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 equilibrium. The first variation of such a functional will vanish 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 functional j with the requisite properties, an equation for the thirdorder virial (namely, fpi# ) is first established. And from an examination of the conditions, which follow from this equation, for equilibrium, it is found that J= [ +x2Q312+x1( 33 22) ] dx (where , 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 pearshaped 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.
 Publication:

The Astrophysical Journal
 Pub Date:
 November 1962
 DOI:
 10.1086/147457
 Bibcode:
 1962ApJ...136.1048C