| d | · 1⁄2 (Ap2 + Bq2 + Cr2) = Lp + Mq + Nr, |
| dt |
(3)
which is (virtually) the equation of energy.
As a first application of the equations (2) take the case of a solid constrained to rotate with constant angular velocity ω about a fixed axis (l, m, n). Since p, q, r are then constant, the requisite constraining couple is
L = (C − B) mnω2, M = (A − C) nlω2, N = (B − A) lmω2.
(4)
If we reverse the signs, we get the “centrifugal couple” exerted by the solid on its bearings. This couple vanishes when the axis of rotation is a principal axis at O, and in no other case (cf. § 17).
If in (2) we put, L, M, N = O we get the case of free rotation; thus
| A | dp | (B − C) qr, |
| dt |
| B | dq | (C − A) rp, |
| dt |