One-dimensional Lagrangians generated by a quadratic form

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 $t=t(s)$.