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Putting sand on the oscillating elastic plate, you can see the formation of Chladni figures . They often serve as an example of the “natural beauty” of physical phenomena, although behind them is the rather simple physics of the resonant excitation of standing waves. And few people pay attention to the curious peculiarity of these figures: the lines on them avoid intersections, as if they are repelled by a certain force. Let's try to understand what kind of physics is hidden behind this repulsion and how it is connected with the quantum theory of chaos.

As we know, elastic bodies can make quite complex vibrations under which they shrink, stretch, bend and twist. Nevertheless, the vibrations of any elastic body can be represented as a combination of simpler

Each normal oscillation is represented by a

Let us now look at a two-dimensional system, an example of which is a thin elastic membrane stretched over a rigid frame. Normal vibrations of a circular membrane look more complicated than in the case of a string, and instead of individual point-nodes there are

In a round membrane, the nodal lines, which are circles and segments along the radii, can intersect at right angles. If the edges of the membrane have an arbitrary shape, finding the frequencies of normal vibrations and pictures of their nodes and antinodes turn into a task that can be solved only with the help of a computer.

The equations describing the vibrations of a thin elastic plate differ from the equations of the oscillation of the membrane, since the plate has its own rigidity, while the membrane is soft and springy only due to tension by external forces. However, there also exist sets of normal vibrations, the drawings of which essentially depend on the shape of the boundaries.

As mentioned above, in the general case, body vibrations are a combination of a whole set of normal vibrations excited in it. The phenomenon of

The two videos below show a typical scheme for obtaining Chladney figures: an elastic plate is attached in the center to a generator of mechanical vibrations, the frequency of which is gradually increased. Normal oscillations of the plate with their pictures of nodes and antinodes are excited when the oscillator frequency resonates with the natural frequencies of these oscillations (the natural frequencies are shown in the video in the lower left corner).

We see pictures of knots and antinodes due to the fact that the air flows near the oscillating plate blow sand particles to the nodal lines of a standing wave.

Another example of normal waves is standing waves on the surface of the water. They are described by an equation that differs from the equations of vibrations of plates and membranes, but they follow the same qualitative regularities, and with their help, analogs of Chladney figures can be obtained.

So, we have seen that in the case of a circular membrane, the nodal lines are theoretically! - remarkably intersect, at the same time, on the Chladni figures on square or more complex plates, the nodal lines avoid intersections. To understand the reason for these patterns, we will have to make a short excursion into the theory of chaos.

Classical chaos is a property of mechanical systems, consisting in an extremely strong dependence of the trajectory of their movement on changes in the initial conditions. This relationship is also known as the “ butterfly effect ”. A striking example of chaotic behavior can be found when trying to predict the weather: a system of equations describing the movement of the atmosphere and oceans does not allow for sufficiently accurate predictions at large times due to exponentially increasing errors due to small inaccuracies in the source data

The phenomenon of chaos was discovered and popularized by the meteorologist and mathematician Edward Lorenz , who discovered that the two weather forecast calculations, starting with very close initial conditions, are almost indistinguishable from each other at first, but from some point begin to radically diverge.

The simplest systems, by the example of which it is convenient to study chaos, are billiards - areas of a flat surface, along which a ball can roll without friction, absolutely elastically bouncing off rigid walls. In

Mechanical systems that are not chaotic are called

Rectangular and round billiards are integrable due to their symmetrical shape.

Billiards of more complex shape, not possessing as high symmetry as a circle or a rectangle, are chaotic

Here you can already guess that the presence of intersections between the lines on the Chladni figures is determined by whether the plate has the form of an integrable or chaotic billiard. This is clearly seen in the photos below.

How to understand why the presence of intersections between the nodal lines due to the integrability of billiards? To do this, turn to the

As in the case of oscillations of membranes and plates, the Schrödinger equation, which describes quantum billiards, allows us to find normal oscillations in the form of standing waves, which have a characteristic pattern of node lines and antinodes that are individual for each oscillation and depend on the shape of the boundaries.

Figures of standing waves in integrable and chaotic quantum billiards are qualitatively different: integrable billiards show symmetric, ordered patterns of standing waves, while in chaotic billiards standing wave patterns are very intricate and do not show any visible patterns (at the end of the article it will be shown that some interesting patterns there still exist).

The qualitative difference can also be seen in the pictures of nodal lines: in the case of integrable quantum billiards, we see ordered families of

Why do the nodal lines in chaotic billiards not intersect? In 1976, the mathematician Karen Uhlenbeck proved a theorem according to which the nodal lines of standing waves of quantum billiards, generally speaking, should not intersect.

In a simplified form, this can be shown as follows: suppose that two nodal lines intersect at the point (

1) It must be zero at the point (

2) If we move from the point (

3) If we move from the point (

Total we have three conditions (or three equations) imposed on the function of two variables

In integrable billiards such exceptions just arise. As we saw above, their special properties — predictability of motion, absence of chaos, regular patterns of standing waves — are a consequence of their high symmetry. The same symmetry ensures the simultaneous fulfillment of three conditions necessary for the intersection of the nodal lines.

Let us now take a closer look at examples of Chladney figures typical of integrable and chaotic billiards. The figure below shows three characteristic cases . The plate on the left has the shape of a circle, so the corresponding quantum billiard is integrable, and the nodal lines intersect with each other. The plate is rectangular in the center, which also corresponds to an integrable system, but the round fixture in the center slightly violates the symmetry of the rectangle, therefore the nodal lines intersect not everywhere. On the right is an example of a purely chaotic system: a plate in the form of a quarter of Sinai billiards (in the upper right corner there is a circular notch), the node lines on which are no longer intersecting.

Thus,

It’s not so easy to get beautiful Chladney figures with intersecting lines on a round plate. When oscillations are excited with central fastening, the circular symmetry of the entire system prohibits the formation of radial nodal lines, so we will see only a boring set of circles (this difficulty can be circumvented by exciting vibrations not from the center, but from the edge of the plate using the bow from the violin) If the plate is not fixed in the center, the figures of Chladni will become more interesting, but due to the violation of circular symmetry, the system will cease to be integrable.

Finally, an attentive reader may notice: but I see that sometimes the nodal lines intersect even on “chaotic” plates. How so, if their intersection is prohibited by the Uhlenbeck theorem?

First, the nodal lines can avoid the intersection, but before that they come so close together that, because of the finite width of the sand track, it will seem to us that there is an intersection. Secondly, between integrable and chaotic systems there is in fact no sharp boundary.

In the classical theory of chaos, the famous Kolmogorov-Arnold-Moser theory is devoted to this question. She says that if you slightly break the symmetry of the integrable system, it will not immediately show chaotic behavior, and, for the most part, will retain its property of predictability of movement. At the level of quantum theory of chaos and Chladni figures, this is manifested in the fact that in some places the intersections of the nodal lines are preserved. This happens either at particularly symmetrical points of the billiard, or far from the source of disturbance that violates the symmetry of the integrable system.

What else is interesting quantum theory of chaos? For the interested reader, I will mention three additional questions that are no longer directly related to the Chladni figures.

Source: https://habr.com/ru/post/406637/