Question

A rigid body of any given shape rotates freely under zero torque. Show by using Euler’s equations
that the rotational kinetic energy and the magnitude of the angular momentum are
constant.

Answer 1

Euler solid body rotation equations are

Where are the body moments of inertia around the principal axes. Multiplying both sides of (1) by and both sides of (2) by and both sides of (3) by gives

Adding (1A,2A,3A) gives (lots of terms cancel, that has in them)

(4)

But (4) is the same thing as

where is the angular momentum vector

Hence

Therefore, and since the are constant, we ?nd

Comparing (5) and (4), we see they are the same. This means that or is a constant. Which implies or the angular momentum is a constant vector.

To show that rotational kinetic energy is constant, we need to show that (which is the kinetic energy) is constant, where is the angular velocity vector. But

But we found that since is constant. Hence the above becomes

(6)

If we can show that then we are done. To do this, we go back to Euler equations (1,2,3) and now instead of multiplying by as before, we now multiply by just each equation. This gives

Adding gives (lots of terms cancel, that has in them)

(7)

But the above is the same as (6), with a factor of . This means or or that the rotational kinetic energy is constant. Which is what we are asked to show.