the Tire Track Problem and

Electrons in a Parabolic Magnetic Field

Institut für Theoretische Physik

Ruprecht-Karls-Universität Heidelberg

After stating the three problems considered in the paper physics/0701241 (where also references can be found)

(i) the solution is described, where nearly no use of mathematics is required, and

(ii) animations to some of these solutions are shown.

Some mathematical formulae and supplements are given in ps-file and pdf-file, resp.

However, we report shortly that more problems are solved by the solutions, before we go through this program.

The linear limit described below yields the shape of the Elastica calculated by James Bernoulli in 1691-1695, who assumed the curvature of a rod to be proportional to the bending moment. In 1772 Daniel Bernoulli (nephew of James) proposed in a letter to Leonhard Euler that the potential energy of a bent rod is proportional to the square of its curvature integrated over the length of the rod. Euler, master in variation techniques, performed a thorough calculation of the elastica in 1774. Both approaches are equivalent.

There are more problems related to the elastica. A nice survey can be found in the PhD thesis by Raph Levien The elastica: A mathematical history.

The closed curves are the shapes of buckled rings. They are elastica under homogeneous pressure from outside. Lévy in 1884, Halphen in 1884-1888, and Greenhill in 1892-1899 investigated them.

A survey of these problems and solutions can be found in my paper "From Elastica to floating bodies of equilibrium" arXiv: 1909.12596

A right cylinder is a body, which is bounded by two parallel, plane areas (base and top areas) and a cylinder surface generated by parallel straight lines, which are perpendicular to the base and top areas.

The magnetic field changes sign at r

Instead of rotating the body, I will keep it fixed and assume that the direction of gravitation is rotated. Then the various water-lines are shown in green and cyan. The red envelope of the water-lines touches in the middle. The part inside the water-line is always above or below the water (provided inside and outside is well defined, which is not the case, if the density of the solid is half the density of water).

In the case of the bicyclist one will generally assume that he/she rides in one direction; in particular that this applies for the rear wheel. If however, he/she is very artistic and the bicycle allows, then there are also cases, where he/she moves back and forth. For some solutions it would even be necessary, that the steerer could be rotated by more than 180 degrees. Nevertheless it may be that parts of the curve can be traversed without problems. After all, bicyclists are not supposed to ride forever.

For the boundary of the floating body one has to require that it is closed and the cross-section is sufficiently convex so that the water-lines intersect the boundary exactly twice, not more often.

I call this the property of constant distance.

If A

In the example with different rotation angles δχ this is the case for δχ=90

It is this property, which made it possible, to obtain a differential equation for the curves by using it for an arbitrary small angle δχ. (An animation for two rotating angles is shown below).

Similarly we have two special limits here: One in which a circle is solution or asymptotically approached, the other one is the linear limit, where most of the curves repeat periodically in linear direction.

The shape of conal sections can be desribed by one number, which can be varied continuously. For ellipses this can be the ratio between the two main axes. (Generally the shape of a conal section is characterized by its numerical eccentricity). The shape of our curves is given by two numbers, which can be varied continuously. Most of the curves repeat after rotations around a certain angle ψ

In arxiv: 0803.1043 with the title "Floating Bodies of Equilibrium in Three Dimensions. The central symmetric case" the conditions for bodies with central symmetry and relative density different from 1/2 are considered. It is, however, shown by an example that there are central symmetric solutions at this density, if one skips the requirement of star shape.

In arxiv: 0902.3538 with the title "Floating Bodies of Equilibrium at Density 1/2 in Arbitrary Dimensions" it is shown that also in dimension three (and higher) there is a large manifold of solutions for density 1/2. To any given envelope of the water-surfaces, which has to have the property that there is only one tangent at the envelope parallel to any given plane, there are solutions.

Presently I consider bodies, which are not central symmetric with densities different from 1/2.

Of course these are also solutions to the tire track problem.

One may compare with the examples given by David L. Finn.

We remark that there is no solution for the chord of constant length, if n-m=1, which is the case here.

The other two figures show pairs of

For reasons I will not explain here, also negative m can be used. An example for m/n=-1/7 with three different lengths of chords is shown.

Here we use that the circle is one of the two curves or that the curve approaches asymptotically the circle. Thus we can use the property of constant distance between the circle and the non trivial curve.

It should be emphasized that these solutions can be described by exponential and trigonometric functions (see section 8 of the paper), whereas in the general case double-periodic functions are needed.

In the following three websites the first figure shows this construction, the second one the movement of the chord. These are no longer solutions to the floating body problem, but the parts of the black and red curves which evade the loops are solutions of the tire track problem.

ε is the ratio of the length of the line to the radius of the magenta circle. If it is larger then two, then the black curves are periodic and three εs were chosen which yield closed curves.

m/n=1/3, ε=5/2=2.5

m/n=-1/3, ε=sqrt(49/10)=2.213594

m/n=-2/1, ε=sqrt(64/15)=2.065591

If ε is less then two, then one obtains a curve inside and another one outside the circle. In this case the curves approach asymptotically the magenta circle. We show curves for ε=1.6.

In the limit case ε=2

one obtains one curve approaching asymptotically the circle.

Again there are various possibilities. One may have

In the limit case of extreme stretching there is no solution for a chord along the same curve. However, there are such straight lines between two curves transferred against each other.

The curve may also be generated by moving the cyan end of our line of length 2

In the linear limit one may also have

These types of curves have appeared in a paper by Evers, Mirlin, Polyakov, and Wölfle (see figs. 1 and 2) as trajectories of electrons in a magnetic field perpendicular to the plane, which increases linearly in one direction of the plane.

In the carousels I the vertices of the pentagon run along five eights and along closed curves with m/n=1/7 and 1/12. The area of the pentagon stays constant.

In the carousels II five copies of a curve with ε=1 are shown, which are rotated against each other by two different angles supporting the vertices of the pentagon. In addition a set of five identical curves of the linear case is shown, along which the pentagon crawls. If the pentagon intersects, then the difference of the two areas is constant.

The boundaries need not have corners. One of Auerbach's examples has the shape of a heart. The other one is without corners. The examples consist of two or three straight lines and the same number of curves lines vis-a-vis. These are pieces of the lines, which already appeared above as the limit case of extreme stretching. Here are further examples without corners..

In general the solutions for density 1/2 are no longer solutions for the trajectories of electrons in a parabolic magnetic field. They are still solutions to the tire track problem, if the cyclist is able to go back and forth in an artistic way. Back to the beginning