New physical device - the heart

The slender tower located in the Italian city of Pisa is well known to everyone from numerous paintings and photographs. She is known not only for her proportions and grace, but also for the misfortune hanging over her. The tower slowly but noticeably deviates from the vertical, as if bowing.

The "leaning" Leaning Tower of Pisa is located in the city where the contemporary great Italian scientist was born and carried out many scientific studies. Galileo Galilei. In his hometown, Galileo became a university professor. A professor of mathematics, although he was engaged not only in mathematics, but also in optics, astronomy, and mechanics.

Let's imagine that on one of the beautiful summer days in those distant years we stand near the Leaning Tower of Pisa, raise our heads and see on the upper gallery ... Galileo. A scientist admiring a beautiful view of the city? No, he, like a playful schoolboy, throws various objects down!

The openwork Leaning Tower of Pisa was an involuntary witness to the experiments of Galileo Galilei.

Perhaps our surprise will increase even more if someone at this time says that we are present at one of the most important physical experiments in the history of science.

Aristotle, a broad-minded thinker who lived in the 4th century BC, argued that a light body falls from a height more slowly than a heavy one. The authority of the scientist was so great that this statement was considered absolutely true for thousands of years. Our everyday observations, moreover, often seem to confirm Aristotle's thought - light leaves slowly and smoothly fly off the trees in the autumn forest, heavy and fast heavy hail knocks on the roof ...

But it was not for nothing that Galileo once said: "... in the sciences, thousands of authorities are not worth one modest and true statement." He doubted the correctness of Aristotle.

Careful observation of the swinging of the lamps in the cathedral helped Galileo to establish the laws of the movement of pendulums.

How will both bodies behave - light and heavy, if they are fastened together? Having asked himself this question, Galileo reasoned further: a light body should slow down the movement of a heavy one, but together they make up an even heavier body and, therefore, must (according to Aristotle) ​​fall even faster.

Where is the way out of this logical impasse? It remains only to assume that both bodies must fall at the same speed.

The experiments are noticeably affected by the air - a dry leaf of a tree slowly sinks to the ground thanks to the gentle breezes of the wind.

The experiment must be carried out with bodies of different weights, but of approximately the same streamlined shape, so that the air does not make its own “corrections” to the phenomenon under study.

And Galileo drops from the Leaning Tower of Pisa at the same moment a cannonball weighing 80 kilograms and a much lighter musket bullet - weighing only 200 grams. Both bodies hit the ground at the same time!

Galileo Galilei. He harmoniously combined the talents of a theoretical physicist and an experimenter.

Galileo wanted to study the behavior of bodies when they were not moving so fast. He made a rectangular chute with well-polished walls from long wooden blocks, set it at an angle and let heavy balls down it (carefully, without pushing).

Good clocks did not yet exist, and Galileo judged the time it took for each experiment, weighing the amount of water flowing out of a large barrel through a thin tube.

With the help of such "scientific" instruments, Galileo established an important regularity: the distance traveled by the ball is proportional to the square of time, which confirmed the idea that he had matured about the possibility of a body moving with constant acceleration.

Once in the cathedral, watching how lamps of different sizes and lengths sway, Galileo came to the conclusion that all lamps suspended on threads of the same length have the period of swing from one top point to another and the height of the rises are the same and constant - regardless of weight! How to confirm an unusual and, as it turned out later, absolutely correct conclusion? With what to compare the oscillations of pendulums, where to get the standard of time? And Galileo came up with a solution that for many generations of scientists will serve as an example of the brilliance and wit of physical thought: he compared the oscillations of a pendulum with the frequency of the beating of his own heart!

Appearance and device of the first pendulum clock invented by Christian Huygens.

Only more than three hundred years later, in the middle of the 20th century, another great Italian, Enrico Fermi, will set up an experiment reminiscent of Galileo's achievements in simplicity and accuracy. Fermi will determine the force of the explosion of the first experimental atomic bomb by the distance at which the blast wave will carry paper petals from his palm ...

The constancy of oscillations of lamps and pendulums of the same length was proved by Galileo, and on the basis of this remarkable property of oscillating bodies, Christian Huygens in 1657 created the first pendulum clock with a regular course.

We all are well aware of the cozy clock with the "talking" cuckoo living in it, which arose thanks to Galileo's powers of observation, which did not leave him even during divine services in the cathedral.

Christian Huygens is a Dutch scientist, mathematician, astronomer and physicist, one of the founders of wave optics. In 1665-81 he worked in Paris. Invented (1657) a pendulum clock with an escapement, gave their theory, established the laws of oscillation of a physical pendulum, laid the foundations for the theory of impact. Created (1678, published 1690) the wave theory of light, explained birefringence. Together with Robert Hooke, he established the constant points of the thermometer. Improved the telescope; designed an eyepiece named after him. Discovered the ring of Saturn and its satellite Titan. Author of one of the first works on the theory of probability (1657).

Early awakening of talents

The ancestors of Christian Huygens occupied a prominent place in the history of his country. His father Konstantin Huygens (1596-1687), in whose house the future famous scientist was born, was a well-educated person, knew languages, was fond of music; after 1630 he became an adviser to Wilhelm II (and later William III). King James I elevated him to the rank of knight, and Louis XIII granted him the Order of Saint Michael. His children - 4 sons (the second - Christians) and one daughter - also left a good mark on history.

