A solution of the ‘converse canonical problem’ for multiple integrals in the calculus of variations

Carathéodory, C.,Die Methode der geodätischen Äquidistanten und das Problem von Lagrange, Acta Math.47, 199–236 (1926). Lipshitz, S. P.,A Canonical Formalism for the Non-Parametric Multiple-Integral Problem of Lagrange in the Calculus of Variations, Aequationes Math.5, 85–102 (1970). Lipshitz, S. P.,The Multiple Integral Problem of Lagrange in the Calculus of Variations (Ph.D. Thesis, University of the Witwatersrand, Johannesburg, 1970).