140 research outputs found
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 -body problem that Julian Palmore published in 1975.Comment: 4 page
On the force fields which are homogeneous of degree
The dynamics defined by a force field which is positively homogeneous of
degree 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 . 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
We show that critical points at infinity in the 3-body problem in
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 .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
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