Dynamics of Coin Tossing
The standard example of a random event we give in probability courses is the tossing of a coin. On the other hand, in mechanics courses we teach that rigid body mechanics is one of the few examples of an integrable system. Not only are its equations deterministic, there is an explicit solution in terms of elliptic functions. In other words, a rigid body is one of the most predictable of all mechanical systems.
But there is a certain tension between these two statements: a coin is a rigid body. If the motion of a rigid body is so completely predictable, why can’t you predict the outcome of a coin toss? There have been several studies of this in the literature, most notably
The basic conclusion is that if we carefully control the initial conditions of a toss, one can indeed predict with certainty the outcome. Diaconis even commissioned a machine that would throw coins in precisely the same fashion each time to verify this fact experimentally.
But there is an aspect of this system that seems not to have been taken into account by these studies. A coin is an unstable system. Small changes in initial conditions-in certain directions- grow with time. This explains why in practice, a coin toss is unpredictable even though under perfect conditions it might be.
There is also another kind of motion of a coin that is stable and hence predictable even in practice. To understand this, imagine throwing a frisbee. Our common sense and everyday experience tells us that this is quite a predictable system. Geometrically, a frisbee is just like a coin only with a bigger diameter. What is the difference? It has to do with the direction of rotation. A tossed coin rotates along an axis parallel to its plane; a thrown frisbee rotates around an axis perpedicular to its plane. It turns out the the latter is a stable motion while the former is not! So common sense and a detailed analysis do agree in the end. But somehow this simple explanation seems to have been missed by earlier discussions.
An example of an unstable system is a pendulum balanced on its head: it is impossible in practice to predict which direction it will fall. The slightest push in either direction will make it fall over. Not knowing which such push will come first we have to think of it as a random event. For the pendulum this only happens at one point in its configuration space. Everywhere else a pendulum is very regular and predictable. That is why we regulate clocks with it!
But suppose a system is unstable at every point in its configuration space. Then two initial velocities that point in slightly different directions will lead to paths that diverge from each other for ever. If the configuration space is bounded, this will lead to unpredictability. This is not a new idea. Arnold has suggested that the notorious unpredictability of the weather can be traced to such an instability in the equations of fluid mechanics. He showed that for all initial configurations of the fluid, small changes to the velocity will grow with time. Thus any small uncertainty in the initial measurement will, in time grow to enormous differences in the predicted outcome. The proper statement of this uses some ideas of geometry.
The Riemannian Geometry of Rigid Mechanics
Based on work in collaboration with Masha Gordina of the University of Connecticutt.
There is a remarkable similarity in the equations of a rigid body and those of an ideal fluid (beyond the fact that they were both discovered by Euler): they simply say that the system moves along the curve of shortest length in the configuration space. In the case of the rigid body this space is the rotation group ; for the fluid it is a much larger space, the space of volume preserving diffeomorphisms. The rate at which two different initial conditions diverge from each other is given by the sectional curvature of the metric, in the plane determined by the two directions. Arnold showed that for the case of the fluid the sectional curvature is mostly negative, which is the sign required or instability.
What about the much simpler case of the rigid body (of which the tossed coin is an example) ? The standard metric of the rotation group has positive curvature, which means geodesics are stable under small perturbations. Actually under the standard metric is just the three dimensional sphere with anti-podes identified.
But this is not the correct metric to use.
Each rigid body has a positive symmetric tensor associated to it, the moment of inertia. Such a tensor defines an inner product on the tangent space of at the identity. This can be transported to any point on
by the right mutliplication, yielding a unique homogenous Riemann metric on the rotation group. The Euler equations of a rigid body are the geodesic equations of this metric. Only for an isotropic rigid body (one that has equal moments of inertia in all directions, like a perfect sphere) does this agree with the standard metric. In general the metric is homogenous but not isotropic: it is the same for all initial orientations ( configurations) of the body, but not in all directions of the initial angular velocity.
A paper by Milnor summarizes the mathematical tools necessary to calculate the curvature tensor of such metrics.
J. Milnor Curvatures of Left Invariant Metrics on Lie Groups Adv. Math. 21, 293-329 (1976).
The standard basis in the Lie algebra of the rotation group is . If the principal moments of inertia are
, these vectors are orthogonal, and
. The metric is determined by the kinetic energy,
where
are the components of angular momentum in this basis.
To use Milnor’s formulas, it is useful to transform to an orthonormal basis etc.Then the commutation relations become
with structure constants
,
.
The non-zero components of the curvature tensor are then,
with the remaining ones given by the symmetries of the Riemann tensor.
When all the principal moments of inertia are equal, these are all positive: as is correct for the standard metric on the rotation group. In the case of interest to us there is an axis of symmetry, so that . The plane of the coin (or frisbee) is spanned by the 1 and 2 axes and the axis orthogonal to this plane is labelled 3. Then
and
. Thus we see that sectional curvature in the directions
is negative if
.
Now, for a cylinder of radius and height
the moments of inertia are given by ( see the book by Landau and Lifshitz; or the article by Diaconis et. al. above)
and
. ( We use units where the mass is equal to one. In any case, it is only an overall constant multiplying these quantities.)
So there is an instability in one pair of directions if . For an American half-dollar coin (the one used by Diaconis et. al.)
both in millimeters. Clearly we are in the unstable regime.
Note that the other components remain positive. Thus the instability is only in the
plane in the tangent space of the configurations. For example, an initial angular momentum along axis 1 will be unstable with respect to perturbations towards the second axis. These are the two possible `tossing’ motions. The rotations around the third axis is what a frisbee does when it is thrown. There is no instability that involves this direction!
Thus we see that the unpredictability of the motion of a rigid body can indeed be explained by its instability.
Diaconis et. al. note that the coin they used has a small bias towards head for another reason: there is more metal on that side. We have ignored that effect in our analysis; it does not affect our explanation for the randomness. We have also ignored the effect of gravity, but an analysis including it doesn’t change the basic conclusion.
