(a) A rigid body has principal moments of inertia , , about its centre of mass, with…
PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames · Oscillations and normal modes
Action and Symmetry, Ch Small oscillations: stability, degeneracy, symmetry and spectra
Problem
(a) A rigid body has principal moments of inertia I1=2.0kgm2, I2=3.0kgm2, I3=5.0kgm2 about its centre of mass, with I1<I2<I3. No external torque acts. It is set spinning freely about the intermediate axis (axis 2) at Ω=4.0rads−1. Linearise Euler's torque-free equations for a small perturbation (ω1,ω3) away from this spin, write the pair as an eigenvalue problem for a rate λ such that the perturbation behaves as exp(λt), find λ, and state whether the spin is stable. (b) In the rotating restricted three-body problem the effective potential's Hessian at the Lagrange point L4 has two negative eigenvalues (a maximum) for every mass ratio μ2=m2/(m1+m2), yet L4 is observed to hold test masses (the Trojan asteroids) when μ2 is small enough. The stability condition derived for L4 in Chapter 10 places the stability boundary at μ2,crit=0.0385, and the Sun-Jupiter system has μ2=9.5×10−4. Decide whether L4 is stable there, and explain why a maximum of the effective potential does not by itself decide the answer for this system. (c) A free rigid molecule that is not linear and has no continuous internal symmetry has 3N coordinates and, by the zero-mode count for a free system, 6 zero-frequency normal modes: 3 translational and 3 rotational. State which conserved quantity, identified by Noether's theorem, is associated with the three translational zero modes, and which with the three rotational zero modes.