What is the difference between those and others? Why is it worth talking about?

It's hard to avoid talking about coordinate systems and reference systems, if we are discussing any space-time problems. And in everyday life, the presentation of a wide variety of information in certain coordinates is the most familiar thing for us., even if we often do not give ourselves an account of this. You don't have to look far for examples. Year, month, day, time, minute, second — what is it for you? Coordinates. Time coordinates. A country, city, Street, House number? Spatial coordinates. Want to meet someone? Can't get anywhere from that, to somehow describe the time and place of the meeting, and in the form, understandable to you and your partner. Also check your watch, if time is important to you. Do you want to send a letter – take care of the right, known mail, addresses. You will not send a letter “to the village, grandfather”. Although these are also coordinates. And even the right ones.

Why do I say such banal things? And to emphasize, what, one side, coordinate systems have long been an integral part of our life, on the other hand, rather careful handling is required, so as not to be trapped. Science with coordinates is certainly not so careless., how we sometimes allow ourselves in everyday life, but even in the scientific approach to coordinate systems, there are some things that are not mentioned., which people who are not very meticulous can sometimes be misleading. And meticulous too. For every sage, simplicity is enough. This is not because of the malice of mathematicians or physicists., but in a simple fact, what “not very good coordinates” can be extremely convenient in many cases, suitable for describing a particular phenomenon. And their “not quite nice” kind of known a long time ago, so what to remember about her? But you need to remember. Oh right. Often, if you remember something about the coordinate system used, much can become clearer.

Let's start again with the things of our loved ones, which seem clear to everyone. The same sheet of paper. Let's draw a straight line on it. What coordinate systems are regular, ordinary – admissible – on this line? What is a coordinate system on a line? A point is needed to define a coordinate system, taken as the origin, zero. And a unit of measurement, with the help of which we assign coordinates to all other points in both directions from the origin. As measured by this unit the distance to the starting point (to which the coordinate zero is assigned). Available on line and direction, as a number sign, setting coordinate. This sign indicates, where is the point in relation to zero – left or right. Everything coordinate systems, units differing only in size, choice of initial (null) point and choosing a positive direction (left or right) are quite regular, admissible coordinate systems. They are allow you to describe all points our direct equally complete. And what coordinate system would be inadmissible in this sense? There are no such? there is. If, contrary to common sense, we decide to choose a unit of measurement with zero length, then we will not be able to describe all the points of a straight line. Only one point we can describe, start countdown. Yes, this example is very far from common sense, but I brought him for that, to draw attention to the fact, what “admissibility” (regularity) or “inadmissibility” (irregularity, or otherwise, singularity) coordinate system is associated precisely with the absence of degeneration of this space into a space of a smaller number of dimensions only due to the choice of some properties of the measurement procedure, generating coordinate system. In the given example, this is the degeneration of a straight line into a point due to the wrong choice of the unit of measurement.. Now I will show, that my example is not so naive, as it might seem at first glance. Suppose we have one good coordinate system on our line (ie. reference point selected, unit and positive direction). Consider all sorts of other coordinate systems on this line, which differ from this one only by the unit of measurement. If the new unit is the same for all points of the line, and differs from zero, then the new coordinate system will be good too, permissible. And if its value can change from point to point? By the way, not such a strange possibility. for instance, we want to have a logarithmic scale on the line. It happens? It happens. This is where danger lies in wait for us. We must track, so that nowhere on the line the new unit of measurement in relation to the old one does not turn into zero, not to infinity. And if, nevertheless, at some point this can happen? You can use such a coordinate system? It seems that in other points it is good? The logarithmic scale is just such! And we use it often. The answer is clear. You can use something, but class it as perfectly good, admissible, regular coordinate systems cannot. it singular coordinate system. What is important is, what no point on the line is special in itself. The peculiarity of this or that point on the line is artificial, due to the specific choice of the unit of measurement in it. Therefore, it is the coordinate system that is called singular. And this must be remembered. In this way, already in the one-dimensional case, we encounter cases of using not only regular, but also singular coordinates.

Now let's see the two-dimensional case. One point is selected as the origin. It is assigned zero values ​​for both coordinates.. Two orthogonal coordinate lines selected, two units, two positive directions. This is our regular coordinate system.. Obvious enough, what if we need in some cases uneven scales, one by one, or along both coordinate lines, then among the many coordinate systems obtained in this way there are also singular at some points of the plane. Whose singularity, as in the one-dimensional case, due to the degeneration to zero of the unit of measurement at these points. but, in the two-dimensional case, there is another possibility of the degeneration of the two-dimensional space into a space of a smaller number of dimensions due to “bad” selection of measurement procedure, generating coordinate system. Quite often we use non-orthogonal coordinate systems., and such, whose coordinate lines converge at a point at some arbitrary angle. Such coordinate systems are sometimes called curved or curved.. A little later I will dwell on others., very commonly used non-Cartesian coordinates. Now I want to draw your attention to the obvious fact. There, where the angle of convergence of the coordinate lines vanishes, degeneration takes place again. Two-dimensional space is depicted as one-dimensional, just one coordinate. Because the zero angle between the coordinate lines means, that there is only one line in this place. That is, at such a point, the coordinate system will be singular due to the fact, that instead of two different units of measurement necessary for its regularity, two copies are used (may differ in size) the same unit of measure.

