Remember the orange plastic reflectors - reflectors attached to the spokes of a bicycle wheel? Let's attach a reflector to the very rim of the wheel and follow its trajectory. The resulting curves belong to the cycloid family. The wheel is called the generating circle (or circle) of the cycloid. But let's go back to our century and switch to more modern technology. On the way of the bike, a pebble got stuck in the wheel tread.

After spinning a few laps with the wheel, where will the stone fly when it pops out of the tread? Against the direction of the motorcycle or in the direction? As you know, the free movement of a body begins tangentially to the trajectory along which it moved. The tangent to the cycloid is always directed in the direction of motion and passes through the top point of the generating circle. Our pebble will also fly in the direction of movement. Remember how you rode through the puddles on a bike without a rear fender as a child? A wet strip on your back is an everyday confirmation of the result just obtained.

The 17th century is the age of the cycloid. The best scientists have studied its amazing properties. What trajectory will bring a body moving under the influence of gravity from one point to another in shortest time? This was one of the first tasks of the science that is now called the calculus of variations. You can minimize (or maximize) different things - path length, speed, time. In the brachistochrone problem, it is time that is minimized (which is emphasized by the name itself: Greek βράχιστος - the smallest, χρόνος - time). The first thing that comes to mind is a straight line. Let's also consider an inverted cycloid with a cusp at the top of the given points. And, following Galileo Galilei, a quarter of a circle connecting our points. Let's make bobsleigh tracks with the considered profiles and see which of the bobs arrives first. The history of bobsledding originates in Switzerland. In 1924 in French city Chamonix hosts the first Winter Olympics. They already host bobsleigh competitions for crews of twos and fours.

The only year when Olympic Games the crew of the bean consisted of five people, it was 1928. Since then, men's crews of twos and fours have always competed in bobsleigh. There are many interesting things in the rules of bobsleigh. Of course, there are restrictions on the weight of the bob and the team, but there are even restrictions on the materials that can be used in the skates of the bob (the front pair of them is movable and connected to the steering wheel, the back is rigidly fixed). For example, radium cannot be used in the manufacture of skates.

Let's start our fours. Which bean will be the first to reach the finish line? Bean Green colour, playing for the Mathematical Etudes team and rolling down the cycloidal slide, comes first! Why did Galileo Galilei consider a quarter of a circle and consider that this is the best descent trajectory in terms of time? He entered broken lines into it and noticed that with an increase in the number of links, the descent time decreases. From here, Galileo naturally moved to a circle, but made the wrong conclusion that this trajectory is the best among all possible. As we have seen, the best trajectory is a cycloid. A single cycloid can be drawn through two given points provided that the cusp of the cycloid is at the top point. And even when the cycloid has to rise to pass the second point, it will still be a steepest descent curve! Another beautiful problem related to the cycloid is the tautochrone problem. Translated from the Greek ταύτίς means "the same", χρόνος, as we already know - "time". Let's make three identical slides with a profile in the form of a cycloid, so that the ends of the slides coincide and are located at the top of the cycloid. Let's put three beans at different heights and give the go-ahead.

Amazing fact - all the beans will come down at the same time! In winter, you can build an ice slide in the yard and check out this property live. The task of the tautochrone is to find such a curve that, starting from any initial position, the time of descent to a given point will be the same. Christian Huygens proved that the only tautochrone is the cycloid. Of course, Huygens was not interested in descending the ice slides. At that time, scientists did not have the luxury of doing science for the love of art. The tasks that were studied came from the life and demands of technology of that time. In the 17th century, long-distance sea voyages were already made. The sailors were already able to determine the latitude quite accurately, but it is surprising that they did not know how to determine the longitude at all. And one of the proposed methods for measuring latitude was based on the availability of accurate chronometers. The first person to think of making pendulum clocks that would be accurate was Galileo Galilei. However, at the moment when he begins to realize them, he is already old, he is blind, and for the remaining year of his life, the scientist does not have time to make a clock. He bequeaths this to his son, but he hesitates and begins to deal with the pendulum, too, only before his death and does not have time to realize the plan.

