Geodesic
One-dimensional Lagrangians generated by a~quadratic form
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2009), pp. 33-44

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In this paper we apply the classical variational calculus to the Lagrangians of particular type. Namely, we establish formulas for the 1$^\mathrm{st}$ integrals and propose a technique for obtaining invariant 1$^\mathrm{st}$ integrals. We also deduce the differential homogeneity conditions for Lagrangians with respect to multiplication of the path $x(t)$ by the function $c(t)$ and with respect to the change of the parameter $t=t(s)$.
Keywords: conformally connected manifold, integral functionals, isotropic extreme curve, one-dimensional Lagrangians, path, quadratic form
Mots-clés : Euler–Lagrange equation, orthogonal group, scalar product. 
@article{IVM_2009_5_a3,
     author = {V. A. Luk'yanov},
     title = {One-dimensional {Lagrangians} generated by a~quadratic form},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {33--44},
     publisher = {mathdoc},
     number = {5},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_5_a3/}
}
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V. A. Luk'yanov. One-dimensional Lagrangians generated by a~quadratic form. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2009), pp. 33-44. http://geodesic.mathdoc.fr/item/IVM_2009_5_a3/