Alain CHENCINER
Astronomie et Systèmes Dynamiques, IMCCE e Departement de Mathematiques, Universite Paris VII
The shape of a triangle

Abstract: Heron's formula, maybe due to Archimedes, expresses the squared area of a triangle as a polynomial in the squared lengths of its sides. A true understanding of this formula and its generalizations to more bodies, comes from an adequate coding of the shape defined by n points in an euclidean space up to translations, rotations and reflexions. The resulting object is a quadratic form which, when masses are added to the picture, turns into an intrinsic (i.e. invariant under isometries) inertia form of the solid body defined by the n punctual masses. If time allows, applications to the n-body problem will be given : for example, Betti's reduced equations of the n-body problem (which generalize Lagrange's ones for 3 bodies) become more tractable in this setting.