Couples, Moments and Euler’s Equations

There is a proof given here of Euler’s equations, but you can really just take them on trust. They tell you that if the principal moments of inertia are all different, the equations become non-linear, with the rate-of-change of each component of the angular velocity depending on the product of the other two. Solving such equations is very difficult indeed, but setting up a simulation to represent them is no problem at all, as you can see in the simulation of the ‘dancing T-handle’. If two of the principal moments are equal, everything becomes much simpler. One of the rates-of-change becomes zero, so the angular velocity about that axis is constant and the equations become linear once again. That describes the case of the gyroscope, where its spin is assumed to remain constant (even though it will actually slow down unless powered). We can add in any couple that is applied to it and get equations for its precession. Another way to think of the gyroscope is as a carrier of a huge angular momentum about its spin axis. When an orthogonal couple C is applied, momentum C.dt is added to that momentum for each small time-step dt. The new momentum vector is the sum of the original vector plus the small C.dt at right-angles to it, so its direction will be rotated slightly. The result is the precession. But the gyroscope has a chapter all to itself.