CARTAN FORMS AND SECOND VARIATION FOR CONSTRAINED VARIATIONAL PROBLEMS

Pedro L. García César Rodrigo

Department of Mathematics, University of Salamanca
37008 Salamanca, Spain
Department of Geometry and Topology, University of Seville
41012 Sevilla, Spain

Abstract

Using the Cartan form of first order constrained variational problems introduced by the authors in [3], we define the second variation. This definition coincides in the unconstrained case with the usual one in terms of the double Lie derivative of the Lagrangian density, an expression, the last, that in the constrained case does not work. The Hessian metric and other associated concepts introduced in this way are compared with those obtained through the Lagrange multiplier rule. The theory is illustrated with an example of isoperimetric problem.

Procceedings of the 7th Internacional Conference on Geometry, Integrability and Quantization (to appear). 14 Pages