Christian's giftedness manifested itself at an early age. At the age of eight, he already studied Latin and arithmetic, learned to sing, and at the age of ten he became acquainted with geography and astronomy. In 1641, his tutor wrote to the child's father: "I see and almost envy the remarkable memory of Christian," and two years later: "I confess that Christian must be called a miracle among boys."

And Christian at this time, having studied Greek, French and Italian and having mastered the game on the harpsichord, became interested in mechanics. But not only that: he willingly engages in swimming, dancing and horseback riding. At the age of sixteen, Christian Huygens, together with his older brother Konstantin, entered the University of Leiden for training in law and mathematics (the latter was more willing and successful; the teacher decides to send one of his works to Rene Descartes).

After 2 years, the elder brother begins to work for Prince Frederik Henrik, and Christian and his younger brother move to Breda, to the Orange College. His father also prepared Christian for public service, but he had other aspirations. In 1650, he returned to The Hague, where his scientific work was hindered only by headaches that had haunted him for some time.

The more difficult the task of determining by reasoning what seems indeterminate and subject to chance, the more the science that achieves the result seems surprising.

Huygens Christian

First scientific works

The range of scientific interests of Christian Huygens continued to expand. He is fond of the works of Archimedes on mechanics and Descartes (and later of other authors, including the English Newton and Hooke) on optics, but does not stop studying mathematics. In mechanics, his main research relates to the theory of impact and to the problem of designing clocks, which at that time was of exceptionally important applied importance and always occupied one of the central places in Huygens's work.

His first achievements in optics can also be called "applied". Together with his brother Constantine Christian Huygens is engaged in the improvement of optical instruments and achieves significant success in this area (this activity does not stop for many years; in 1682 he invents a three-lens eyepiece, which still bears his name. While improving telescopes, Huygens, however, in the Diopter ” wrote: “... a person: who could invent a telescope, based only on theory, without the intervention of chance, would have to have a superhuman mind”).

New instruments allow important observations to be made: On March 25, 1655, Huygens discovers Titan, the largest satellite of Saturn (whose rings he had been interested in for a long time). In 1657, another work by Huygens appeared, “On Calculations when Playing Dice,” one of the first works on the theory of probability. He writes another essay "On the Impact of Bodies" for his brother.

In general, the fifties of the 17th century were the time of the greatest activity of Huygens. He gains notoriety in the scientific world. In 1665 he was elected a member of the Paris Academy of Sciences.

"Huygens principle"

H. Huygens studied Newton's optical works with unflagging interest, but did not accept his corpuscular theory of light. Much closer to him were the views of Robert Hooke and Francesco Grimaldi, who believed that light has a wave nature.

But the concept of light-wave immediately gave rise to many questions: how to explain the rectilinear propagation of light, its reflection and refraction? Newton gave seemingly convincing answers to them. Rectilinearity is a manifestation of the first law of dynamics: light corpuscles move uniformly and rectilinearly if no forces act on them. Reflection was also explained as an elastic rebound of corpuscles from the surfaces of bodies. The situation with refraction was somewhat more complicated, but even here Newton offered an explanation. He believed that when a light corpuscle flies up to the boundary of the body, an attraction force from the side of the substance begins to act on it, imparting acceleration to the corpuscle. This leads to a change in the direction of the velocity of the corpuscle (refraction) and its magnitude; therefore, according to Newton, the speed of light in glass, for example, is greater than in vacuum. This conclusion is important, if only because it allows for experimental verification (experiment later refuted Newton's opinion).

Christian Huygens, like his predecessors mentioned above, believed that all space is filled with a special medium - ether, and that light is waves in this ether. Using the analogy with waves on the surface of water, Huygens came up with the following picture: when the front (i.e., the leading edge) of the wave reaches a certain point, i.e., the oscillations reach this point, then these oscillations become the centers of new waves diverging in all directions , and the movement of the envelope of all these waves gives a picture of the propagation of the wave front, and the direction perpendicular to this front is the direction of wave propagation. So, if the wave front in the void at some point is flat, then it always remains flat, which corresponds to the rectilinear propagation of light. If the front of a light wave reaches the boundary of the medium, then each point on this boundary becomes the center of a new spherical wave, and, having constructed the envelopes of these waves in space both above and below the boundary, it is easy to explain both the law of reflection and the law of refraction (but at In this case, one has to accept that the speed of light in a medium is n times less than in vacuum, where it is n - the same refractive index of the medium, which is included in the law of refraction recently discovered by Descartes and Snell).

It follows from the Huygens principle that light, like any wave, can also go around obstacles. This phenomenon, which is of fundamental interest, does exist, but Huygens considered that the "side waves" that arise during such an envelope do not deserve much attention.

