The frequency map for billiards inside ellipsoids
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Cita com:
hdl:2117/9126
Tipus de documentReport de recerca
Data publicació2010-04
Condicions d'accésAccés obert
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Reconeixement-NoComercial-SenseObraDerivada 3.0 Espanya
Abstract
The billiard motion inside an ellipsoid Q Rn+1 is completely integrable. Its
phase space is a symplectic manifold of dimension 2n, which is mostly foliated with Liouville
tori of dimension n. The motion on each Liouville torus becomes just a parallel translation
with some frequency ! that varies with the torus. Besides, any billiard trajectory inside Q is
tangent to n caustics Q 1 ; : : : ;Q n, so the caustic parameters = ( 1; : : : ; n) are integrals
of the billiard map. The frequency map 7! ! is a key tool to understand the structure of
periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the
caustic parameters.
We present four conjectures, fully supported by numerical experiments. The last one gives
rise to some lower bounds on the periods. These bounds only depend on the type of the
caustics. We describe the geometric meaning, domain, and range of !. The map ! can
be continuously extended to singular values of the caustic parameters, although it becomes
“exponentially sharp” at some of them.
Finally, we study triaxial ellipsoids of R3. We compute numerically the bifurcation curves
in the parameter space on which the Liouville tori with a fixed frequency disappear. We
determine which ellipsoids have more periodic trajectories. We check that the previous lower
bounds on the periods are optimal, by displaying periodic trajectories with periods four, five,
and six whose caustics have the right types. We also give some new insights for ellipses of R2.
Forma part[prepr201006RamS]
URL repositori externhttp://www.ma1.upc.es/~casas/trabajos/Casas_RamirezRos_2010.pdf
Fitxers | Descripció | Mida | Format | Visualitza |
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Casas_RamirezRos_2010.pdf | 1,762Mb | Visualitza/Obre |