Yacov Kantor poses monthly
physics questions and to the extend they are solved also the solutions.
The question of 06/98 is the problem 19 in the Scottish Book, which
contains a large number of mathematical problems posed by mathematicians
in Lviv (Lemberg) in the nineteen thirties.
Recently I considered the question (a): Are there long logs of non-circular
cross-section with half the density of water which can float in any position
without tendency to rotate?
It turns out, that there are non-circular solutions to this problem.
However, after finding the solution Yacov Kantor informed me, that this part
of the problem was already solved by Auerbach in 1938.
Next I considered the problem (b) with densities ρ different from one
half. There are one-parameter families of cross-sections for ρ not equal 1/2
which have a p-fold rotation axis. For given p they exist for p-2 densities
ρ. There are strong indications, that for all p-2 densities one has the
same family of cross-sections.
All this can be found in
Meanwhile I have found the differential equation which describes the boundary
curve. The conjecture that one has the same family of cross-sections for all
p-2 densities proved true.
A short account can be found here , and more in
The main results of physics/0203061 and physics/0205059 are published in
Studies in Applied Mathematics 111 (2003) 167-183
Deborah Oliveros-Braniff and Luis Montejano Peimbert looked for special
solutions of this problem.
Revista Ciencias 55-56 (1999) 46-53
They restricted themselves to
the case where the edges of an equilateral pentagon constitute five
waterlines. They call these 'volantines'. They imagine five cyclists each in
the middle of an edge moving in the direction of the edge, and the five
corners moving along the boundary (figure 1 and 2). Thus they restrict
themselves to the ratio of circumference over length of the boundary line
below water equal to five. They found a solution with p=7 which, however, was
not sufficiently convex (figure 9 of their paper).
Recently Sergei Tabachnikov
informed me that there is an equivalent problem which goes by the name of
problem. The problem consists in determining the direction of bicycle
motion from the tire tracks of the bicycle wheels. If this is impossible then
the curves are called bicycle curves. The track of the front wheel corresponds
to the boundary of the floating log, and the track of the rear wheel to the
envelope of the waterlines.
David L Finn has considered this
problem and gives
construction if a certain piece of the track of the rear wheel is given.
However, he does not give a solution for a closed path which would be necessary
for the floating-body problem.
On the other hand the differential equation derived in my paper yields solutions
for bicycle curves which are not closed, too.
In physics/0603160 the
explicit solution in terms of the differential equation derived earlier is
given. It is shown explicitly, that this yields solutions to the floating body
and the tire track problem.
A more elegant version is given in physics/0701241 which
contains figures of a large number of curves. Meanwhile I have prepared an
introduction under the title Three Problems - One
solution nearly without mathematics, but with many animations.
Es gibt auch eine deutsche Version dieser Einführung unter dem Titel Drei Probleme - Eine Lösung fast ohne
Mathematik, aber mit vielen Animationen.