| C | dr | (A − B) pq. |
| dt |
(5)
These equations are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated. If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap2 + Bq2 + Cr2 and A2p2 + B2q2 + C2r2 are both constant. The former is, in fact, equal to 2T, and the latter to Γ2, where T is the kinetic energy and Γ the resultant angular momentum.
To complete the solution of (2) a third integral is required; this involves in general the use of elliptic functions. The problem has been the subject of numerous memoirs; we will here notice only the form of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875), If we write
| u = ∫φ0 | dφ | , Δφ = √(1 − k2 sin2 φ), |
| Δφ |
we have, in the notation of elliptic functions, φ = am u. If we assume
p = p0 cos am (σt + ε), q = q0sin am (σt + ε), r = r0Δ am (σt + ε),
(7)
we find
| ṗ = − | σp0 | qr, q̇ = | σq0 | rp, ṙ = | k2σr0 | pq. |
| q0r0 | r0p0 | p0q0 |