Christian Huygens' ideas about light were far from modern. So, he believed that light waves are longitudinal, i.e. that the directions of oscillations coincide with the direction of wave propagation. This may seem all the more strange since Huygens himself apparently already had an idea of ​​the phenomenon of polarization, which can only be understood by considering transverse waves. But this is not the main thing. Huygens' principle had a decisive influence on our ideas not only about optics, but also about the physics of any oscillations and waves, which now occupies one of the central places in our science. (V. I. Grigoriev)

More about Christian Huygens:

Christian Huygens von Zuylichen - the son of the Dutch nobleman Constantine Huygens "Talents, nobility and wealth were, apparently, hereditary in the family of Christian Huygens," wrote one of his biographers. His grandfather was a writer and dignitary, his father was a secret adviser to the princes of Orange, a mathematician, and a poet. Faithful service to their sovereigns did not enslave their talents, and it seemed that Christian was destined for the same enviable fate for many. He studied arithmetic and Latin, music and versification. Heinrich Bruno, his teacher, could not get enough of his fourteen-year-old pupil:

“I confess that Christian must be called a miracle among boys ... He deploys his abilities in the field of mechanics and construction, makes amazing machines, but hardly necessary.” The teacher was wrong: the boy is always looking for the benefits of his studies. His concrete, practical mind will soon find schemes of machines that people really need.

However, he did not immediately devote himself to mechanics and mathematics. The father decided to make his son a lawyer and, when Christian reached the age of sixteen, he sent him to study law at the University of London. Being engaged in legal sciences at the university, Huygens at the same time is fond of mathematics, mechanics, astronomy, and practical optics. A skilled craftsman, he grinds optical glasses on his own and improves the pipe, with the help of which he will later make his astronomical discoveries.

Christian Huygens was the immediate successor of Galileo-Galilei in science. According to Lagrange, Huygens "was destined to improve and develop the most important discoveries of Galileo." There is a story about how for the first time Huygens came into contact with the ideas of Galileo. Seventeen-year-old Huygens was going to prove that bodies thrown horizontally move along parabolas, but, having found the proof in the book of Galileo, he did not want to "write the Iliad after Homer."

After graduating from the university, Christian Huygens becomes an adornment of the retinue of the Count of Nassau, who, on a diplomatic mission, is on his way to Denmark. The count is not interested in the fact that this handsome young man is the author of curious mathematical works, and he, of course, does not know how Christian dreams of getting from Copenhagen to Stockholm to see Descartes. So they will never meet: in a few months Descartes will die.

At the age of 22, Christian Huygens publishes Discourses on the Square of the Hyperbola, Ellipse, and Circle. In 1655, he builds a telescope and discovers one of Saturn's satellites, Titan, and publishes New Discoveries in the Size of a Circle. At the age of 26, Christian writes notes on dioptrics. At the age of 28, his treatise “On Calculations when Playing Dice” was published, where one of the first ever research in the field of probability theory is hidden behind a seemingly frivolous title.

One of Huygens' most important discoveries was the invention of the pendulum clock. He patented his invention on July 16, 1657 and described it in a short essay published in 1658. He wrote about his watch to the French king Louis XIV: “My automata, placed in your apartments, not only amaze you every day with the correct indication of time, but they are suitable, as I hoped from the very beginning, for determining the longitude of a place on the sea.” Christian Huygens was engaged in the task of creating and improving clocks, especially pendulum clocks, for almost forty years: from 1656 to 1693. A. Sommerfeld called Huygens "the most brilliant watchmaker of all time."

At the age of thirty, Christian Huygens reveals the secret of Saturn's ring. The rings of Saturn were first noticed by Galileo as two lateral appendages "supporting" Saturn. Then the rings were visible, like a thin line, he did not notice them and did not mention them again. But Galileo's pipe did not have the necessary resolution and sufficient magnification. Watching the sky with a 92x telescope. Christian discovers that the ring of Saturn was taken as side stars. Huygens solved the riddle of Saturn and for the first time described its famous rings.

At that time Christian Huygens was a very handsome young man with large blue eyes and a neatly trimmed mustache. The reddish curls of the wig, coolly curled in the fashion of that time, fell to the shoulders, lying on the snow-white Brabant lace of an expensive collar. He was friendly and calm. No one saw him especially agitated or confused, in a hurry somewhere, or, on the contrary, immersed in slow thoughtfulness. He did not like to be in the “light” and rarely appeared there, although his origin opened the doors of all the palaces of Europe to him. However, when he appeared there, he did not look at all awkward or embarrassed, as often happened to other scientists.

But in vain the charming Ninon de Lanclos seeks his company, he is invariably friendly, no more, this convinced bachelor. He can drink with friends, but not much. Sneak a little, laugh a little. A little bit of everything, a very little bit, so that as much time as possible is left for the main thing - work. Work - an unchanging all-consuming passion - burned him constantly.

Christian Huygens was distinguished by extraordinary dedication. He was aware of his abilities and sought to use them to the fullest. “The only entertainment that Huygens allowed himself in such abstract works,” one of his contemporaries wrote about him, “was that he was engaged in physics in between. What for an ordinary person was a tedious task, for Huygens was entertainment.

In 1663 Huygens was elected a Fellow of the Royal Society of London. In 1665, at the invitation of Colbert, he settled in Paris and the following year became a member of the newly organized Paris Academy of Sciences.

In 1673, his work "Pendulum Clock" was published, where the theoretical foundations of Huygens' invention were given. In this work, Huygens establishes that the cycloid has the property of isochronism, and analyzes the mathematical properties of the cycloid.

