Variational equations on the Möbius strip

• Autorzy: Urban Z. , Volná J.

Warianty tytułu

CS

Variační rovnice na Möbiově pásce

• Języki publikacji: EN

Abstrakty

EN

In this paper, systems of second-order ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by the canonical embedding of the two-dimensional Möbius strip into the Euclidean space, are considered in the class of variational equations. For a given non-variational system, the conditions assuring variationality (Helmholtz conditions) for the induced system on the Möbius strip are formulated. The theory contributes to variational foundations of geometric mechanics.

CS

V tomto clánku je studována variacnost systému obycejných diferenciálních rovnic (dynamických forem v geometrické mechanice) druhého rádu, kterou indukuje kanonické vložení dvojrozmerné Möbiovy pásky do Euklidova prostoru. Pro daný nevariacní systém rovnic jsou formulovány nutné a postacující podmínky variacnosti (Helmholtzovy podmínky). Práce je príspevkem k variacním základum geometrické mechaniky na Möbiove pásce.

Słowa kluczowe

EN

Lagrangian; Euler-Lagrange equation; Helmholtz conditions; fibered manifold; Möbius strip

PL

równanie Eulera-Lagrange'a; warunki Helmholtza; wstęga Möbiusa

• Wydawca: STE GROUP
• Czasopismo: Systemy Wspomagania w Inżynierii Produkcji
• Rocznik: 2017
• Tom: Vol. 6, iss. 4
• Strony: 325--333
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Urban Z.

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

autor

Volná J.

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

Bibliografia

• 1. D. Krupka. “On the local structure of the Euler–Lagrange mapping of the calculus of variations”, in: Proc. Conf. Diff. Geom. Appl., Charles University, Prague, 1981, p. 181–188; arXiv:math-ph/0203034.
• 2. D.Krupka. “Variational sequences in mechanics”, Calc. Var., Vol. 5, 1997, p. 557–583.
• 3. D. Krupka, D. Saunders (Eds.). Handbook of Global Analysis, Elsevier, Amsterdam, 2008.
• 4. D. Krupka, Z. Urban, J. Volná. “Variational submanifolds of Euclidean spaces”, submitted, 2017.
• 5. J.M. Lee. Introduction to Smooth Manifolds, 2nd Edition, Springer-Verlag, New York, 2012.
• 6. W. Sarlet. “The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics”, J. Phys. A: Math. Gen., Vol. 15, 1982, p. 1503–1517.
• 7. F.Takens. “A global version of the inverse problem of the calculus of variations”, J. Diff. Geom., Vol. 14, 1979, p. 543–562.
• 8. J. Volná, Z. Urban. “First-order Variational Sequences in Field Theory”, in: D. Zenkov (Ed.), The Inverse Problem of the Calculus of Variations, Local and Global Theory, Atlantis Press, Amsterdam–Beijing–Paris, 2015, p. 215–284.
• 9. F.W. Warner. Foundations of Differentiable Manifolds and Lie Groups, 2nd Ed., Springer, New York, 1983.