Differential-geometric structure associated with Lagrangian and its dynamic interpretation

The paper is devoted to investigation of differential-geometric structure associated with Lagrangian $L$ depending of $n$ functions of one variable $t$ and their derivatives by means of Cartan–Laptev method. We construct a fundamental object of a structure associated with Lagrangian. We also construct a covector $E_i$ $(i=1,\dotsc,n)$ embraced by prolonged fundamental object so that the system of equalities $E_i=0$ is an invariant representation of the Euler equations for the variational functional. Due to this, there is no necessity to connect Euler equations with the variational problem. Moreover, we distinguish by invariant means the class of special Lagrangians generating connection in the bundle of centroaffine structure over the base $M$. In case when Lagrangian $L$ is special, there exist a relative invariant $\Pi$ defined on $M$ which generates covector field on $M$ and fibered metric in the bundle of centroaffine structure over the base $M$.