What has not (yet) been understood                                                          Latest Change 2023-06-12

In brief:
Some aspects of the Foucault Pendulum which I have not(yet) understood, for which I have not(yet) found a satisfying explanation, for which different sources in the literature give different or sometimes conflicting opinions.
Also, in my data processing I have room for improvement on the subject of fitting a dataset to an ellipse.
I present the problems in the form of study tasks.

Schumacher derives that when a repelling impulse is given at a certain moment in the pendulum's period that the precession of the ellipse (not the ellipse itself) will be perfectly suppressed.

Study task:
- Check the derivation of Schumacher, make a description of all steps-in-between, such that a less matematically trained person also can follow it.
- Check that the neglections which S. applies are justifyable and acceptable.
- Predict what happens if the impulse is to early or to late, or to strong or to weak.
- In my spreadsheet is a section for the Schumacher calculation where manual iteration should be done. Try to automate that iteration process, or better, make it superfluous by making a direct derivation.
Note: My experiments so far tend to confirm the Schumacher Criterium. Obeying the condition I found that the Foucault Precession was reasonably regular although there was a substantial ellipse which periodically changed direction.

Crane recommends both attracting and repelling drive with the argument that it would eliminate effects of certain asymmetries in the setup. However he hardly explains which asymmetries these are.
Note that Schumacher has some comment on this.

Study task:
- Give a mathematical derivation that the story about the center magnet is correct.
- Indicate what happens when the magnet is to weak/far off  or  to strong/close by
- Indicate/derive which asymetries can be suppressed by the combination of attracting and repelling drive and indicate what happens if the attracting and repelling impulses are not equal.

Szostak recommends a repelling drive to counteract the forming of an ellipse. T.m.h.o. his figure 3 indicates just the opposite (Or I do not understand it)
Study task:
- Explain that Sz is right / wrong with his argument in favour of repelling drive.

Permanent magnet in the center:
Crane  recommends a repelling magnet in the center to eliminate the precession of the elipse,
Intuitively it feels that an attracting magnet in the center will reduce the amount of ellipse. After all, when the Bob passes straight over it will not have a net effect, because the effect of the atracting force will be cancelled by that of the repelling force after the pass. However, if there is an ellipse, there will  always be an attracting force towards the center.
The contra is: If it is so simple, why is it not done always?

Study task:
- Explain whether or not a permanent magnet in the center of the base unit will reduce the ellipticity. The Bob does have a permanent magnet in the lowest central position.
- Explain what will happen when the magnet is to strong or to weak or has the wrong polarity.

Alternating ellipse:
Kamerlingh-Onnes has mayhematically derived this periodial change in direction, but my knowledge of math is insufficient to follof his derivation.

Study task:
- Explain why the ellips periodically changes direction using the simplest math.
- Explain why the changes occur when the pendulum swings East-West or North-South.
- Explain why the amplitude of the short axis varies more like a squarewave than like a sinewave. That is, when it changes it does so quite rapidly, within a few minutes. Is this real or is it an errort in the way the amplitude is determined in the software I use?
(If you want to study this problem I 'll be happy to provide the necessary information).

Fitting a dataset to an ellipse.
In my dataprocessing I use a rather rude method to derive the ellips parameters (long axis, short axis, angle of long axis and direction) from a set of co-ordinate pairs.
There may be a beter way.

Study task:
Produce an algorithm, preferably in C or Pascal that:
Given a array of co-ordinate pairs (x, y), assumed they lay on the circumference of an ellipse with some small measurement errors, calculates the long axis, the short axis, the angle of the long axis w.r.t. the horizontal and the direction in which the elliptical path goes.
The long axis of the ellipse can have any direction 0 .. pi. Angles > pi are not distinguishable from angles < pi. The short axis will never be as long as the long axis (circle). The data in the array always starts near the center, immediately after a centerpass has been detected.

Make the pendulum swing time independent of the amplitude.
It is known that the period of a pendulum depends slightly on the amplitude, even for very small amplitudes. 
With my backgound as electronic engineer i am familiar with the figures of Lissajous, a.o. an ellipse, if you plot two sine shaped signals of the same frequency on an X-Y oscilloscope. The shape and orientation of the ellipse depend on the amplitude and phase of the two signals. If they have exacly the same frequency the ellipse will be stable. If the frequency of the signals slightly differs the ellipse will change shape and orientation in a regular way.

My feeling is that the precession of the Foucault Pendulum's ellipse is connected to the difference in swingtime between the long and short axis' of the ellipse.  

Christian Hiygens made the period of his pendulum clocks nearly independent of the amplitude by clamping the top of the "wire" between two cycloidally shaped brackets. The cyclo´d is known to have this property for a pendulum with a specific length.

The question is: If we make the swing time of our pendulum independent of the amplitude, will the ellipse precession disapear?
The Huiygens trick can be applied using a properly shaped tube around the pivoting poin.t (kind of trumpet shape).

The contra is that the formula for the ellipse precession tells me that the precession increases with a larger short-axis. This is contray to the thought where the difference between the short- and long axis' swingtimes is responsible for the precession. In that case we should see less precession when the swingtimes are closer to each other.

Study task:
Pull me out of this swamp.