# Why is periodic motion always represented as a sine function

Why was periodic motion started to be represented as a sine function at the beginning? Is it because it is perhaps the simplest periodic function? Is there certainly any reason behind this representation?

• Where did you find such a statement? Jul 29, 2021 at 17:16
• I was introduced to periodic motion through an equation y=asinwt......so question popped why is it that way
– MSKB
Jul 29, 2021 at 17:26
• It isn't - but it is represented as a series of sin funcs. There is no other "fundamental" function which repeats itself. -- and is also everywhere continuous and differentiable Aug 3, 2021 at 13:02

Periodic functions can be also represented by cosine functions, linear combinations of sine and cosine, and more generally by linear combinations of exponential functions $$e^{int}$$ which represents a circular motion with fixed angular speed.

Of course, they can be also represented differently, like for example zig-zag functions used in ancient Babylon. The main mathematical advantage of exponentials come from the fundamental fact that they are eigenfunctions of the shift operator: $$e^{in(t+a)}=e^{ina}e^{int}.$$ This fundamental property implies all other useful properties.

The idea that "every (celestial) motion can be represented in terms of uniform circular motions" was promoted by Greek philosophers in 4th century BC, and it was implemented in practice by Ptolemy and later astronomers.

Added. In my earlier answer I said that Fourier was not influenced by astronomy. He was solving different problems, related to heat equation. But the prominent role of sine and cosine in his work is explained by the SAME fundamental reason: they are eigenfunctions of the shift operator. And this fundamental property explains their omnipresence in mathematical physics and other areas of mathematics.

• I am curious about what did trigger them to intiate a representation using trigonometric function
– MSKB
Jul 29, 2021 at 18:59
• I asked about epicycles and Fourier expansion in a previous question. Could it be that Fouries was influenced by epicycles? The answer said he didnt. I doubt it that sine functions were introduced by eigen- whatevers. Jul 29, 2021 at 19:10
• Fourier apparently was not influenced by astronomy. But of course the exponential (or sin and cos) play the same role in his work for the SAME reason: because these are eigenfunctions of the shift operator. Jul 30, 2021 at 1:44
• But was Fourier aware of shift operators and eigenvalues? Jul 30, 2021 at 7:19
• @Deschele Schilder: Yes, he was, except he did not use this name. Eigenvalues and eigenfunctions naturally occur when one solves the heat equation by Fourier method of separation of the variables. Aug 2, 2021 at 17:31

All periodic motions can be represented by (finite or infinite) sums of sines and cosines (Fourier).

You could have started with other basis functions. Like zigzag functions with different frequencies. The fact is though that most waves in Nature are caused by vibrations obeying Hooke's law, which has sine or cosine solutions.

There are other kinds of periodic processes which obey different (non-linear) equations. These processes can't be approximated by a superposition of sines or cosines ( no superposition). Like the waveforms resembling a curved mountain with a sharp peak. I forgot their name.

I'm not sure if this is a history question but it's a fact that physics stayed in the linear realm a long time. Non-linear phenomena are much more interesting. But much more complex. If mechanics had started out non-linearly then maybe today you would have learned about non-linear signals already at high school. Although... Computers were not available in the time of Newton and Huygens. And you need these to do the intensive numerical calculations going along with non-linearity. To do this by hand was even for Newton too much.(sorry for eventual spelling mistakes; I type on a phone...).

By the way, the non-linear "mountain" waves are called peakons.

Already in the 17th century Hooke invented his law. It is a linear law (pull a spring twice as far and the opposing spring force gets twice that big). Solutions to this law are sines and cosines. They can both involve time and distance so they can represent waves, which were thought to be linear phenomena (which they are not in reality but the appriximation showed to hold very well).

• Why is this downvoted? It's almost a copy of the first one. Aug 6, 2021 at 17:09