by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid, and if A, B, C denote the corresponding moments of inertia, the kinetic energy is given by
2T = A (θ̇ sin φ − sin θ cos φψ̇)2 + B (θ̇ cos φ + sin θ sin θψ)2
+ C (φ̇ + cos θψ̇)2.
(14)
If A = B this reduces to
2T = A (θ̇2 + sin2 θ ψ̇2) + C (φ̇ + cos θ ψ̇)2;
(15)
cf. § 20 (1).
§ 22. Equations of Motion in Generalized Co-ordinates.—Suppose we have a dynamical system composed of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are subject to mutual forces and also to the action of any given extraneous forces. The configuration of such a system can be completely specified by means of a certain number (n) of independent quantities, called the generalized co-ordinates of the system. These co-ordinates may be chosen in an endless variety of ways, but their number is determinate, and expresses the number of degrees of freedom of the system. We denote these co-ordinates by q1, q2, ... qn. It is implied in the above description of the system that the Cartesian co-ordinates x, y, z of any particle of the system are known functions of the q’s, varying in form (of course) from particle to particle. Hence the kinetic energy T is given by
| 2T = Σ {m (ẋ2 + ẏ2 + ż2) } |
| = a11q̇12 + a22q̇22 + ... + 2a12q̇1q̇2 + ..., |
(1)