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 st integrals and propose a technique for obtaining invariant 1 st 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 ).