Investigating the curvilinear motion of a heavy point, Huygens, continuing to develop the ideas expressed by Galileo, shows that a body, when falling from a certain height along various paths, acquires a finite velocity that does not depend on the shape of the path, but depends only on the height of the fall, and can rise to a height equal (in the absence of resistance) to the initial height. This proposition, which essentially expresses the law of conservation of energy for motion in a gravitational field, is used by Huygens for the theory of the physical pendulum. He finds an expression for the reduced length of the pendulum, establishes the concept of the swing center and its properties. He expresses the formula of a mathematical pendulum for cycloidal motion and small oscillations of a circular pendulum as follows:

"The time of one small oscillation of a circular pendulum is related to the time of falling down twice the length of the pendulum, as the circumference of a circle is related to the diameter."

It is significant that at the end of his essay the scientist gives a number of proposals (without a conclusion) about the centripetal force and establishes that the centripetal acceleration is proportional to the square of the speed and inversely proportional to the radius of the circle. This result prepared the Newtonian theory of the motion of bodies under the action of central forces.

From the mechanical research of Christian Huygens, in addition to the theory of the pendulum and centripetal force, his theory of the impact of elastic balls is known, which he presented for a competitive task announced by the Royal Society of London in 1668. Huygens' impact theory is based on the law of conservation of living forces, momentum and Galileo's principle of relativity. It was not published until after his death in 1703. Huygens traveled quite a lot, but he was never an idle tourist. During the first trip to France, he studied optics, and in London he explained the secrets of making his telescopes. Fifteen years he worked at the court of Louis XIV, fifteen years of brilliant mathematical and physical research. And in fifteen years - only two short trips to his homeland to heal

Christian Huygens lived in Paris until 1681, when, after the repeal of the Edict of Nantes, he returned to his homeland as a Protestant. While in Paris, he knew Römer well and actively assisted him in the observations that led to the determination of the speed of light. Huygens was the first to report Römer's results in his treatise.

At home, in Holland, again not knowing fatigue, Huygens builds a mechanical planetarium, giant seventy-meter telescopes, describes the worlds of other planets.

Huygens' work in Latin appears on light, corrected by the author and republished in French in 1690. Huygens' Treatise on Light entered the history of science as the first scientific work on wave optics. This "Treatise" formulated the principle of wave propagation, now known as Huygens' principle. Based on this principle, the laws of reflection and refraction of light were derived, and the theory of double refraction in Icelandic spar was developed. Since the speed of propagation of light in a crystal is different in different directions, the shape of the wave surface will not be spherical, but ellipsoidal.

The theory of propagation and refraction of light in uniaxial crystals is a remarkable achievement of Huygens' optics. Christian Huygens also described the disappearance of one of the two rays when they pass through the second crystal with a certain orientation relative to the first. Thus, Huygens was the first physicist to establish the fact of light polarization.

Huygens' ideas were highly valued by his successor Fresnel. He ranked them above all discoveries in Newton's optics, arguing that Huygens' discovery "is perhaps more difficult to make than all Newton's discoveries in the field of light phenomena."

Huygens does not consider colors in his treatise, as well as the diffraction of light. His treatise is devoted only to the justification of reflection and refraction (including double refraction) from the wave point of view. This circumstance was probably the reason why Huygens' theory, despite its support in the 18th century by Lomonosov and Euler, did not receive recognition until Fresnel resurrected the wave theory on a new basis in the early 19th century.

Christian Huygens died on June 8, 1695, when KosMoteoros, his last book, was being printed in the printing house. (Samin D.K. 100 great scientists. - M .: Veche, 2000)

More about Christian Huygens:

Huygens (Christian Huyghensvan Zuylichem) is a mathematician, astronomer, and physicist whom Newton recognized as great. His father, signor van Zuylichem, secretary of the princes of Orange, was a remarkable writer and scientifically educated.

Christian Huygens began his scientific activity in 1651 with an essay on the quadrature of the hyperbola, ellipse and circle; in 1654 he discovered the theory of evolute and involute, in 1655 he found the satellite of Saturn and the type of rings, in 1659 he described the system of Saturn in a work he published. In 1665, at the invitation of Colbert, he settled in Paris and was accepted as a member of the Academy of Sciences.

Clocks with wheels driven by weights have been in use for a long time, but the regulation of such clocks was unsatisfactory. Since the time of Galileo, the pendulum has been used separately for accurate measurement of small periods of time, and it was necessary to count the number of swings. In 1657, Christian Huygens published a description of the design of the clock he invented with a pendulum. Later, published by him in 1673, in Paris, the famous work Horologium oscillatorium, sive de mota pendulorum an horologia aptato demonstrationes geometrica, which contains a presentation of the most important discoveries in dynamics, in its first part also contains a description of the structure of the clock, but with the addition improvements in the way the pendulum gains, making the pendulum cycloidal, which has a constant swing time, regardless of the magnitude of the swing. In order to explain this property of the cycloidal pendulum, the author devotes the second part of the book to the derivation of the laws of falling of bodies free and moving along inclined straight lines, and finally along a cycloid. Here, for the first time, the beginning of the independence of motions is clearly expressed: uniformly accelerated, due to the action of gravity, and uniform due to inertia.

Christian Huygens proves the laws of uniformly accelerated motion of freely falling bodies, based on the beginning that the action imparted to the body by a force of constant magnitude and direction does not depend on the magnitude and direction of the speed that the body already possesses. Deriving the relationship between the height of the fall and the square of time, Huygens makes the remark that the heights of the falls are related as the squares of the acquired velocities. Further, considering the free movement of a body thrown upwards, he finds that the body rises to the greatest height, having lost all the speed communicated to it and acquires it again when returning back.

