Spinning gas clouds with precession: a new formulation

Abstract

We consider Dyson's model (Dyson F J 1968 J. Math. Mech. 18 91) of an ellipsoidally stratified ideal gas cloud expanding adiabatically into a vacuum, in the Liouville integrable case where the gas is monatomic (γ = 5/3) and there is no vorticity (Gaffet B 2001a J. Phys. A: Math. Gen. 34 2097; Paper I). In the cases of rotation about a fixed axis the separation of variables can be achieved, and the separable variables are linearly related to a set of three variables denoted by ρ, R, W (Gaffet B 2001b J. Phys. A: Math. Gen. 34 9195; Paper II). We show in the present work that these variables admit a natural generalization to cases of rotation about a movable axis (precessing motion). The present study is restricted to the consideration of the so-called degenerate cases (see Gaffet B 2006 J. Phys. A: Math. Gen. 39 99; Paper III), but we hope to generalize our results in the future to the non-degenerate ones as well. We also present a new, compact and generally valid formulation of one of the integrals of motion, of the sixth degree in the momenta, denoted by I6.