3.4.2 The Solberg—H0iland conditions
In order to discuss the implications of this stability criterion, we shall assume that the Eulerian changes Sp and S V can be neglected in the configuration. The hypothesis Sp = 0 is valid whenever the characteristic time of the disturbances exceeds the travel time of a sound wave in the perturbed domain. (This amounts to filtering out the p-modes of oscillation.) By virtue of Eq. (3.72), the hypothesis 8 V = 0 implies that we restrict our analysis to disturbances having many nodes, that is, perturbations with wavelengths much shorter than the star's radius.
Given these assumptions, it is immediately apparent from Eq. (3.79) that the stability of an equilibrium with respect to axially symmetric motions depends on the character of the quadratic form £ ■ M£. By virtue of Eq. (3.73), a self-gravitating system is dynamically stable with respect to short-wavelength perturbations if and only if £ ■ M£ is positive definite. Indeed, if this condition is not satisfied, it is always possible to find a Lagrangian displacement £ for which the second variation,
Jv is negative. If so, then, the total energy E fails to be an absolute minimum, thus indicating an unstable state of equilibrium.
For further use, let us define the following vectors:
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