The angular momentum of a relative equilibrium

There are two main reasons why relative equilibria of N point masses under the influence of Newton attraction are mathematically more interesting to study when space dimension is at least 4: On the one hand, in a higher dimensional space, a relative equilibrium is determined not only by the initial configuration but also by the choice of a complex structure on the space where the motion takes place; in particular, its angular momentum depends on this choice; On the other hand, relative equilibria are not necessarily periodic: if the configuration is "balanced" but not central, the motion is in general quasi-periodic. In this exploratory paper we address the following question, which touches both aspects: what are the possible frequencies of the angular momentum of a given central (or balanced) configuration and at what values of these frequencies bifurcations from periodic to quasi-periodic relative equilibria do occur ? We give a full answer for relative equilibrium motions in dimension 4 and conjecture that an analogous situation holds true for higher dimensions. A refinement of Horn's problem given by Fomin, Fulton, Li and Poon plays an important role. P.S. The conjecture is now proved (see Alain Chenciner and Hugo Jimenez Perez, Angular momentum and Horn's problem, arXiv:1110.5030v1 [math.DS]).Comment: 17 pages, 3 figure

Non-avoided crossings for n-body balanced configurations in R^3 near a central configuration

The balanced configurations are those n-body configurations which admit a relative equilibrium motion in a Euclidean space E of high enough dimension 2p. They are characterized by the commutation of two symmetric endomorphisms of the (n-1)-dimensional Euclidean space of codispositions, the intrinsic inertia endomorphism B which encodes the shape and the Wintner-Conley endomorphism A which encodes the forces. In general, p is the dimension d of the configuration, which is also the rank of B. Lowering to 2(d-1) the dimension of E occurs when the restriction of A to the (invariant) image of B possesses a double eigenvalue. It is shown that, while in the space of all dxd-symmetric endomorphisms, having a double eigenvalue is a condition of codimension 2 (the avoided crossings of physicists), here it becomes of codimension 1 provided some condition (H) is satisfied. As the condition is always satisfied for configurations of the maximal dimension (i.e. if d=n-1), this implies in particular the existence, in the neighborhood of the regular tetrahedron configuration of 4 bodies with no three of the masses equal, of exactly 3 families of balanced configurations which admit relative equilibrium motion in a four dimensional space.Comment: 35 pages, 1 diagram, 6 figures Section 1.5.2 is new: it introduces the condition (H) which had been overlooked in the first versio

A remarkable periodic solution of the three-body problem in the case of equal masses

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a fixed eight-shaped curve. Setting aside collinear motions, the only other known motion along a fixed curve in the inertial plane is the ``Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every ``Euler configuration" in which one of the bodies sits at the midpoint of the segment defined by the other two (Figure 1). Numerical computations by Carles Sim\'o, to be published elsewhere, indicate that the orbit is ``stable" (i.e. completely elliptic with torsion). Moreover, they show that the moment of inertia I(t) with respect to the center of mass and the potential U(t) as functions of time are almost constant.Comment: 21 pages, published versio

Between two moments

In this short note, we draw attention to a relation between two Horn polytopes which is proved in [Chenciner-Jim\'enez P\'erez] as the result on the one side of a deep combinatorial result in [Fomin,Fulton, Li,Poon], on the other side of a simple computation involving complex structures. This suggested an inequality between Littlewood-Richardson coefficients which we prove using the symmetric characterization of these coefficients given in [Carr\'e,Leclerc].Comment: 9 pages, 3 figure

Morse 2-jet space and h-principle

A section in the 2-jet space of Morse functions is not always homotopic to a holonomic section. We give a necessary condition for being the case and we discuss the sufficiency

The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium

22 pages, 2 figuresInternational audienceFrom a normal form analysis near the Lagrange equilateral relative equilibrium, we deduce that, up to the action of similarities and time shifts, the only relative periodic solutions which bifurcate from this solution are the (planar) homographic family and the (spatial) P12 family with its twelfth-order symmetry. After reduction by the rotation symmetry of the Lagrange solution and restriction to a center manifold, our proof of the local existence and uniqueness of P12 follows that of Hill's orbits in the planar circular restricted three-body problem. Indeed, near the Lagrange solution, the restrictions of constant energy levels of the reduced flow to a center manifold (actually unique) turn out to be three-spheres. In an annulus of section bounded by relative periodic solutions of each family, the normal resonance along the homographic family entails that the Poincaré return map is the identity on the corresponding connected component of the boundary. Using the reflexion symmetry with respect to the plane of the relative equilibrium, we prove that, close enough to the Lagrange solution, the return map is a monotone twist map

Éloge de Poincaré

Henri Poincaré (photo A. Gerschel et fils, Livre du Centenaire de l'École polytechnique, 1894, Collections EP) Cimetière du Montparnasse, 9 juillet 2012 Mesdames, Messieurs, membres de la famille d’Henri Poincaré, représentants d’institutions ou de sociétés savantes, mathématiciens, philosophes, physiciens, journalistes, passants, curieux, poètes… C’est un homme encore jeune, 58 ans, un homme en pleine possession de son génie, je le vois, assis à sa table de travail, une main posée sur le re..

Éloge de Poincaré

Henri Poincaré (photo A. Gerschel et fils, Livre du Centenaire de l'École polytechnique, 1894, Collections EP) Cimetière du Montparnasse, 9 juillet 2012 Mesdames, Messieurs, membres de la famille d’Henri Poincaré, représentants d’institutions ou de sociétés savantes, mathématiciens, philosophes, physiciens, journalistes, passants, curieux, poètes… C’est un homme encore jeune, 58 ans, un homme en pleine possession de son génie, je le vois, assis à sa table de travail, une main posée sur le re..