The next iconic figure was Christian Huygens. He noticed that the period of oscillation of an ordinary pendulum, considered by Galileo, depends on the initial position, i.e. from amplitude. Thinking about what should be the trajectory of the movement of the load, so that the time of rolling along it does not depend on the amplitude, he solves the problem of the tautochrone. But how to make the load move along the cycloid? Translating theoretical research into a practical plane, Huygens makes "cheeks" on which the rope of the pendulum is wound, and solves several more mathematical problems. He proves that the "cheeks" must have the profile of the same cycloid, thereby showing that the evolute of a cycloid is a cycloid with the same parameters. In addition, the construction of the cycloidal pendulum proposed by Huygens allows one to calculate the length of the cycloid. If the blue thread, the length of which is equal to four radii of the generating circle, is deflected as much as possible, then its end will be at the intersection point of the “cheek” and the trajectory cycloid, i.e. at the top of the cycloid - "cheeks". Since this is half the length of the cycloid arc, the total length is equal to eight radii of the generating circle. Christian Huygens made a cycloidal pendulum, and watches with it were tested on sea voyages, but did not take root. However, just like a clock with a conventional pendulum for these purposes. Why, however, do clockworks with an ordinary pendulum still exist? If you look closely, then with small deviations, like a red pendulum, the "cheeks" of the cycloidal pendulum have almost no effect. Accordingly, the movement along the cycloid and along the circle for small deviations almost coincide.

Literature:
G. N. Berman. Cycloid. Moscow: Nauka, 1980.
S. G. Gindikin. Stories about physicists and mathematicians. M.: MTsNMO, 2006.

Comments: 1

Vladimir Zakharov

Lecture by Academician of the Russian Academy of Sciences, Doctor of Physical and Mathematical Sciences, Chairman of the Scientific Council of the Russian Academy of Sciences on Nonlinear Dynamics, Head. Department of Mathematical Physics at the Physical Institute of the Russian Academy of Sciences. Lebedev, professor at the University of Arizona (USA), twice winner of the State Prize, winner of the Dirac medal Vladimir Evgenievich Zakharov, read on May 27, 2010 at the Polytechnic Museum as part of the Polit.ru Public Lectures project.

Sergey Kuksin

International Scientific Conference"Days classical mechanics» Moscow, MIAN, st. Gubkina, 8 January 26, 2015

Chaos is a mathematical film consisting of nine chapters, thirteen minutes each. This is a film for the general public, dedicated to dynamical systems, the butterfly effect and chaos theory.

Alexandra Skripchenko

Mathematician Alexandra Skripchenko about billiards as a dynamical system, rational angles and Poincaré's theorem.

Yuli Ilyashenko

The Kolmogorov–Arnold–Moser theory answers questions like “Can planets fall into the Sun? If yes, with what probability? And after what time? Mathematical statement of the problem: suppose that the masses are so small that their attraction to each other can be neglected. Then the trajectories of the planets can be calculated; Newton did it. If we go over to the real case, when the mutual attraction of the planets affects their orbits, we get a small perturbation of the integrable, i.e. exactly solvable system. Poincare considered the study of small perturbations of integrable systems of classical mechanics to be the main problem in the theory of differential equations. The lectures will tell, at a level accessible to older students, about the main ideas of the KAM theory. We will not go up to the n-body problem and classical mechanics, but we will discuss the diffeomorphisms of the circle and the main step of the induction process proposed by Kolmogorov for problems of celestial mechanics.

Olga Romaskevich

If you act very cruelly and take away a pencil and paper from a mathematician, he will look at the sky in search of new problems. The question of planetary motion (in the mathematical world, codenamed "The n-body problem") is extremely complex - so complex that even for special subcases of the n=3 case, a huge number of papers are published every year. It is impossible to analyze all aspects of this task even for a semester course. We, however, will not be afraid, and will try to play around with the mathematics that arises here. The main motivation for us will be the two-body problem: the problem of the movement of one planet around the Sun, assuming that there are no other planets in the vicinity.

Dmitry Anosov

The book talks about differential equations. In some cases, the author explains how differential equations, and in others, how geometric considerations help to understand the properties of their solutions. (This is what the words “we solve, we draw” in the title of the book are connected with.) Several physical examples are considered. At the most simplified level, some of the achievements of the 20th century are described, including the understanding of the mechanism for the emergence of "chaos" in the behavior of deterministic objects. The book is intended for high school students interested in mathematics. They only need to understand the meaning of the derivative as instantaneous speed.

