The axis that will not hold
A book has three axes it can be spun about. Two of them it keeps. The third — the middle one, neither the longest nor the shortest — it cannot keep: throw it that way and it turns itself over, and over, at an interval you can compute. Nothing acts on it. No force does this. It is what the equations of a free body say, and everything below is measured on this page and checked against the closed form that has been in print since Jacobi.
The throw
What is conserved while it does that
Where omega is allowed to be
Two numbers survive the tumble, so the tip of the spin vector is trapped on the intersection of two surfaces and cannot leave the curve it starts on. Here is the whole family of those curves for this throw's energy, the body's three axes marked, and the one curve the body is actually on. The separatrix is the one that crosses at the middle axis: the only curve of the family that is not a loop.
The closed form, laid over the arithmetic
The line is a Runge–Kutta integration, stepping forward blind. The dots are Jacobi's elliptic functions evaluated at the same instants, which know the answer without stepping at all. If either is wrong they part.
Only ever the middle one
Twelve bodies of random shape. For each axis of each, the rate at which a wobble grows, from the linearised equations and again from actually integrating one. A positive rate is an axis that cannot be held — and there is never more than one.
The price of aiming better
Postponing the flip is the only thing a careful throw can buy. Halving the wobble does not halve the tumble; it adds a fixed number of seconds, the same number every time, because the interval is a logarithm.
Explorer 1
Nothing above lets the body change which curve it is on. A real body can: it flexes, and flexing costs energy while leaving the angular momentum alone. Turn the leak on and watch where every start ends up.
Checks
Why the middle one is different
Spin the body about an axis and give it a small wobble. The wobble obeys two linear equations that feed each other, and the whole question is the sign of the product of the two coefficients. About the long axis and about the short axis that product is negative, the pair of them go round in a circle, and the wobble stays the size it was — the body nods slightly and keeps nodding, forever, which is what a stable spin looks like. About the middle axis the product is positive, and a pair of equations that feed each other with the same sign is a hyperbola, not a circle. The wobble does not go round. It grows, like eσt, and σ is a number you can read off the three moments: it is on the page above, next to every shape in the table.
That is the whole theorem, and it is worth noticing how little it depends on. Not the shape, beyond the ordering of three numbers. Not the mass, not the material, not gravity — the body in the first panel has nothing acting on it at all, and still turns over. Not even, in the end, the size of the wobble: the growth is exponential, so a wobble a thousand times smaller is not a thousand times better, it is eleven seconds better, and the chart above is that sentence drawn.
There is a floor under that chart, and it is not a physical one. Which side of the separatrix a throw is on is settled by one subtraction, M² − 2T·I₂, whose value is of order the square of the wobble while the two numbers going into it are of order one. At a wobble of a thousandth that subtraction still has ten good digits; at a millionth, four; at 1e-8, none — the machine reports the throw as exactly balanced and the flip as never coming. The arithmetic loses the distinction some way before the physics does, which is why the chart stops where it stops, and the checks above measure that rather than take my word for it.
What holds it to the truth
A page that both simulates a thing and grades its own simulation is grading its homework, so the arithmetic here is checked against three things it cannot talk to. The first is the closed form: a free body's spin is an elliptic function of time exactly, written down in the nineteenth century, and the dots on the third panel are that formula evaluated directly, with no time-stepping anywhere in it. The second is the pair of conserved quantities, which are conserved by the differential equations and not by my code — if a step is wrong, the energy moves. The third is Poinsot's picture, which says the spin vector's tip must lie on the intersection of two fixed ellipsoids and also that its shadow on the fixed momentum direction never moves at all; the page measures both while it animates, in the panel above, and they hold to fourteen digits.
The satellite
Explorer 1 went up in January 1958, spinning about its long axis, which is the axis of smallest moment and — by everything above — a perfectly stable one. Within a day it was tumbling end over end. The equations on this page were not wrong; they were incomplete. Its four whip antennas flexed, and flexing turns rotational energy into heat while leaving the angular momentum untouched, so the body could not stay on the curve it started on. It slid to the lowest energy available at that momentum, which is the spin about the axis of largest moment: end over end. Every spin-stabilised satellite since is spun about its fattest axis, and the last panel is that lesson, with the leak as a slider.
Second opinion in
tests/test_tumble.py, which builds the
elliptic functions again by quadrature and integrates the body again in
Python. Entry 0009.