Bernard Gaffet

We consider the class of ellipsoidal gas clouds expanding into a vacuum [1, 2] which has been shown to be a Liouville integrable Hamiltonian system [3]. This system presents several interesting features, such as the Painlevé property [4, 5], the existence of Bäcklund transformations and the separability of variables, all shown to be present in at least several sub-cases.
A remarkable result that emerged from the study of the cases of rotation around a fixed principal axis, was that the Liouville torus, which is the locus of trajectories of the representative point of the cloud when all the constants of motion are fixed, could be assimilated with a quartic surface presenting 16 conic point singularities. The geometry of such surfaces is entirely determined by the datum of a 6th degree polynomial in one variable, and the consideration of the corresponding natural coordinate system then led to the separation of variables for these cases [6]. Further, the equation of the surface takes the form of a $4 \times 4$ determinant, which constitutes a generalization of Stieltjes $4 \times 4$ determinant formulation of the addition formula for elliptic functions; and the corresponding matrix also defines the system of the equations of motion; so that it can be said that the differential system is completely determined by the surface’s geometry.
Forsaking now the assumption of a fixed rotation axis, in cases where the energy constant takes its minimum value compatible with the other constants of motion, we found that the Liouville torus was still reducible to the form of a quartic surface, presenting 15 conic points only instead of 16 (16 conic points were indeed present originally, but one of them had to disappear in the process of reducing the surface to the 4th degree). The geometry of these surfaces is entirely determined by the datum of a plane unicursal quartic (which is the transformed version of the missing conic point). The system can be reduced to the form of a differential equation of second degree, the coefficients of which are polynomials of degree 7, which are determined by the surface’s geometry, except for their quadratic dependence on a single free parameter, $z$. Defining $u$ the (time-like) independent variable, and $\Phi$ the integration constant (which are functions defined on the Liouville torus), it is found that $\Phi$ depends linearly on the parameter: $\Phi = \Phi(z)$ and then u may be taken to coincide with $\Phi(z')$, for any value of $z'$ distinct from $z$. Solving the system for one particular value of $z$ therefore also solves it for all other values of the parameter. It appears that the geometry alone does not specify in this case any particular value of $z$, but then any two values lead to differential systems which (although their solutions differ) turn out to be equivalent. It may also be worth pointing out that changing $z$ may be viewed as exchanging the roles of $u$ and $\Phi$.
Finally, in degenerate cases the Liouville torus presents a double line of self-intersection, and the separation of variables can be achieved. Sections by planes through the double line are conic sections, which may be labeled by a parameter w, say. Denoting α the eccentric anomaly on the conic, the differential system in fact takes a remarkably simple form: $da/dw = f(w)$, and involves an elliptic inteqgral.