2.8: Multivariable First Integral

Following and generalizing the one-variable derivation, multiplying the above equations one by one by the corresponding $y_{i}^{\prime}=d y_{i} / d x$ we have the n equations

$$\dfrac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}} \dfrac{d y_{i}}{d x}-\dfrac{d}{d x}\left(\dfrac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}^{\prime}}\right) y_{i}^{\prime}=0$$

Since $f$ doesn’t depend explicitly on $x$, we have

$$\dfrac{d f}{d x}=\sum_{i=1}^{n}\left(\dfrac{\partial f}{\partial y_{i}} \dfrac{d y_{i}}{d x}+\dfrac{\partial f}{\partial y_{i}^{\prime}} \dfrac{d y_{i}^{\prime}}{d x}\right)$$

and just as for the one-variable case, these equations give

$$\dfrac{d}{d x}\left(\sum_{i=1}^{n} y_{i}^{\prime} \dfrac{\partial f}{\partial y_{i}^{\prime}}-f\right)=0$$

and the (important!) first integral $\sum_{i=1}^{n} y_{i}^{\prime} \dfrac{\partial f}{\partial y_{i}^{\prime}}-f=\mathrm{constant.}$