The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems

• Autores: Mike Crampin, David J. Saunders
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 30, Nº 3, 2004, págs. 657-690
• Idioma: inglés
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• Resumen

  • We consider higher-order homogeneous multiple-integral variational problems defined in the context of m-frame bundles, and construct an m-form (the Hilbert-Caratheodory form) with the same extremals as the Lagrangian. We show that for second-order problems, and for problems where m = 1, this m-form is projectable to the corresponding sphere bundle. Where a second-order problem of this kind has been obtained from a problem on an affine jet bundle, the projected m-form is a generalization of the Caratheodory form used in the study of first-order problems.