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Alain Albouy

Publications and source records attributed to Alain Albouy.

At least 19 recordsLinked to original sources
arXiv2025

On the eigenvalues of a central configuration

The equations of the Newtonian $n$-body problem have a matrix form, where an $n\times n$ matrix depending on the masses and on the mutual distances appears as a factor. The $n$ eigenvalues of this matrix are real and nonnegative. In a motion of relative equilibrium, the configuration, called {\it central}, has constant mutual distances. The matrix is constant. We prove that in a relative equilibrium of 5 bodies the two nontrivial eigenvalues are strictly greater than the three trivial ones. This result improves published inequalities about the central configurations, which belong to two independent lines of research. One starts with Williams in 1938 and concerns constraints on the shape of the configuration. The other concerns the Hessian of the potential and its index, and applies to the linear stability of the self-similar motions and to the possible bifurcations. We also considerably clarify the very useful identities with which Williams discusses his inequalities.

math-ph↗
arXiv2025

Does Newtonian dynamics need Euclidean space?

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 plane is replaced by some function with the same homogeneity. We show how several convexity properties of the generalized Keplerian orbits appear in this context. We describe the generalized hodographs.

physics.class-ph↗
arXiv2024

Note on the attraction of an ellipsoid in a spherical universe

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 exterior point if they have the same total mass. We extend this theorem to the spherical geometry by adapting a forgotten proof by Chasles of the classical theorem. Compared to the existing results, this is a slightly stronger statement with a much shorter proof.

math-ph↗
arXiv2024

Bounded orbits for 3 bodies in $\mathbb{R}^4$

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 tend to its infimum on an unbounded sequence of states. Consequently the infimum of the energy is its minimum. This completes our previous work \cite{AD19} on the existence of Lyapunov stable relative periodic orbits in the 3-body problem in $\mathbb{R}^4$.

math.DS↗
arXiv2023

A limit of nonplanar 5-body central configurations is nonplanar

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. 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.

math-ph↗
arXiv2023

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.

math-ph↗
arXiv2022

Darboux inversions of the Kepler problem

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 the striking property that all the orbits are periodic on open sets of the phase space. We give a description of this family which explains why they have this property: they are the Darboux inverses of the Kepler problem on constant curvature surfaces. What we call the Darboux inverse was briefly introduced by Darboux in 1889 as an alternative approach to the conformal maps that Goursat had just described.

math-ph↗
arXiv2022

An antimaximum principle for periodic solutions of a forced oscillator

Consider the equation of the linear oscillator $u"+u=h(θ)$, where the forcing term $h:\mathbb R\to\mathbb R$ is $2π$-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.

math.CA↗
arXiv2020

Lambert's Theorem: Geometry or Dynamics?

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 transparent proofs of it (see our timeline §9). We give two new proofs. The first one (§4) goes along the lines of Hamilton's variational proof in his famous paper of 1834, but we shorten his computation in such a way that the hypothesis is now used without redundancy. The second (§6) is among the few which are close to Lambert's geometrical proof. It starts with the new remark that two Keplerian arcs related by the hypothesis of Lambert's theorem correspond to each other through an affine map. We also show (§7) that despite the singularities due to the occurrence of collisions, the classes of arcs related by Lambert's theorem all have the same topology. We give (§8) some simple related results about conic sections and affine transformations.

math.DS↗
arXiv2020

Relative equilibria of the 3-body problem in $\mathbb{R}^4$

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 prove that in dimension 4, for three arbitrary positive masses, and for an arbitrary value (of rank 4) of the angular momentum, the energy possesses a minimum, which corresponds to a motion of relative equilibrium which is Lyapunov stable when considered as an equilibrium of the reduced problem. The nearby motions are nonsingular and bounded for all time. We also describe the full family of relative equilibria, and show that its image by the energy-momentum map presents cusps and other interesting features.

math.DS↗
arXiv2020

Some simple results about the Lambert problem

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 also prove in some cases that the elapsed time is a convex function of natural parameters. Our statements and proofs do not distinguish between the three types of Keplerian conic section, elliptic, parabolic and hyperbolic. We also prove non-uniqueness results and non-convexity results. We do not develop any algorithm of resolution, limiting ourselves to such obviously useful a priori questions: How many solutions should we expect? Can we be sure that the Newton method will converge?

math-ph↗
arXiv2019

Lambert's theorem and projective dynamics

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 prove that the Hooke problem has a property similar to Lambert's theorem, which also extends to the spaces of constant curvature.

math-ph↗
arXiv2014

Projective dynamics and first integrals

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 mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms.

math-ph↗
arXiv2014

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

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 Knörrer's remark on the correspondence between the Neumann potential on a sphere and the geodesic motion on an ellipsoid.

math-ph↗
arXiv2014

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.

math.CA↗
arXiv2013

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.

math-ph↗
arXiv2009

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.

math-ph↗