Alexey Belov

The Olympiad task is known: Coins (convex figures) lie on a flat table. Then one of them can be pulled off the table without hitting the others. For a long time, mathematicians tried to prove the spatial analogue of this statement, until a counterexample was constructed! An idea arose: often there is no crack in a small grain, the crack does not grow beyond the grain boundary, and the cracks hold each other. This idea theoretically makes it possible to create composites in which cracks do not grow, in particular, ceramic armor.

Alexey Sosinsky

One of the most important concepts of mechanics and theoretical physics - the concept of the configuration space of a mechanical system - for some reason remains unknown not only to schoolchildren, but also to most mathematics students. The lecture deals with a very simple but very meaningful class of mechanical systems - flat hinged mechanisms with two degrees of freedom. We will find that in the "general case" their configuration spaces are two-dimensional surfaces, and we will try to understand which ones. (Here are Dima Zvonkin's final results ten years ago.) Next, unsolved mathematical problems related to hinged mechanisms are discussed. (Including two hypotheses, or rather, unproved theorems, by the American mathematician Bill Thurston.)

Vladimir Protasov

Calculus of variations - the science of finding the minimum of a function in an infinite-dimensional space. Unlike the minimum problems we are used to, when we need to optimally choose a number (parameter), or, say, a point on a plane, in variational problems we need to find the optimal function. At the same time, the same set of tools solves the problems of the different origin: from classical mechanics, geometry, mathematical economics, etc. We will start with the old problems known since the 17th century, and by jumping bridges from one problem to another, we will quickly get to modern results and unsolved problems.

5. Parametric equation of the cycloid and the equation in Cartesian coordinate X

Suppose we have a cycloid formed by a circle of radius a centered at point A.

If we choose as a parameter that determines the position of the point, the angle t=∟NDM, by which the radius, which had a vertical position AO at the beginning of the rolling, has managed to turn, then the x and y coordinates of the point M will be expressed as follows:

x \u003d OF \u003d ON - NF \u003d NM - MG \u003d at-a sin t,

y= FM = NG = ND - GD = a - a cos t

So the parametric equations of the cycloid have the form:

When changing t from -∞ to +∞, you get a curve consisting of an innumerable set of such branches, which is shown in this figure.

Also, in addition to the parametric equation of the cycloid, there is also its equation in Cartesian coordinates:

Where r is the radius of the circle forming the cycloid.

6. Problems for finding parts of a cycloid and figures formed by a cycloid

Task number 1. Find the area of ​​a figure bounded by one arc of a cycloid whose equation is given parametrically

and axis Oh.

Solution. To solve this problem, we use the facts known to us from the theory of integrals, namely:

The area of ​​the curvilinear sector.

Consider some function r = r(ϕ) defined on [α, β].

ϕ 0 ∈ [α, β] corresponds to r 0 = r(ϕ 0) and, therefore, the point M 0 (ϕ 0 , r 0), where ϕ 0 ,

r 0 - polar coordinates of the point. If ϕ changes, “running through” the whole [α, β], then the variable point M will describe some curve AB given by

the equation r = r(ϕ).

Definition 7.4. A curvilinear sector is a figure bounded by two rays ϕ = α, ϕ = β and a curve AB given in polar

coordinates by the equation r = r(ϕ), α ≤ ϕ ≤ β.

The following

Theorem. If the function r(ϕ) > 0 and is continuous on [α, β], then the area

curved sector is calculated by the formula:

This theorem was proved earlier in the topic of a definite integral.

Based on the above theorem, our problem of finding the area of ​​​​a figure bounded by one arch of the cycloid, the equation of which is given by the parametric x= a (t - sin t) , y= a (1 - cos t) , and the Ox axis, is reduced to the following solution .

Solution. From the curve equation dx = a(1−cos t) dt. The first arch of the cycloid corresponds to the change in the parameter t from 0 to 2π. Hence,

Task number 2. Find the length of one arc of the cycloid

The following theorem and its corollary were also studied in integral calculus.

