Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Lagrangian; Euler-Lagrange form; dynamical form; Helmholtz-type form; Helmholtz form; Helmholtz conditions
Lagrangian; Euler-Lagrange form; dynamical form; Helmholtz-type form; Helmholtz form; Helmholtz conditions
Summary:
Helmholtz conditions in the calculus of variations are necessary and sufficient conditions for a system of differential equations to be variational ‘as it stands’. It is known that this property geometrically means that the dynamical form representing the equations can be completed to a closed form. We study an analogous property for differential forms of degree 3, so-called Helmholtz-type forms in mechanics ($n=1$), and obtain a generalization of Helmholtz conditions to this case.
References:
[1] Anderson, I.: The Variational Bicomplex. (Technical Report, Utah State University, 1989)
[2] Crampin, M., Prince, G. E., Thompson, G.: A geometric version of the Helmholtz conditions in time dependent Lagrangian dynamics. J. Phys. A: Math. Gen. 17 (1984) 1437–1447 DOI 10.1088/0305-4470/17/7/011 | MR 0748776
[3] Dedecker, P., Tulczyjew, W. M.: Spectral sequences and the inverse problem of the calculus of variation. Proc. Internat. Coll. on Diff. Geom. Methods in Math. Phys., Salamanca 1979, In: Lecture Notes in Math. 836 (Berlin: Springer, 1980) 498–503 MR 0607719
[4] Helmholtz, H.: Über der physikalische Bedeutung des Princips der kleinsten Wirkung. J. Reine Angew. Math. 100 (1887) 137–166
[5] Klapka, L.: Euler-Lagrange expressions and closed two-forms in higher order mechanics. In: Geometrical Methods in Physics, Proc. Conf. on Diff. Geom. and Appl. Vol. 2, Nové Město na Moravě, 1983, Krupka, D., Ed. (J. E. Purkyně Univ. Brno, Czechoslovakia, 1984) 149–153 MR 0793205 | Zbl 0552.70011
[6] Krupka, D.: Lepagean forms in higher order variational theory. In: Modern Developments in Analytical Mechanics I: Geometrical Dynamics, Proc. IUTAM-ISIMM Symposium, Torino, Italy, 1982 (Accad. Sci. Torino, Torino, 1983) 197–238 MR 0773488 | Zbl 0572.58003
[7] Krupka, D.: Some Geometric Aspects of Variational Problems in Fibered Manifolds. Folia Fac. Sci. Nat. Univ. Purk. Brunensis, Physica 14, Brno, Czechoslovakia, 1973; ArXiv:math-ph/0110005
[8] Krupka, D.: Variational Sequence on Finite Order Jet Spaces. In: Differential Geometry and its Applications, Proc. Conf., Brno, Czechoslovakia, 1989, Janyška, J. and Krupka, D., Eds. (World Scientific, Singapore, 1990) 236–254 MR 1062026
[9] Krupka, D.: Variational Sequences in Mechanics. Calc. Var. 5, 557–583(1997) DOI 10.1007/s005260050079 | MR 1473308 | Zbl 0892.58001
[10] Krupka, D.: Global variational principles: Foundations and current problems. In: Global Analysis and Applied Mathematics (AIP Conference Proceedings 729, American Institute of Physics, 2004) 3–18 MR 2215681 | Zbl 1121.58019
[11] Krupka, D., Šeděnková, J.: Variational Sequences and Lepage Forms. In: Proceedings of Conference Differential Geometry and its Applications, Prague, 2004, Ed. by Bureš, J., Kowalski, O., Krupka, D., Slovák, J. (Charles Univ., Prague, Czech Republic, 2005), pp. 605–615 Zbl 1115.35349
[12] Krupka, D., Krupková, O., Prince, G., Sarlet, W.: Contact symmetries of the Helmholtz form. Differential Geometry and its Applications 25 (2007) 518–542 MR 2351428
[13] Krupková, O.: Lepagean $2$-Forms in Higher Order Hamiltonian Mechanics, I. Regularity. Arch. Math. (Brno) 22 (1986) 97–120 MR 0868124
[14] Krupková, O.: The Geometry of Ordinary Variational Equations. Lecture Notes in Math. 1678 (Springer, Berlin, 1997) MR 1484970
[15] Krupková, O., Prince, G.E.: Lepage Forms, Closed $2$-Forms and Second-Order Ordinary Differential Equations. Russian Mathematics (Iz. VUZ), 2007, Vol. 51, No. 12, pp. 1–16 MR 2402204
[16] Krupková, O., Prince, G.E.: Second Order Ordinary Differential Equations in Jet Bundles and the Inverse Problem of the Calculus of Variations. In: Handbook of Global Analysis (Elsevier, 2008) 841–908 MR 2389647 | Zbl 1236.58027
[17] Lepage, Th.: Sur les champs géodésiques du Calcul des Variations. Bull. Acad. Roy. Belg., Cl. des Sciences 22 (1936) 716–729 Zbl 0016.26201
[18] Saunders, D. J.: The Geometry of Jet Bundles. (Cambridge University Press, 1989) MR 0989588 | Zbl 0665.58002
[19] Takens, F.: A Global Version of the Inverse Problem of the Calculus of Variations. J. Diff. Geom. 14, 543–562(1979) MR 0600611 | Zbl 0463.58015
Search
Partner of
