On homogeneous functions in second-order field theory

• Autorzy: Brajercík J. , Urban Z.

Warianty tytułu

SK

O homogenních funkcích v teorii pole druhého řádu

• Języki publikacji: EN

Abstrakty

EN

The classical concept of a homogeneous function is introduced and extended within the theory of differential groups, known in the theory of differential invariants. Invariance under reparametrizations of solutions of partial differential equations is studied. On this basis the wellknown generalizations of the Euler theorem are obtained (the Zermelo conditions). The positive homogeneity concept is then applied to second-order variational equations in field theory.

SK

Standardní koncept homogenní funkce je zaveden a zobecnen pomocí užití diferenciálních grup, známých v teorii diferenciálních invariantu. Studujeme invarianci vzhledem k reparametrizacím integrálních krivek parciálních diferenciálních rovnic. Na základe tohoto prístupu obdržíme známé zobecnení Eulerova teorému, tzv. Zermelovy podmínky. Koncept pozitivní homogenity aplikujeme na variacní rovnice druhého rádu v teorii pole.

Słowa kluczowe

EN

Lagrangian; Euler-Lagrange equation; Zermelo condition; jet; differential groups; field theory

PL

równanie Eulera-Lagrange'a; warunek Zermelo; grupy różnicowe; teoria pola

• Wydawca: STE GROUP
• Czasopismo: Systemy Wspomagania w Inżynierii Produkcji
• Rocznik: 2017
• Tom: Vol. 6, iss. 4
• Strony: 230--236
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Brajercík J.

• University of Prešov, Department of Physics, Mathematics and Techniques, 17. novembra 1, 081 16, Prešov, Slovak Republic, tel.: [phone]

autor

Urban Z.

• VŠB - Technical University of Ostrava, Department of Mathematics and Descriptive Geometry, 17. listopadu 15, 708 33, Ostrava, Czech Republic, tel.: [phone]

Bibliografia

• 1. M. Crampin, D.J. Saunders. “The Hilbert-Carathéodory form for parametric multiple integral problems in the calculus of variations”, Acta Appl. Math., Vol. 76, 2003, p. 37-55.
• 2. D. R. Grigore, D. Krupka. “Invariants of velocities and higher-order Grassmann bundles”, J. Geom. Phys., Vol. 24, 1998, p. 244-264.
• 3. M. Kawaguchi. “An Introduction to the Theory of Higher Order Spaces I: The Theory of Kawaguchi Spaces”, RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Sciences by Means of Geometry, Vol. III, (K. Kondo, ed.), Gakujutu Bunken Fukyu-Kai, Tokyo, 1962, p. 718-734.
• 4. K. Kondo. “On the physical meaning of the Zermelo conditions of Kawaguchi space”, Tensor, N.S., Vol. 14, 1963, p. 191-215.
• 5. D. Krupka and J. Janyška. Lectures on Differential Invariants, J. E. Purkyne University, Faculty of Science, Brno, Czechoslovakia, 1990, p. 193.
• 6. R. Matsyuk. “Autoparallel variational description of the free relativistic top third order dynamics”, in: Diff. Geom. Appl., Proc. Conf., Opava, Czech Republic, August 2001, Silesian University in Opava, Czech Republic, 2001, p. 447-459.
• 7. M. A. McKiernan. “Sufficiency of parameter invariance conditions in areal and higherorder Kawaguchi spaces”, Publ. Math. Debrecen, Vol. 13, 1966, p. 77-85.
• 8. Z. Urban, D. Krupka. “The Zermelo conditions and higher order homogeneous functions”, Publ. Math. Debrecen, Vol. 82, No. 1, 2013, p. 59-76.