Theorem. If the curve AB is given by the equation y = f(x), where f(x) and f ’ (x) are continuous on , then AB is rectifiable and

Consequence. Let AB be given parametrically

L AB = (1)

Let the functions x(t), y(t) be continuously differentiable on [α, β]. Then

formula (1) can be written as

Let's make a change of variables in this integral x = x(t), then y'(x)= ;

dx= x'(t)dt and hence:

Now let's get back to solving our problem.

Solution. We have and therefore

Task number 3. It is necessary to find the surface area S formed from the rotation of one arc of the cycloid

L=((x,y): x=a(t - sin t), y=a(1 - cost), 0≤ t ≤ 2π)

In integral calculus, there is the following formula for finding the surface area of ​​a body of revolution around the x-axis of a curve defined parametrically on a segment: x=φ(t), y=ψ(t) (t 0 ≤t ≤t 1)

Applying this formula to our cycloid equation, we get:

Task number 4. Find the volume of the body obtained by rotating the arch of the cycloid

Along the axis Ox.

In integral calculus, when studying volumes, there is the following remark:

If the curve bounding the curvilinear trapezoid is given by parametric equations and the functions in these equations satisfy the conditions of the theorem on the change of variable in a certain integral, then the volume of the body of rotation of the trapezoid around the Ox axis will be calculated by the formula

Let's use this formula to find the volume we need.

Problem solved.

Conclusion

So, in the course of this work, the main properties of the cycloid were clarified. They also learned how to build a cycloid, found out the geometric meaning of the cycloid. As it turned out, the cycloid has a huge practical application not only in mathematics, but also in technological calculations, in physics. But the cycloid has other merits. It was used by scientists of the 17th century in developing methods for studying curved lines - those methods that eventually led to the invention of differential and integral calculus. It was also one of the "touchstones" on which Newton, Leibniz and their first researchers tested the power of powerful new mathematical methods. Finally, the problem of the brachistochrone led to the invention of the calculus of variations, which is so necessary for today's physicists. Thus, the cycloid was inextricably linked with one of the most interesting periods in the history of mathematics.

Literature

1. Berman G.N. Cycloid. - M., 1980

2. Verov S.G. Brachistochrone, or another secret of the cycloid // Kvant. - 1975. - No. 5

3. Verov S.G. Secrets of the cycloid// Kvant. - 1975. - No. 8.

4. Gavrilova R.M., Govorukhina A.A., Kartasheva L.V., Kostetskaya G.S., Radchenko T.N. Applications of a definite integral. Guidelines and individual assignments for 1st year students of the Faculty of Physics. - Rostov n / a: UPL RSU, 1994.

5. Gindikin S.G. Star age of a cycloid // Kvant. - 1985. - No. 6.

6. Fikhtengolts G.M. Course of differential and integral calculus. T.1. - M., 1969

Such a line is called an "envelope". Every curved line is the envelope of its tangents.

Matter and motion, and the method that they constitute, enable everyone to realize their potential in the knowledge of truth. The development of a methodology for the development of a dialectical-materialistic form of thinking and the mastery of a similar method of cognition is the second step towards solving the problem of development and realizing the possibilities of Man. Fragment XX Opportunities...

The situation can get sick with neurasthenia - neurosis, the basis clinical picture which constitutes the asthenic state. Both in the case of neurasthenia and in the case of decompensation of neurasthenic psychopathy, the essence of mental (psychological) protection is manifested by a departure from difficulties into irritable weakness with vegetative dysfunctions: either a person unconsciously “fights back” more from an attack ...

Various kinds activities; development of spatial imagination and spatial representations, figurative, spatial, logical, abstract thinking of schoolchildren; the formation of skills to apply geometric and graphic knowledge and skills to solve various applied problems; familiarization with the content and sequence of stages of project activities in the field of technical and ...

Arcs. Spirals are also involutes of closed curves, such as the involute of a circle. The names of some spirals are given by the similarity of their polar equations with the equations of curves in Cartesian coordinates, for example: parabolic spiral (a - r)2 = bj, hyperbolic spiral: r = a/j. Rod: r2 = a/j si-ci-spiral, whose parametric equations look like: , )