Galileo allowed without proof that when falling along differently inclined straight lines from the same height, bodies acquire equal speeds. Christian Huygens proves this as follows. Two straight lines of different inclination and equal height are attached with their lower ends one to the other. If a body launched from the upper end of one of them acquires a greater speed than that launched from the upper end of the other, then it can be launched along the first of such a point below the upper end so that the speed acquired below is sufficient to lift the body to the upper end of the second straight line, but then it would turn out that the body rose to a height greater than the one from which it fell, but this cannot be.

From the motion of a body along an inclined straight line, H. Huygens proceeds to motion along a broken line and then to motion along any curve, and he proves that the speed acquired when falling from any height along the curve is equal to the speed acquired in free fall from the same height in a vertical line, and that the same speed is required to lift the same body to the same height, both in a vertical straight line and in a curve.

Then, passing to the cycloid and considering some of its geometric properties, the author proves the tautochronism of the motions of the heavy point along the cycloid. In the third part of the work, the theory of evolutes and evolvents, discovered by the author as early as 1654, is presented; here Christians find the form and position of the evolution of the cycloid.

The fourth part presents the theory of the physical pendulum, here Christian Huygens solves the problem that was not given to so many contemporary geometers - the problem of determining the center of swings. It is based on the following proposition: “If a complex pendulum, having left rest, completed some part of its swing, a larger half-swing, and if the connection between all its particles is destroyed, then each of these particles will rise to such a height that their common center of gravity at the same time will be at the height at which it was when the pendulum came out of rest. This proposition, not proved by Christian Huygens, appears to him as a basic principle, while now it represents the application of the law of conservation of energy to the pendulum. The theory of the pendulum of the physical is given by Huygens in a completely general form and in application to bodies of various kinds. In the last, fifth part of his work, the scientist gives thirteen theorems on centrifugal force and considers the rotation of a conical pendulum.

Another remarkable work of Christian Huygens is the Theory of Light, published in 1690, in which he expounds the theory of reflection and refraction and then of double refraction in Icelandic spar, in the same form as it is now presented in the textbooks of physics. Of the others discovered by H. Huygens, we will mention the following.

Discovery of the true appearance of Saturn's rings and its two satellites, made with a ten-foot telescope, which he himself arranged. Together with his brother Christian Huygens, he was engaged in the manufacture of optical glasses and significantly improved their production. Open theoretically the ellipsoidal form of the earth and its compression at the poles, as well as an explanation of the influence of centrifugal force on the direction of gravity and on the length of the second pendulum at different latitudes. Solution of the issue of collision of elastic bodies simultaneously with Wallis and Brenn.

Christian Huygens owns the invention of the clock spiral, which replaces the pendulum, the first clock with a spiral was made in Paris by the watchmaker Thuret in 1674. He also owns one of the solutions to the question of the form of a heavy homogeneous chain in equilibrium.

Christian Huygens - quotes

The more difficult the task of determining by reasoning what seems indeterminate and subject to chance, the more the science that achieves the result seems surprising.

4.1.3. Tasks for the experiment

1. By choosing various initial conditions and parameter values, follow the bifurcations (qualitative changes in the structure) of the phase portrait. Explore the trigger mode separately by changing the initial value u .

2. Choose the parameter values ​​so that they fall into the area on the plane ( E ,R ) corresponding to excitation

self-oscillations. Experimentally find out the dependence of the period of self-oscillations on the parameters, build the appropriate graphs.

4.2. Galileo–Huygens clock

4.2.1. Model

The mathematical model of small oscillations of a conventional pendulum, taking into account viscous friction, is a model of a linear oscillator:

viscous friction coefficient, ω is the frequency of free oscillations of the pendulum in the absence of viscous friction (ω 2 = g l , where g is the acceleration of its own

free fall, l is the length of the pendulum thread). Equation (4.2) defines the operator of a dynamical system whose state (vector of phase variables) is the vector (ϕ , ϕ & ). At δ = 0 (from-

the absence of viscous friction), the pendulum performs free undamped sinusoidal oscillations, the period of which does not depend on the initial conditions (angle ϕ and angular velocity ϕ & ). Constant-

The value of the oscillation period of a pendulum (for small deviations) was first established by G. Galileo.

However, in reality, viscous friction is always present.

(δ > 0), and the solution of Eq. (4.2) for small δ (δ 2< ω 2 ) имеет видзатухающих синусоидальных колебаний с частотой

Ω = ω 2 − δ 2 (under any initial conditions, the phase trajectories of the system tend as t → +∞ to a stable equilibrium state (ϕ = 0, ϕ & = 0 )). To be able to use ma-

dutnik as a clock, you need to count its fluctuations and show them (for example, with an arrow on the dial). In addition, it is necessary not to allow the oscillations of the pendulum to die out, i.e. required to turn fading free vibrations into undamped self-oscillations. Both of these problems were solved by H. Huygens, who proposed a device called clockwork. The simplest version of the clock is shown in Fig. 4.4.

