2.8: Multivariable First Integral

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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

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

Since \(f\) doesn’t depend explicitly on \(x\), we have

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

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

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

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