Classical Mechanics (Goldstein) · Problem 5

The Rigid Body Equations of Motion — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · liouville-equations-nonrigid-motion

Problem

Equation (5.38) holds for the motions of systems that are not rigid, relative to a chosen rotating set of coordinates. For general nonrigid motion, if the rotating axes are chosen to coincide with the (instantaneous) principal axes of the continuous system, show that Eqs. (5.39) are to be replaced by d(Iiωi)dt+ϵijkωjωkIk−ωidIidt=Ni,i=1,2,3,\frac{d(I_i \omega_i)}{dt} + \epsilon_{ijk} \omega_j \omega_k I_k - \omega_i \frac{d I_i}{dt} = N_i, \quad i = 1, 2, 3, where Ii=∫dV ρ(r) ϵijk xjvk′I_i = \int dV\, \rho(\mathbf{r})\, \epsilon_{ijk}\, x_j v'_k with ρ(r)\rho(\mathbf{r}) the mass density at point r\mathbf{r}, and v′\mathbf{v}' the velocity of the system point at r\mathbf{r} relative to the rotating axes. These equations are sometimes known as the Liouville equations and have applications for discussing almost-rigid motion, such as that of Earth including the atmosphere and oceans.

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