(a) Consider a primed set of axes coincident in origin with an inertial set of axes but rotating with respect to the inertial frame with fixed angular velocity ω0. If a system of mass points is subject to forces derived from a conservative potential V depending only on the distance to the origin, show that the Lagrangian for the system in terms of coordinates relative to the primed set can be written as
L=T′+ω0⋅L′+21ω0⋅I′⋅ω0−V,
where primes indicate the quantities evaluated relative to the primed set of axes. What is the physical significance of each of the two additional terms?
(b) Suppose that ω0 is in the x2′x3′ plane, and that a symmetric top is constrained to move with its figure axis in the x3′x1′ plane, so that only two Euler angles are needed to describe its orientation. If the body is mounted so that the center of mass is fixed at the origin and V=0, show that the figure axis of the body oscillates about the x3′ axis according to the plane-pendulum equation of motion and find the frequency of small oscillations. This illustrates the principle of the gyro compass.