zbMATH — the first resource for mathematics

The regularization of the second order Lagrangians in example. (English) Zbl 1369.49070
This paper deals with a geometric formulation of the regular third order Hamiltonian systems for a given second order Lagrangian. A regularization method is illustrated on concrete examples of the Lagrangian which is quadratic in second derivatives. The arguments combine Dedecker-Hamilton extremals, Lepagean equivalents and Poincaré-Cartan forms.
49S05 Variational principles of physics
35R01 PDEs on manifolds
53Z05 Applications of differential geometry to physics
70H30 Other variational principles in mechanics
Full Text: Link
[1] Dedecker, P.: On the generalization of symplectic geometry to multiple integrals in the calculus of variations. In: Lecture Notes in Math., 570, Springer, Berlin, 1977, 395-456. | · Zbl 0352.49018
[2] Gotay, M. J.: A multisymplectic framework for classical field theory and the calculus of variations, I. covariant Hamiltonian formalism. In: M., Francaviglia, D. D., Holm (eds.): Mechanics, Analysis and Geometry: 200 Years After Lagrange, North-Holland, Amsterdam, 1990, 203-235. · Zbl 0741.70012
[3] Krupka, D.: Some geometric aspects of variational problems in fibered manifolds. Folia Fac. Sci. Nat. 14 (1973), 1-65.
[4] Krupka, D.: Lepagean forms in higher order variational theory. In: S., Benenti, M., Francaviglia, A., Lichnerowitz (eds.): Modern Developments in Analytical Mechanics I: Geometrical Dynamics, Proc. IUTAM-ISIM Symp., Accad. delle Scienze di Torino, Torino, 1983, 197-238. | · Zbl 0572.58003
[5] Krupková, O.: Hamiltonian field theory. J. Geom. Phys. 43 (2002), 93-132. | | · Zbl 1016.37033
[6] Krupková, O.: Hamiltonian field theory revisited: A geometric approach to regularity. In: Steps in Differential Geometry, Proc. of the Coll. on Diff. Geom., University of Debrecen, Debrecen, 2001, 187-207. | · Zbl 0980.35009
[7] Krupková, O., Smetanová, D.: Legendre transformation for regularizable Lagrangians in field theory. Letters in Math. Phys. 58 (2001), 189-204. | | · Zbl 1005.70025
[8] Saunders, D. J.: The Geometry of Jets Bundles. Cambridge University Press, Cambridge, 1989. · Zbl 0665.58002
[9] Smetanová, D.: On regularization of second order Hamiltonian systems. Arch. Math. 42 (2006), 341-347. · Zbl 1164.35304
[10] Smetanová, D.: On regularization of second order Lagrangians. In: Global Analysis and Applied Mathematics, American Institut of Physics, Proc. 729, Ankara, 2004, 289-296. | · Zbl 1119.35328
[11] Smetanová, D.: The second order lagrangians-regularity problem. In: 14th Conference on Applied Mathematics, APLIMAT 2015, STU Bratislava, Bratislava, 2015, 690-697.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.