Singular at some points can become quite ordinary, “orthogonal” everywhere coordinates, not only at the expense “wrong” the choice of the magnitude or direction of the scales, but just due to some properties of the space itself (quite regular!), do not allow to describe all this space the only one regular coordinate system. An example of such a space in the one-dimensional case is the closed line, and in two-dimensional – spherical surface. Exactly closed space and is that property, which prevents the possibility of introducing a single regular coordinate system, covering the entire space entirely. When trying to get by with a single coordinate system, all coordinates or some part of them acquire a limited base range of change, period. Special points can also appear, in which coordinates degenerate again, as, eg, at the poles on the sphere. The grid of parallels and meridians works great everywhere, except for two points, where the parallels contract to a point, simulating the disappearance of one of the two units of measurement, necessary for the correct description of a two-dimensional surface. In this way, coordinate system, based on the description of the sphere using parallels and meridians (exactly that, which we use for orientation on the surface of the Earth) is singular in nature. This property does not prevent us from using it quite successfully in everyday life.. And due to the fact, that the earth also rotates, we have a very attractive opportunity to attribute to these fictitious special points, the poles have a mystical meaning. And even organize expeditions, to reach them. Because, from the point of view of the Earth's rotation, there really are two special points, through which passes imaginary axis of rotation. And besides, these points are hard to reach. However, then, that the poles of the coordinate system, based on parallels and meridians are also placed precisely at these points of the earth's surface, for the coordinate system itself, the fact is not essential. The poles of such a coordinate system on a sphere could well be placed at any two points, at the ends of the same diameter.

Now let's look at one more class of singular coordinate systems, used very widely. I want to talk about polar coordinates on the plane and spherical polar coordinates in three-dimensional space. These coordinates are very widely used in physics.. And they are quite familiar to an ordinary person.. Each of us quite often considers ourselves as the center point of the coordinate system, in which everything that he sees is located at some distance (radius) from him, and, Maybe, in different directions, which are marked by turning at some angle from some chosen direction. Such a coordinate system is singular already because, that one (or two) coordinates, corners, are periodic, since at some turn (period) the direction is again the same as the initially selected, from which the angle of rotation is measured. But there is also a more significant singular point in it, just the origin. At this point, ie. when the value of the radius is equal to zero, all directions degenerate, they cannot be defined unambiguously. Any direction can be attributed to this point. For the simplest reason – for an unambiguous choice of direction, you need to have at least two points, and not one. Then, that this feature is purely coordinate, owes its existence only to the method of constructing the coordinate system, obvious enough. Polar coordinate systems are very useful and effective in those cases, when the main role is played only by the distance between objects. As a matter of fact, this is a way of describing, which highlighted naturally inscribes a one-dimensional description into the outer world of a larger number of dimensions. At the same time, attention is focused on the only essential coordinate – distance, radius.

Understanding such properties of singular coordinate systems allows not only avoiding misinterpretation of some “special” phenomena in the resulting description of the world. It also allows you to better understand the laws of nature.. Take the law of gravity for example Newton. What it says? That there is an attractive force between massive bodies, which depends only on the masses of the bodies and on the distance. Why does it depend on only one quantity in three-dimensional space?? In polar coordinates – only from one of three coordinates? Yes, for the simplest reason. In Newton's approximation, bodies are considered as points. And if you only have two points, then you don't really have any three-dimensional space. You only have one dimensional space, in which two points are highlighted. And if you consider the general movement of two points, then you automatically get a plane, swept out by the line connecting them (and one of the laws Kepler in addition). And that's it. In a system of two points, there is only one essential coordinate, the distance between these points. Strength just has nothing to depend on anymore. Respectively, in Newton's approximation, it is the polar coordinates that will be especially convenient for describing, let's say systems, consisting of a star and one planet. Despite their obvious singularity for three-dimensional space. Naturally, upon rejection of this simplest approximation, considering the influence of the rest of the world, the gravitational force at any given point will be determined by a much more complex structure, curvature space (space-time) at this point. And polar coordinates, quite possibly, will cease to be more comfortable, how, let's say, Cartesian. If only because, which have a built-in singularity.

© Gavryusev V.G.
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