Editing Appendix/Ramblings/BordeauxPostDefense (section)

====Joel's Initial Response Regarding Shapes====
<font color="darkgreen">My "shape" question was in the context of especially Figures 4.6 and 4.7 (pp. 81 - 82) in Baptiste's dissertation.  I was interested to hear Baptiste elaborate on his sentence near the top of p. 83:  "The effect of binarity tends to modify the shape with respect to single body figures."  Focusing on the central "spheroid", it seems clear that a (non-differentiable?) cusp appears at the surface of the spheroid in the configuration that marks the end of the [https://ui.adsabs.harvard.edu/abs/2003MNRAS.339..515A/abstract M. Ansorg, A. Kleinw&auml;chter &amp; R. Meinel (2003)] sequence.  I presume that **just past** this critical model, as the spheroid becomes detached from the surrounding ring, the surface of the spheroid is everywhere smooth (contains no cusp).  I was curious to know (from Baptiste) whether he had closely examined this "detachment".  

(A) There must be a local ''effective-potential'' maximum in the equatorial plane; where is this maximum with respect to the location of the cusp?  And where is it located immediately after detachment; is it in the "gap", or is it located inside one of the two objects?

(B)  I was also curious to know whether the **mathematical** topological transition -- from a single, distorted object to a detached pair -- is in any way reflected in the **physics** of the system at this critical point along the model sequence.  (This is potentially a crazy question, but I was nevertheless curious how Baptiste would respond to it.)</font>

Further elaboration &hellip; Off and on over the past few decades I have given some thought to a closely related example of topological-transition.  If a binary system is to form from the "fission" of a rotating ellipsoidal (or dumb-bell-shaped) configuration, then that will also involve a transition from a figure whose surface encloses a single object and (suddenly?) to a figure whose surface encloses two disconnected objects.  As we attempt to model the time-dependent processes associated with binary fission, a discussion of topological transition is likely to be most relevant &#8212; if at all &#8212; in the case of incompressible fluids because shape evolution can be reduced to a 2D-surface (rather than 3D volume) problem.  

NOTE:  Even though I have **thought** about this issue in the past, I have not made any concrete progress.  Perhaps an approach along the following lines is relevant:  [https://www.semanticscholar.org/paper/A-hybrid-variational-level-set-approach-to-handle-Walker/7ee624ab9ffe45241cb0e5a0ce0898a6da201e7b S. W. Walker (2007)] &#8212; ''A hybrid variational-level set approach to handle topological changes''

In the context of Baptiste's axisymmetric models, the relevant technical hurdles might be reduced even further because, presumably, the shape evolution can be reduced to a 1D problem.  Someone ought to be able to write down the set of time-dependent "1D" equations (for the surface of an axisymmetric, incompressible fluid) that would allow us to follow the evolution of a spheroidal configuration as it makes a transition to a spheroid/ring system.