After each oscillation of the pendulum back and forth, the ratchet wheel (ratchet), under the influence of a wound spring or a falling load, rotates one tooth and simultaneously imparts a pushing impulse to the pendulum. Thus, the speed of rotation of the ratchet wheel is determined by the oscillation frequency of the pendulum, and the teeth of the ratchet at the moment of its rotation push the pendulum, supporting its oscillations. Thus, with the help of the clock stroke in the pendulum, automatic control(status feedback).

The mathematical model of a pendulum with viscous friction and a clock stroke that imparts an instantaneous pushing impulse (bump) to the pendulum has the form:

				ϕ = 0,
	2 δϕ
ϕ += ϕ −

where ϕ & − is the pre-shock angular velocity, and ϕ & + is the post-shock one (the angle ϕ does not have time to change). The impact occurs at some ϕ = α (in particular, α can be equal to zero, which corresponds to the lower position of the pendulum) and ϕ & > 0 .

Let us draw a phase trajectory corresponding to one complete oscillation from the value ϕ = α again to ϕ = α . Let M 0 be

initial point, and M 1 is the point of repeated value ϕ = α at

provided that the movement occurs in accordance with the differential equation (4.2). At the moment of arrival at the point M 1, the momentum p is transferred, and the point M 1 moves along the axis ϕ & by a distance p to the point M 2 (Fig. 4.5).

P M2

α ϕ

Denote by u the value of ϕ & at the point M 0 , by u ~ the value of ϕ & at the point M 1 , and by u the value of ϕ & at the point M 2 . Then, solving the differential equation (4.2) under the initial conditions

(ϕ = α ,ϕ = 0)	and considering	that the period of one full sine wave
long-range oscillation is 2π /Ω (Ω = ω 2 − δ 2 ) , we get:
			−2 πδ/Ω
Since at point M 1				momentum transfer p , we have

u = u ~ + p . From here we get the formula dot mapping(or sequence functions) are mappings of the line ϕ = α into

itself along the phase trajectory of the dynamical system (4.3); this formula relates the values ​​u and u:

u = e− 2 πδ / Ω u+ p.

On fig. 4.6 depicted Koenigs–Lamerey diagram(or simply Lamerey diagram) showing the sequence

values ​​u ,u ,u , ..., obtained from the initial value u due to the point mapping (4.4) in accordance with the graph

function (4.4) and the bisector of the angle (ray u = u ). It can be seen from the Lamerey diagram that this sequence of values ​​tends to a stable fixed point(the point passing into itself under the mapping (4.4))u * , corresponding to self-oscillations of the clock. This fixed point is globally stable, i.e. the system enters the self-oscillatory regime under any initial conditions. The value u * is found from equation (4.4) if both instead of u and instead of u substitute u * :

*=
		− e −2 πδ/ Ω

4.2.2. Implementation in AnyLogic

The work is implemented in the Part3\clock.alp file (Fig. 4.7).

AT the animation window shows oscillations of the pendulum clock

With by the hour. Phase portrait on the plane (ϕ ,ϕ & )

Xia in a separate window that is not part of the animation (to display it on the screen, select the “root.x_(root.x)” tab in the animation window). It is recommended to first run the model for one step, then on the AnyLogic toolbar you

Christian Huygens von Zuylichen - the son of the Dutch nobleman Constantine Huygens, was born on April 14, 1629. “Talents, nobility and wealth were, apparently, hereditary in the family of Christian Huygens,” wrote one of his biographers. His grandfather was a writer and dignitary, his father was a secret adviser to the Princes of Orange, a mathematician, and a poet.

Faithful service to their sovereigns did not enslave their talents, and it seemed that Christian was destined for the same enviable fate for many. He studied arithmetic and Latin, music and versification. Heinrich Bruno, his teacher, could not get enough of his fourteen-year-old pupil:

“I confess that Christian must be called a miracle among boys ... He deploys his abilities in the field of mechanics and construction, makes amazing machines, but hardly necessary.” The teacher was wrong: the boy is always looking for the benefits of his studies. His concrete, practical mind will soon find schemes of machines that people really need.

However, he did not immediately devote himself to mechanics and mathematics. The father decided to make his son a lawyer and, when Christian reached the age of sixteen, he sent him to study law at the University of London.

Being engaged in legal sciences at the university, Huygens at the same time is fond of mathematics, mechanics, astronomy, and practical optics. A skilled craftsman, he grinds optical glasses on his own, improves the pipe, with the help of which he will later make his astronomical discoveries.

Christian Huygens was Galileo's immediate successor in science. According to Lagrange, Huygens "was destined to improve and develop the most important discoveries of Galileo." There is a story about how for the first time Huygens came into contact with the ideas of Galileo. Seventeen-year-old Huygens was going to prove that bodies thrown horizontally move along parabolas, but, having found the proof in the book of Galileo, he did not want to "write the Iliad after Homer."

After graduating from the university, he becomes an adornment of the retinue of the Count of Nassau, who, on a diplomatic mission, is on his way to Denmark. The count is not interested in the fact that this handsome young man is the author of curious mathematical works, and he, of course, does not know how Christian dreams of getting from Copenhagen to Stockholm to see Descartes. So they will never meet: in a few months Descartes will die.

