Dictionary of Mathematical Eponymy: The Kovalevskaya Top

When I was about eight, my parents bought, as a Christmas gift for my brother and me, a “Jungle Gym”, plastic tubes and connectors that fit together to make whatever the imagination came up with, a sort of large-scale Meccano. My brother went out into the garden to build castles and trains; I stayed indoors with the cheap gyroscope from my stocking.

I love things that spin, and have made every effort to introduce my children to spinning things from an early age, in the hope that they’ll understand more about the maths of them than I do.

As I understand it, there are three relatively well-behaved sorts of top:

  • The Euler top, which has no particular symmetry and spins around its centre of gravity.
  • The Lagrange top, which has rotational symmetry; its fixed point is where this axis meets the ground; and
  • The Kovalevskaya top, which is brilliantly weird.

What is a Kovalevskaya top?

This is going to require you to think in three dimensions. What I describe here is an example of the Kovalevskaya top; I do not believe it to be the only example.

Imagine a rigid pendulum that’s free to swing in any direction it likes. Perpendicular to the pendulum is a cross (the bars at right angles), and the ends are joined in a circle.

There’s a bit more to it on a technical level - the moment of inertia1 about the unweighted cross-beam is the same as the moment of inertia about the pendulum, and double that about the weighted cross-beam; also, the centre of mass lies somewhere on the pendulum axis.

Now, the Euler top just spins. The Lagrange top spins and precesses. The Kovalevskaya top goes all over the shop!</p>

When it’s not moving very fast, gravity is the important thing and the motion is a combination of the pendulum rotating in two possible directions, while the cross itself rotates around the pendulum. That’s neat in itself.

When it’s moving fast, gravity isn’t the dominant force any more. The pendulum rotates merrily in circles around the unweighted axis, which can be anywhere. If the top is also spinning about the pendulum, then gravity causes the body to precess, and if the centre of mass is too high, the motion becomes unstable.

Why is it important?

The Kovalevskaya top, even if it wasn’t important, would be very cool. Look at it! It’s spinning in all sorts of different directions!

But its importance comes from being the first non-obvious top that is mathematically tractable - it allows a solution to Euler’s gyroscopic equations given the initial conditions. (It’s not a nice solution, it involves hyperelliptic functions, but it’s a solution.) Apart from Euler and Lagrange tops, the Kovalevskaya top is2 the only rigid body rotation that can be solved in this way.

Who was Sofia Kovalevskaya?

Sofia Kovalevskaya (she transcribed her name as Sophie Kowalevski) was born in Moscow in 1850. She is best known for her work in analysis, mechanics and in partial differential equations - her doctoral thesis includes the Cauchy-Kovalevskaya theorem, which applies to the existence of local solutions of such equations. I say ‘thesis’ - she wrote three papers while studying under Weierstrass in Berlin, which were collectively accepted as a doctoral thesis by Gottingen (she graduated summa cum laude).

After her husband’s death, she became first a privat-docent and later a professor at Stockholm University. She died in 1891 of pneumonia.

* I would be remiss not to point you at Evelyn Lamb’s article on the top, and at the German wikipedia page, which is far more detailed than the English-language one.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. A measure of how hard an object is to accelerate in a circle – it is to angular motion what mass is to linear. []
  2. I gather []

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