The points of Bifurcation along the Maclaurin, the Jacobi, and the Jeans sequences

Abstract

The role which the second- and the third-order virial equations governing equilibrium can play in isolating points of neutral stability along equilibrium sequences is discussed and clarified. It is shown that a necessary condition for the occurrence of a neutral point is that a non-trivial Lagrangian displace ment exists for which the first variations of all of the integral relations (five in the second order and fifteen in the third order), provided by the virial equations, vanish. By using this condition, it is possible, for example, to isolate the point of bifurcation along the Jacobian sequence without any prior specification of the nature of the sequence which follows bifurcation, As further illustrations of the method, the known points of neutral stability along the Maclaurin and the Jeans sequences are also derived.