At the age of 22, Huygens published Discourses on the Square of the Hyperbola, Ellipse, and Circle. In 1655, he builds a telescope and discovers one of Saturn's satellites, Titan, and publishes New Discoveries in the Size of a Circle. At the age of 26, Christian writes notes on dioptrics. At the age of 28, his treatise “On Calculations when Playing Dice” was published, where one of the first ever research in the field of probability theory is hidden behind a seemingly frivolous title.

One of Huygens' most important discoveries was the invention of the pendulum clock. He patented his invention on July 16, 1657 and described it in a short essay published in 1658. He wrote about his watch to the French king Louis XIV: “My automatic machines, placed in your apartments, not only amaze you every day with the correct indication of the time, but they are suitable, as I hoped from the very
beginning, to determine the longitude of a place on the sea. The task of creating and improving clocks, especially pendulum ones. Christian Huygens studied for almost forty years: from 1656 to 1693. A. Sommerfeld called Huygens "the most brilliant watchmaker of all time."

At thirty, Huygens reveals the secret of Saturn's ring. The rings of Saturn were first noticed by Galileo as two lateral appendages "supporting" Saturn. Then the rings were visible, like a thin line, he did not notice them and did not mention them again. But Galileo's pipe did not have the necessary resolution and sufficient magnification. Watching the sky with a 92x telescope. Christian discovers that the ring of Saturn was taken as side stars. Huygens figured it out
riddle of Saturn and for the first time described its famous rings.

At that time Huygens was a very handsome young man with large blue eyes and a neatly trimmed mustache. The reddish curls of the wig, coolly curled in the fashion of that time, fell to the shoulders, lying on the snow-white Brabant lace of an expensive collar. He was friendly and calm. No one saw him especially agitated or confused, in a hurry somewhere, or, on the contrary, immersed in slow thoughtfulness. He did not like to be in the “light” and rarely appeared there, although his origin opened the doors of all the palaces of Europe to him. However, when he appeared there, he did not look at all awkward or embarrassed, as often happened to other scientists.

But in vain the charming Ninon de Lanclos seeks his company, he is invariably friendly, no more, this convinced bachelor. He can drink with friends, but not much. Sneak a little, laugh a little. A little bit of everything, a very little bit, so that as much time as possible is left for the main thing - work. Work - an unchanging all-consuming passion - burned him constantly.

Huygens was distinguished by extraordinary dedication. He was aware of his abilities and sought to use them to the fullest. “The only entertainment that Huygens allowed himself in such abstract works,” one of his contemporaries wrote about him, “was that he studied physics in between. What for an ordinary person was a tedious task, for Huygens was entertainment.

In 1663 Huygens was elected a Fellow of the Royal Society of London. In 1665, at the invitation of Colbert, he settled in Paris and the following year became a member of the newly organized Paris Academy of Sciences.

In 1673, his essay "Pendulum Clock" was published, where the theoretical foundations of Huygens' invention were given. In this essay, Huygens establishes that the cycloid has the property of isochronism, and analyzes the mathematical properties of the cycloid

Investigating the curvilinear motion of a heavy point, Huygens, continuing to develop the ideas expressed by Galileo, shows that a body, when falling from a certain height along various paths, acquires a finite velocity that does not depend on the shape of the path, but depends only on the height of the fall, and can rise to a height equal (in the absence of resistance) to the initial height. This provision, which essentially expresses the law
conservation of energy for movement in a gravitational field, Huygens uses for the theory of the physical pendulum. He finds an expression for the reduced length of the pendulum, establishes the concept of the swing center and its properties. He expresses the formula of a mathematical pendulum for cycloidal motion and small oscillations of a circular pendulum as follows:

"The time of one small oscillation of a circular pendulum is related to the time of falling along twice the length of the pendulum, as the circumference of a circle is related to the diameter"

It is significant that at the end of his essay, the scientist gives a number of proposals (without a conclusion) about the centripetal force and establishes that the centripetal acceleration is proportional to the square of the speed and inversely proportional to the radius of the circle. This result prepared the Newtonian theory of the motion of bodies under the action of central forces.

From the mechanical research of Huygens, in addition to the theory of the pendulum and centripetal force, his theory of the impact of elastic balls is known, which he presented for a competitive task announced by the Royal Society of London in 1668. Huygens' impact theory is based on the law of conservation of living forces, momentum and Galileo's principle of relativity. It was published only after his death in 1703.

Huygens traveled quite a lot, but he was never an idle tourist. During the first trip to France, he studied optics, and in London ~ explained the secrets of making his telescopes. Fifteen years he worked at the court of Louis XIV, fifteen years of brilliant mathematical and physical research. And in fifteen years - only two short trips to his homeland to heal.

Huygens lived in Paris until 1681, when, after the repeal of the Edict of Nantes, he, as a Protestant, returned to his homeland. While in Paris, he knew Römer well and actively assisted him in the observations that led to the determination of the speed of light. Huygens was the first to report Römer's results in his treatise.

At home, in Holland, again not knowing fatigue, Huygens builds a mechanical planetarium, giant seventy-meter telescopes, describes the worlds of other planets.

Huygens' work in Latin appears on light, corrected by the author and republished in French in 1690. Huygens' Treatise on Light entered the history of science as the first scientific work on wave optics. This Treatise formulated the principle of wave propagation, now known as Huygens principle Based on this principle, the laws of reflection and refraction of light are derived, the theory of double refraction in Icelandic spar is developed. Since the speed of light propagation in a crystal is different in different directions, the shape of the wave surface will not be spherical, but ellipsoidal.

