60 citations · 233 across the 21 of their papers we have counts for
21 papers
On the eigenvalues of a central configuration
Alain Albouy, Jiexin Sun
The equations of the Newtonian -body problem have a matrix form, where an matrix depending on the masses and on the mutual distances appears as a factor. The eig…
Does Newtonian dynamics need Euclidean space?
Alain Albouy
We present an elementary deduction of the Newtonian force from Kepler's laws. We relate it to a generalization by Jacobi of the Keplerian motion, where the Euclidean form in the pl…
Note on the attraction of an ellipsoid in a spherical universe
Alain Albouy
A classical theorem states that two confocal ellipsoids, each of them endowed with a surface distribution of mass which is called homeoidal, exert the same Newtonian force on an ex…
Bounded orbits for 3 bodies in
Alain Albouy, Holger R. Dullin
We consider the Newtonian 3-body problem in dimension 4, and fix a value of the angular momentum which is compatible with this dimension. We show that the energy function cannot te…
A limit of nonplanar 5-body central configurations is nonplanar
Alain Albouy, Antonio Carlos Fernandes
Moeckel (1990), Moeckel and Simó (1995) proved that, while continuously changing the masses, a 946-body planar central configuration bifurcates into a spatial central configuration…
How many Keplerian arcs are there between two points of spacetime?
Alain Albouy, Antonio J. Ureña
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 c…
An antimaximum principle for periodic solutions of a forced oscillator
Alain Albouy, Antonio J. Ureña
Consider the equation of the linear oscillator , where the forcing term is -periodic and positive. We show that the existence of a periodic…
Darboux inversions of the Kepler problem
Alain Albouy, Lei Zhao
While extending a famous problem asked and solved by Bertrand in 1873, Darboux found in 1877 a family of abstract surfaces of revolution, each endowed with a force function, with t…
Relative equilibria of the 3-body problem in
Alain Albouy, Holger R. Dullin
The classical equations of the Newtonian 3-body problem do not only define the familiar 3-dimensional motions. The dimension of the motion may also be 4, and cannot be higher. We p…
Some simple results about the Lambert problem
Alain Albouy, Antonio J. Urena
We give simple proofs of some simple statements concerning the Lambert problem. We first restate and reprove the known existence and uniqueness results for the Keplerian arc. We al…
Lambert's theorem and projective dynamics
Alain Albouy, Lei Zhao
We prove that the classical Lambert theorem about the elapsed time on an arc of Keplerian orbit extends without change to the Kepler problem on a space of constant curvature. We pr…
Lambert's Theorem: Geometry or Dynamics?
Alain Albouy
Lambert's theorem (1761) on the elapsed time along a Keplerian arc drew the attention of several prestigious mathematicians. In particular, they tried to give simple and transparen…
Are Palmore's "ignored estimates" on the number of planar central configurations correct?
Alain Albouy
We wish to draw attention on estimates on the number of relative equilibria in the Newtonian -body problem that Julian Palmore published in 1975.
On the force fields which are homogeneous of degree
Alain Albouy
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 b…
Some remarks about Descartes' rule of signs
Alain Albouy, Yanning Fu
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 p…
Some Problems on the Classical N-Body Problem
Alain Albouy, Hildeberto E. Cabral, Alan A. Santos
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 differ…
Relative equilibria of four identical satellites
Alain Albouy, Yanning Fu
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…
Symmetry of Planar Four-Body Convex Central Configurations
Alain Albouy, Yanning Fu, Shanzhong Sun
We 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 i…
Projective dynamics and first integrals
Alain Albouy
We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mech…
Euler configurations and quasi-polynomial systems
Alain Albouy, Yanning Fu
In the Newtonian 3-body problem, for any choice of the three masses, there are exactly three Euler configurations (also known as the three Euler points). In Helmholtz' problem of 3…
Projective dynamics and classical gravitation
Alain Albouy
Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a "field of projective forces", we define a law of dynamics…