The First Birkhoff Coefficient and the Stability of 2-Periodic Orbits on   Billiards
  In this work we address the question of proving the stability of elliptic 2-periodic orbits for strictly convex billiards. Eventhough it is part of a widely accepted belief that ellipticity implies stability, classical theorems show that the certainty of stability relies upon more fine conditions. We present a review of the main results and general theorems and describe the procedure to fullfill the supplementary conditions for strictly convex billiards. 
Sylvie Oliffson Kamphorst, Sonia Pinto de Carvalho (Departamento de   Matematica UFMG, Brasil)