Euler-Lagrange equation

The Euler-Lagrange equation stipulates that functionals in the following form:

$$t(x,y,y^{\prime })={\int }_a^{b}I(x,y,y^{\prime })dx$$

have stationary paths if and only if:

$$\frac{d}{dx}\left(\frac{\partial I}{\partial y^{\prime }}\right)-\frac{\partial I}{\partial y}=0$$

For special cases of the equation, we have the Beltrami identity if the functional does not rely on $x$ and only on $y$ and $y^{\prime }$:

$$I-y^{\prime }\frac{\partial I}{\partial y^{\prime }}=C$$