Are Palmore's "ignored estimates" on the number of planar central configurations correct?

We wish to draw attention on estimates on the number of relative equilibria in the Newtonian $n$-body problem that Julian Palmore published in 1975.Comment: 4 page

On the force fields which are homogeneous of degree −3-3

The dynamics defined by a force field which is positively homogeneous of degree $-3$ can always be reduced, by simply constraining it. The dimension of the phase space is reduced by two dimensions, while it may only be reduced by one dimension if the degree of homogeneity is different from $-3$. This remark is an elegant foundation of Appell's projective dynamics. We show how it relates to Kn\"orrer's remark on the correspondence between the Neumann potential on a sphere and the geodesic motion on an ellipsoid.Comment: 6 pages, 1 figur

Relative equilibria of four identical satellites

We consider the Newtonian 5-body problem in the plane, where 4 bodies have the same mass m, which is small compared to the mass M of the remaining body. We consider the (normalized) relative equilibria in this system, and follow them to the limit when m/M -> 0. In some cases two small bodies will coalesce at the limit. We call the other equilibria the relative equilibria of four separate identical satellites. We prove rigorously that there are only three such equilibria, all already known after the numerical researches in [SaY]. Our main contribution is to prove that any equilibrium configuration possesses a symmetry, a statement indicated in [CLO2] as the missing key to proving that there is no other equilibrium.Comment: 16 pages, 2 figure

Some remarks about Descartes' rule of signs

What can we deduce about the roots of a real polynomial in one variable by simply considering the signs of its coefficients? On one hand, we give a complete answer concerning the positive roots, by proposing a statement of Descartes' rule of signs which strengthens the available ones while remaining as elementary and concise as the original. On the other hand, we provide new kinds of restrictions on the combined numbers of positive and negative roots.Comment: 10 page

Some Problems on the Classical N-Body Problem

Our idea is to imitate Smale's list of problems, in a restricted domain of mathematical aspects of Celestial Mechanics. All the problems are on the n-body problem, some with different homogeneity of the potential, addressing many aspects such as central configurations, stability of relative equilibrium, singularities, integral manifolds, etc. Following Steve Smale in his list, the criteria for our selection are: (1) Simple statement. Also preferably mathematically precise, and best even with a yes or no answer. (2) Personal acquaintance with the problem, having found it not easy. (3) A belief that the question, its solution, partial results or even attempts at its solution are likely to have great importance for the development of the mathematical aspects of Celestial Mechanics.Comment: 10 pages, list of mathematical problem

Critical points at infinity of the 3-body Problem in R4\mathbb{R}^4

We show that critical points at infinity in the 3-body problem in $\mathbb{R}^4$ do not realize the infimum of the energy. This completes our previous work [Journal of Geometric Mechanics, 12, pp323-341 (2020), doi:10.3934/jgm.2020012] on the existence of Lyapunov stable relative periodic orbits in the 3-body problem in $\mathbb{R}^4$.Comment: 6 pages, 1 figur

How many Keplerian arcs are there between two points of spacetime?

We consider the Keplerian arcs around a fixed Newtonian center joining two prescribed distinct positions in a prescribed flight time. We prove that, putting aside the "opposition case" where infinitely many planes of motion are possible, there are at most two such arcs of each "type". There is a bilinear quantity that we call b which is in all the cases a good parameter for the Keplerian arcs joining two distinct positions. The flight time satisfies a "variational" differential equation in b, and is a convex function of b.Comment: 16 pages, 3 figure

A limit of nonplanar 5-body central configurations is nonplanar

Moeckel (1990), Moeckel and Sim\'o (1995) proved that, while continuously changing the masses, a 946-body planar central configuration bifurcates into a spatial central configuration. We show that this kind of bifurcation does not occur with 5 bodies. Question 17 in the list Albouy & al (2012) is thus answered negatively.Comment: 17 pages, 5 figure

Symmetry of Planar Four-Body Convex Central Configurations

International audienceWe study the relationship between the masses and the geometric properties of central configurations. We prove that in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles

An antimaximum principle for periodic solutions of a forced oscillator

Consider the equation of the linear oscillator u '' + u = h(theta), where the forcing term h : R -> R is 2 pi-periodic and positive. We show that the existence of a periodic solution implies the existence of a positive solution. To this aim we establish connections between this problem and some separation questions of convex analysis.Paris Observatory gran