The theory of propagation and refraction of light in uniaxial crystals is a remarkable achievement of Huygens' optics. Huygens also described the disappearance of one of the two rays when they pass through the second crystal with a certain orientation of it relative to the first. Thus, Huygens was the first physicist to establish the fact of light polarization.

Huygens' ideas were highly valued by his successor Fresnel. He ranked them above all discoveries in Newton's optics, arguing that Huygens' discovery "is perhaps more difficult to make than all Newton's discoveries in the field of light phenomena."

Huygens does not consider colors in his treatise, as well as the diffraction of light. His treatise is devoted only to the justification of reflection and refraction (including double refraction) from the wave point of view. This circumstance was probably the reason why Huygens' theory, despite its support in the 18th century by Lomonosov and Euler, did not receive recognition until Fresnel resurrected the wave theory on a new basis in the early 19th century.

Huygens died on June 8, 1695, when KosMoteoros, his last book, was being printed in the printing house.

The founder of the modern theory of theoretical mechanics, Christian Huygens, was born on April 14, 1629 in The Hague. Huygens received the foundations of mathematics and mechanics at the lectures of Professor Frans van Schoten at the University of Leiden. The first scientific work of the young scientist was published in 1651 and was called "Discourses on the quadrature of the hyperbola, ellipse and circle." Of great practical importance were Huygens' works in the field of exact sciences - a description of the foundations of probability theory, mathematical theory of numbers and various curves, and the wave theory of light. He was the first in Holland to receive a patent for a pendulum clock. This shows the breadth of Christian Huygens' scientific outlook.

If your mentor is Descartes, you are destined to become a genius

The breadth of Huygens' interests is striking. During his scientific activity, he wrote dozens of serious scientific papers in mechanics and mathematics and physics. Recognizing the merits of the great Dutchman in understanding the world around him and setting the views on the scientific basis that existed at that time, the royal scientific community honored Christian Huygens by electing him in 1663 as its member - the first of foreign scientists. In 1666 the French founded their Academy of Sciences. Huygens became the first president of the French scientific community.

One of the many branches of science enriched by the works of the Dutch naturalist was astronomy. The friendship of his father, Constantine Huygens, with the founder of the philosophical theory of Cartesianism, Rene Descartes, had a huge impact on the views of the young Christian. Huygens became interested in astronomical research. With the help of his brother, he rebuilt his home telescope in such a way as to achieve the highest possible magnification - 92x.

Mars, Saturn, on and on...

The very first astronomical discovery of Huygens became a scientific sensation. In 1655, observing the vicinity of Saturn through a telescope, the astronomer noticed the same oddities that Galileo Galilei pointed out in his writings. But the Italian could not give a clear justification for this phenomenon. Huygens, on the other hand, correctly determined that these are accumulations of ice of various sizes that surround the planet and do not leave the orbit of Saturn under the influence of its giant attraction. Huygens examined in his telescope and the satellite of Saturn, later named Titan. Four years later, the scientist systematized his discoveries of rings in the orbit of Saturn in a scientific work.

1656 year. The sphere of astronomical interests of Huygens for the first time goes far beyond the solar system. The object of observation is the nebula in the constellation of Orion discovered 45 years earlier by the Frenchman Nicolas de Pereysky. Today, the Orion Nebula is classified in astronomical catalogs under the name Messier 42 (NGC1976). Huygens made the primary classification of nebula objects and the calculation of astronomical coordinates, began calculating the size of the nebula and the distance to the Earth.

Fifteen years later, the Dutchman returned to astronomical observations. The object of his attention was the Red Planet. Observing the South Pole of Mars through a telescope, Huygens found that it was covered with an ice cap. Even then, astronomers were sure that there could be certain conditions on Mars for the existence of living organisms. The astronomer quite accurately calculated the period of revolution of the planet around its own axis.

Huygens' worldview

The last scientific work in the field of astronomy was an article published after his death, in 1698 in The Hague. The treatise is a compilation of philosophy and astronomy in an attempt to understand the basic physical laws of the existence and structure of the universe. Huygens was one of the first European scientists to put forward the hypothesis that other objects outside the Earth were inhabited by intelligent beings. Huygens' posthumous scientific work was translated into English, French, German and Swedish. The scientific testament of Christian Huygens, by personal decree of Emperor Peter I in 1717, was translated into Russian by Jacob (James) Bruce. The work is known to the Russian scientific community as the “Book of the World » .

Summing up many years of observations of various objects in the Universe, Huygens made an attempt to provide a scientific basis for the existence of the Copernican heliocentric system, as well as to learn how to calculate the true distances to stars and nebulae based on their apparent brightness.

Like other major scientists of the Middle Ages, Huygens had talented students. The most famous of them is the German mathematician Gottfried Leibniz.

Christian Huygens died in The Hague on July 8, 1695 at the age of 66. Contemporaries highly appreciated the scientific achievements of the famous Dutchman in the field of astronomy. In 1997, a probe of the European Space Agency, named after him, launched to the satellite of Saturn, Titan, discovered by him. The mission of the spacecraft was as successful as the life of Christian Huygens was long and rich in scientific discoveries.