Correspondence of Marcel Riesz with Swedes. Part II. file: Riesz1.tex

0.4. ALFRED LILJESTRÖM 25
After my return from Copenhagen I made some measurements in order to be able to
master numerically fhe top’s special inertia properties. It turned out that one from the
above equation comes to the following more special equation holding for the nutation limits
of the top:
1 − q 2 = 2mglAu −
R 2 0
(1 − qu)2
1 − u 2 = 0,
provided one assumes that, at the start, the top has its axis nearly horisontal and there is
no translatory motion.
A simple calculation shows that in this formula the fractal expression for u = q takes
its minimal value 1 − q 2 or exactly the value that linear expression in the formula takes
for u = 0. Geometrically, this means that for large values of R 0 , e.g. if the top’s initial
spin is sufficiently large, then the straigt line meets the cubic in two points which lie very
close to the value u = q. This in turn implies that the braking of the top with some
approximation follows the formula
R z = R 0 cos θ
The nutation theory that I presented in Copenhagen can thus beused to follow in
detail the top’s curious brakning process. It gives likewise a striking verification of what
I also said in Copenhagen, among other things, namely that the pivot motion of the top
with its centre of gravity displaced by no means contradicts the existence of two nutation
limits, so that the phenomen evoced by you corresponds to the case when both limits
coincide, the straight line being tangent to the cubic.
I understand now very well that you now, after havig taken part of my results in
my first letter, gladly want to arrive to them along more elementary paths, because you
find the nutation theory complicated. The attempt to an “elementary theory”, as your
son presented in his letter, and which you refer to, gives me some doubts. To start,
without further ado, from formula R z = R 0 cos θ and then say that one can by “se bort
fra tyngdenutationerne, umiddelbart vise at θ vil variere omtrænt lineært med tiden” 10
(neglect the weight nutations immediately show that θ varies inversely linearly with time)
seems to mean that by assuming θ constant one can show that θ varies linearly with time.
Moreover, the presentation, it seems me, is based on the misunderstanding that gravity
alone causes the nutations. This is not the case. If one puts the gravity constant g equal
to zero in the nutations equation, then it is only one little term that vanishes, all the other
remain and witnes that the laws of natures are such that inertia properties of the ball-top
can give rise to complications.
I am tempted to quote Poincaré/’s famous reply to Felix Klein: “Voilà une difficulté
dont on ne triomphe pas en quelques lignes!” (French, Here is a difficulty that one does
not overcome in a few lines!)
10 Editor: This quote is in Danish.