Earlier, I had wondered what it might mean for a Saari spiderweb to be "optimal." I now have a suggestion. 1. Consider a set of N point masses in the xy plane which under Newton's laws of gravity & motion act like a rigid rotator rotating about the center of mass with axis in the z direction. If we write the potential & kinetic energies PE = -G * SUM(0<i<j<=N) m_i * m_j / |x_i - x_j| KE = (w^2 / 2) * SUM(0<i<=N) m_i * |x_i|^2 where G=Newton constant, w=constant angular velocity, m_i = positive mass of ith planet, x_i = position of ith planet in rotating reference frame (which therefore is "stationary") which is a 2-vector... Then one way to find suitable coordinates is MINIMIZE KE-PE by choice of x_1, x_2, ..., x_N. Note, this minimizes the "action." https://en.wikipedia.org/wiki/Principle_of_least_action Also note, KE is a concave-U quadratic, while -PE has spikes to +infinity but is bounded below by 0. That makes it completely obvious that a minimum of KE-PE always exists for ANY number N of planets and ANY set of positive masses. That makes it clear this is a wider class of solutions than Saari spiderwebs because those always involve exactly 2*N*K masses coming in K sets of 2*N equal masses, plus one optional central mass. Any such action-min always is an exact rigid rotator solution of Newton laws because grad(PE) = grad(KE) necessarily happens at any such min, and this causes negated gravitational forces = centrifugal forces to precisely balance so that each body feels zero net force in the rotating frame. The global min is presumably "more optimum" than nonglobal mins and also more optimal than non-min flat spots -- in the sense that it presumably corresponds to less-unstable motion. 2. Now, about stability. If an infinitesimal perturbation of the bodies exists which would decrease the action, then the rigid rotator motion is unstable. But such a perturbation is always available: perturb the x_i almost arbitrarily in the third (z) dimension. This leaves KE unaltered (also leaving the momentum and angular momentum and kinetic energy of each mass unaltered) but increases PE, thus decreasing the action KE-PE per unit time. Also true if we restrict attention to the codimension-1 subclass of these perturbations which leave the center of mass unaltered. Now you might complain that this perturbation alters the total energy KE+PE, but we could if desired rescale all the coordinates by a factor s infinitesimally close to 1 [since KE is proportional to s^2 while PE is proportional to -1/s] to restore the old value of KE+PE while still decreasing KE-PE... we also can simultaneously adjust w to keep the angular momentum unaltered... So in view of these possibilities, it would seem that the rigid-rotator motion for ALL of these configurations is ALWAYS unstable if N>=4. When N=1 or 2 or perhaps 3 you might be able (and presumably are able when N=1 and 2) to argue we have stability because the s and w counter-adjustments sort of reduce the dimension of the space of the perturbations I described, to zero. But you can't hide from instability in that way if N is large enough, and N>=4 seems large enough. Wonderful result, except there is something wrong with that analysis by me: Moeckel showed (correcting J.C.Maxwell 1885) that N planets of equal small-enough masses co-orbiting a unit-mass sun as a rigidly rotating regular N-gon, are stable if N>=7. Richard MOECKEL: Linear stability analysis of some symmetrical classes of relative equilibria, in Hamiltonian Dynamical Systems: History, Theory and Applications, IMA vol.63, Springer Verlag 1995. http://link.springer.com/chapter/10.1007/978-1-4613-8448-9_20 To get stability, the sun mass needs to grow at least like 0.435*N^3 for unit mass planets, and N>6 is necessary: Gareth E. Roberts: Linear stability of the n+1-gon relative equilibrium, Hamiltonian Syst. Celest. Mech., World Scient. Monogr. Ser. Math. 6 (2002) 303-330. http://mathcs.holycross.edu/~groberts/Papers/HAMSYS-98.pdf Richard Moeckel: Linear stability of relative equilibria with a dominant mass Journal of Dynamics and Differential Equations 6,1 (January 1994) 37-51 gives conditions under which the rigid-rotators consisting of N+1 masses in a limit when there is 1 mass much larger than all the others, will be stable. I think my problem was that "being able to decrease the action" is NOT the same thing as "instability" even though some sources act as though it is the same thing. Really, you need to prove all eigenvalues of the time-evolution operator have non-positive real part, to show (a commonly accepted kind of) "stability." GE Roberts: Spectral instability of relative equilibria in the planar n-body problem. Nonlinearity 12 (1999) 757-769. http://mathcs.holycross.edu/~groberts/Papers/EqualMass.pdf showed that if N>24305 then any rigid rotator Newton solution made of N EQUAL-mass stars is spectrally unstable. (And he conjectures N>2 suffices.) That seems to destroy any hope these configurations could be reasonable models of any kind of galaxy. This analysis is purely in the plane, no 3D perturbations are considered. If I understand Roberts EQ8 correctly, then for a rigid-rotator N-body Newton solution, he constructs the 4Nx4N matrix [ wK Minv ] [ S wK ] where each block is 2Nx2N, K=BlockDiag(J,J,...J) and J is the 2x2 matrix [ 0 1] [-1 0] and M is the diagonal matrix with entries m1,m1, m2,m2, m3, m3, ..., mN,mN and Minv is its inverse, and finally S is the symmetric Hessian matrix of the potential energy function of x1,y1,x2,y2,x3,y3,...,xN,yN. He then claims we have spectral stability in 2D if and only if the eigenvalues of this 4Nx4N matrix, aside from the 8 eigenvalues 0,0,,w*i,w*i,w*i,-w*i,-w*i,-w*i all have zero real part. Spectral stability in 2D is a necessary condition for stronger forms of stability. 3. Now, about the 3-body case N=3. Euler realized that collinear solutions exist L. EULER: De moto rectilineo trium corporum se mutuo attahentium, Novi Comm. Acad. Sci. Imp. Petrop., 11 (1767) 144- (and the body-locations arise from solving a quintic) but it has been shown (Andoyer 1906, Meyer 1933, Conley) those solutions are always unstable. Lagrange proved that, for 3 generic masses, the only non-collinear solutions involve the 3 masses lying at the vertices of an EQUILATERAL triangle. It is a quite remarkable theorem of geometry that no matter what the 3 masses are, an equilateral triangle always yields a force balance. (For three non-generic masses such as m1=m2, further solutions can exist, such as certain isoceles triangles.) Gascheau then proved Lagrange's solution is STABLE if 0 < (m1*m2+m2*m3+m3*m1)/(m1+m2+m3)^2 < 1/27. For example, if one of the masses is large and the other two small, we get stability. J. L. LAGRANGE: Essai sur le probleme de trois corps, Oeuvres, 6 (1772) 229-. G. GASCHEAU: Examen d’une classe d’equations differential et applications a un cas particulier du probleme des trois corps, Comp. Rendus. Acad. Sci, 16 (1843) 393-. (E.J.Routh 1875 also proved stability and solved the question for general power-law forces, not just inverse-square law.) 4. It turns out I was not the first to suggest considering "post Newtonian" relativistic corrections. Kei Yamada, Hideki Asada: Triangular solution to general relativistic three-body problem for general masses Phys.Rev.D 86 (2012) 124029 http://arxiv.org/abs/1212.0754 apparently Yamada can show that perturbations of the Lagrange equilateral triangle solution to Newton laws, exist which are exact solutions of the first-order post-Newtonian more-relativistic laws of motion; and that these are stable if a perturbed version of the Gascheau condition is satisfied. 5. It occurs to me Saari's spiderwebs can be generalized to allow the concentric N-gons to be dovetailed if desired; and to allow N to be odd if desired.