On the point of bifurcation along the sequence of the Jacobi ellipsoids

Abstract

In this paper, the known point of bifurcation along the sequence of the Jacobi ellipsoids is isolated by a new method based on equilibrium considerations only. The method consists in finding an integral property (or, more generally, a functional) of the configuration which vanishes as a condition of equilibnum. The first variation of such a functional will vamsh at a point of bifurcation (and only at a point of bifurcation) for a Lagrangian displacement which deforms the body from the shape it has along an equilibrium sequence to the shape it will have in the sequence following bifurcation. For finding a func tional J with the requisite properties, an equation for the third-order virial (namely, ∫ρu_ix_ix_kdx) is first established, And from an examination of the conditions, which follow from this equation, for equilibrium, it is found that J = ∫_vρ [x3 B13+ x2Bl2+ xl (B33 - B22)] dx (where B_ij is the tensor potential of the gravitational field) has all the necessary properties The first variation of J, for the Lagrangian displacement which deforms a Jacobi ellipsoid into a pear-shaped object, is then evaluated, and it is shown that its vanishing determines the point of bifurcation along the Jacobian sequence, in agreement with Darwin's result.