Martedi' 20 dicembre alle ore 14.00 presso la sala conferenze del Centro di Ricerca Matematica Ennio de GIorgi, Collegio Puteano, si terra' un seminario di sistemi dinamici:
Dr. Martin CELLI (S.N.S.)
"The N-body problem, its perverse choreographies, and some properties of systems with vanishing total mass"
ABSTRACT "Newton's equations describe the motion of N punctual particles which interact through gravitation. For some solutions, which are called choreographies, the N bodies chase each other on the same curve with the same phase shifts between two bodies. The first non trivial choreography (the "eight" orbit) was found numerically by C. Moore in 1993, and its existence was proved by A. Chenciner and R. Montgomery in 1999, thanks to considerations on symmetries.
Since this discovery, many choreographies have been found numerically (C. Simó) or analytically (S. Terracini, A. Venturelli, D. Ferrario, K.-C. Chen), thanks to a trick due to C. Marchal. All the choreographies that are actually known require equal masses. Indeed, it has been proved that in the plane, there was no choreography with distinct masses or "perverse" choreography with number of bodies N lower than or equal to five (A. Chenciner, 2000). This happens to be true for any N if we replace the Newtonian potential by a logarithmic potential (M. Celli, 2002). This can be applied to N-vortex systems, which are used to describe planar fluids.
The proof of the latter result is based on properties of the equilibria of N-body systems with vanishing total mass. For systems with vanishing total mass, some strange phenomena may occur. For instance, in the two-body problem, the motion of the vector defined by the two bodies is uniform rectilinear. As another example, in the three-body problem, for any non collinear configuration, we can find a system of masses whose sum vanishes and initial velocities which generate a rigid motion (the distances between the bodies are constant) with dimension three, whereas it can be shown that with positive masses, a rigid motion is always planar.
These properties are due to the fact that, with vanishing total mass, the time-dependent first integral of the center of inertia becomes a vector, which is invariant under translations. Thanks to this, the collinear three-body problem is integrable under an assumption on the velocities.
This property also enables to compute the four-body central configurations for masses x, -x, y, -y (M. Celli, 2005). The study of central configurations (which generate homothetic motions) is a difficult problem in celestial mechanics. Their finiteness for positive masses is the subject of S. Smale's sixth problem for the century, which was recently solved for N=4 (M. Hampton, R. Moeckel, 2004) thanks to a computer assisted proof. The only other known case where four-body central configurations can be computed is the case with equal masses (A. Albouy, 1996)."
Stefano Marmi Scuola Normale Superiore fax (39)050563513 Piazza dei Cavalieri 7 phone (39)050509064 56126 Pisa email: s.marmi@sns.it Italy
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