulum. The period is:

If the oscillating body cannot be represented as a material point, then the pendulum is called physical (Fig. 1.1.6). We write the equation of its motion in the form:

.

In the case of small fluctuations , or =0 , where . This is the equation of motion of a body that performs harmonic oscillations. The oscillation frequency of a physical pendulum depends on its mass, length and moment of inertia about the axis passing through the suspension point.

Let's denote . Value is called the reduced length of the physical pendulum. This is the length of a mathematical pendulum whose period of oscillation coincides with the period of a given physical pendulum. A point on a straight line connecting the suspension point with the center of mass, lying at a distance of the reduced length from the axis of rotation, is called the swing center of the physical pendulum ( O'). If the pendulum is suspended in the center of the swing, then the reduced length and period of oscillation will be the same as at the point O. Thus, the suspension point and the swing center have the properties of reciprocity: when the suspension point is transferred to the swing center, the old suspension point becomes the new swing center.

A mathematical pendulum that swings with the same period as the physical one under consideration is called isochronous to the given physical pendulum.

1.1.4. Addition of vibrations (beats, Lissajous figures). Vector description of vibration addition

The addition of equally directed oscillations can be performed using the method of vector diagrams. Any harmonic oscillation can be represented as a vector as follows. Let's choose an axis X with origin at point O(fig.1.1.7)

From a point O construct a vector that makes up the angle with axle X. Let this vector rotate with angular velocity . Projection of a vector onto an axis X is equal to:

that is, it performs harmonic oscillations with an amplitude a.

Consider two harmonic oscillations of the same direction and the same cyclic small , given by the vectors and . Offsets along the axis X are equal:

the resulting vector has a projection and represents the resulting oscillation (Fig. 1.1.8), according to the cosine theorem Thus, the addition of harmonic oscillations is carried out by adding the vectors.

Let us carry out the addition of mutually perpendicular oscillations. Let the material point make two mutually perpendicular oscillations with a frequency:

.

The material point itself will then move along some curvilinear trajectory.

From the equation of motion follows: ,

. (1.1.9)

From equation (1.1.9) you can get the ellipse equation (Fig.1.1.9):

Consider special cases of this equation:

1. Oscillation phase difference α= 0. At the same time those. or This is the equation of a straight line, and the resulting oscillation occurs along this straight line with amplitude (Fig. 1.1.10). a.

its acceleration is equal to the second derivative of the displacement with respect to time then the force acting on the oscillating point, according to Newton's second law, is equal to

That is, the force is proportional to the displacement X and is directed against the displacement to the equilibrium position. This force is called the restoring force. In the case of a load on a spring, the restoring force is the elastic force, in the case of a mathematical pendulum, it is the component of gravity.

The restoring force by nature obeys Hooke's law F=-kx, where

is the coefficient of the restoring force. Then the potential energy of the oscillating point is:

(the integration constant is chosen equal to zero, so that when X).

Anharmonic Oscillator