Abstract
In this paper we apply the classical variational calculus to the Lagrangians of particular type. Namely, we establish formulas for the 1st integrals and propose a technique for obtaining invariant 1st integrals. We also deduce the differential homogeneity conditions for Lagrangians with respect to multiplication of a path x(t) by a function c(t) and with respect to the change of the parameter t = t(s).
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References
L. N. Krivonosov, Development of the Conformal Field Theory (Nizhegorodsk. Gos. Ped. Univ., Nizhni Novgorod, 1992) [in Russian].
P. Griffiths, Exterior Differential Systems and the Calculus of Variations (Birkhauser, Boston, 1983; Mir, Moscow, 1986).
L. E. El’sgol’tz, Variational Calculus (GITTL, Moscow, 1958) [in Russian].
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Original Russian Text © V.A. Luk’yanov, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 5, pp. 33–44.
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Luk’yanov, V.A. One-dimensional Lagrangians generated by a quadratic form. Russ Math. 53, 28–37 (2009). https://doi.org/10.3103/S1066369X09050041
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DOI: https://doi.org/10.3103/S1066369X09050041