Nous obtenons que la sémi-gerbe canonique d'un Lagrangien régulier d'ordre k est uniquement déterminée par deux un-formes Cartan–Poincaré associées. Autrement dit, la sémi-gerbe canonique est uniquement déterminée par sa structure présimplectique canonique et par une des une-formes Cartan–Poincaré. Nous prouvons que ce champ de vecteurs d'ordre est déterminée par un problème variationel pour lequel seulement la partie verticale de la courbe varie.
We show that the canonical semispray of a regular Lagrangian of order k is uniquely determined by two associated Cartan–Poincaré one-forms. Equivalently, the canonical semispray is uniquely determined by its canonical presymplectic structure and one of the Cartan–Poincaré one-forms. We prove that this order vector field is determined by a variational problem, for which only the vertical part of the curve is varied.
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Ioan Bucataru 1
@article{CRMATH_2007__345_5_269_0, author = {Ioan Bucataru}, title = {Canonical semisprays for higher order {Lagrange} spaces}, journal = {Comptes Rendus. Math\'ematique}, pages = {269--272}, publisher = {Elsevier}, volume = {345}, number = {5}, year = {2007}, doi = {10.1016/j.crma.2007.07.027}, language = {en}, }
Ioan Bucataru. Canonical semisprays for higher order Lagrange spaces. Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 269-272. doi : 10.1016/j.crma.2007.07.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.07.027/
[1] Higher order differential equations and higher order Lagrangian mechanics, Proc. Camb. Philos. Soc., Volume 99 (1986), pp. 565-587
[2] Les prolongements d'une variété différentiable. I. Calcul des jets, prolongement principal, C. R. Acad. Sci. Paris, Volume 233 (1951), pp. 598-600
[3] On the general theory of differentiable manifolds with almost tangent structure, Canad. Math. Bull., Volume 8 (1965), pp. 721-748
[4] Generalized Classical Mechanics and Field Theory, North-Holland Publishing Co., Amsterdam, 1985
[5] The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics, Kluwer Academic Publisher, 1997 (FTPH no. 82)
[6] A new look at second order equations and Lagrangian mechanics, J. Phys. A: Math. Gen., Volume 17 (1984) no. 10, pp. 1999-2009
[7] The Lagrange differential geometry, Bull. Acad. Polon. Sci., Volume 24 (1976), pp. 1089-1096
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