What is a double pendulum

Sexl Physik 8, textbook

55 | p oincaré, 1903: A very small cause that we do not notice causes a considerable effect that we cannot overlook, and then we say that the effect is random ... 55.1 Double pendulum: If you hang two rigid pendulums on each other, one can examine the various forms of oscillation of a double pendulum. Building instructions at physikplus.oebv.at 55.2 Turbulence is a model game for an unpredictable phenomenon that is, however, determined by the laws of nature. You can also observe mixing processes like in the picture when you add milk to the coffee. Turbulence plays an important role in the formation of clouds and influences the climate. 55.3 ECG ventricular fibrillation With ventricular fibrillation, irregular electrical impulses occur, which can lead to "death of the heart". In the process, many thousands of muscle fibers suddenly give up their exact coordination. In the EKG (the recording of the heart currents) a mathematical structure can be recognized (the “period doubling”), which describes the path into deterministic chaos. 1.2 Causality and predictability The connection between cause and effect is called causality. In physics we assume that the same causes have the same effects. In applying the laws of nature in practice, we see that the situation is more complex. The initial situations are never exactly the same, we never find exactly the same causes and we cannot create them artificially. The following applies: Strong causality Similar causes lead to similar effects. This largely corresponds to our experience with natural or technical phenomena. However, there are also phenomena that contradict this idea: If we let a number of balloons start from the same place at the same time, their trajectories will soon diverge widely; if we roll a die, the number is random. Although the movements of the balloons and the cube are determined by the laws of nature known to us, they cannot be predicted. We therefore speak of weak causality. Weak causality Similar causes can lead to completely different effects. We want to investigate the phenomenon mentioned using the example of the double pendulum. Experiment: Double pendulum 55.1 You need: a double pendulum (55.1) What has to be done? Start the double pendulum several times from the same starting position. Can you see differences in movement? Describe. When looking at the swinging double pendulum, one soon sees that the behavior can usually only be predicted for short periods of time. For longer periods of time, at best, probability statements about it are possible. Particularly critical phases arise when one of the pendulums approaches a rollover: will there be enough energy to cross the apex, in which direction will it turn? How is it that, despite known laws of motion, no longer-term prognoses are possible about the motion of the double pendulum? The meteorologist Edward LoreNZ (1917–2008, USA) found an answer to this question. The weather is probably the best known system in which long-term forecasts are not possible. When Lorenz simulated the weather on the computer in 1961 and he happened to repeat a calculation with rounded initial values, he discovered that the system suddenly developed completely differently. The sensitive dependence on the initial conditions has been called the butterfly effect since Lorenz: the flapping of a butterfly's wings can be the cause of a hurricane. The effect can also be seen in completely different contexts. One example is the earth's magnetic field, the irregular polarity of which was mentioned in Physics 7 (p. 80). Even pushing a rigid pendulum or a child's swing regularly can lead to chaotic behavior. Electronic oscillating circuits or dripping water taps can also show chaotic behavior. Physical systems that are particularly sensitive to slightly different initial conditions are called chaotic. Small differences in the initial conditions can lead to completely different behavior. With chaotic systems it is impossible to predict the behavior of the system for any length of time. For testing purposes only - owned by Verlag öbv

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