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5.6: Euler’s Integral Equation

  • Page ID
    9591
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    An integral form of the Euler differential equation can be written which is useful for cases when the function \(f\) does not depend explicitly on the independent variable \(x\), that is, when \( \frac{\partial f}{\partial x}=0.\) Note that

    \[\frac{df}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} \frac{dy}{dx}+\frac{\partial f}{\partial y^{\prime }}\frac{dy^{\prime }}{dx}\]

    But

    \[\frac{d}{dx}\left( y^{\prime }\frac{\partial f}{\partial y^{\prime }}\right) =\frac{\partial f}{\partial y^{\prime }}\frac{dy^{\prime }}{dx}+y^{\prime } \frac{d}{dx}\frac{\partial f}{\partial y^{\prime }}\]

    Combining these two equations gives

    \[\frac{d}{dx}\left( y^{\prime }\frac{\partial f}{\partial y^{\prime }}\right) =\frac{df}{dx}-\frac{\partial f}{\partial x}-y^{\prime }\frac{\partial f}{ \partial y}+y^{\prime }\frac{d}{dx}\frac{\partial f}{\partial y^{\prime }}\]

    The last two terms can be rewritten as

    \[y^{\prime }\left( \frac{d}{dx}\frac{\partial f}{\partial y^{\prime }}-\frac{ \partial f}{\partial y}\right)\]

    which vanishes when the Euler equation is satisfied. Therefore the above equation simplifies to

    \[\frac{\partial f}{\partial x}-\frac{d}{dx}\left( f-y^{\prime }\frac{\partial f}{\partial y^{\prime }}\right) =0 \label{5.24}\]

    This integral form of Euler’s equation is especially useful when \( \frac{\partial f}{\partial x}=0,\) that is, when \(f\) does not depend explicitly on the independent variable \(x\). Then the first integral of Equation \ref{5.24} is a constant, i.e.

    \[f-y^{\prime }\frac{\partial f}{\partial y^{\prime }}=\text{constant}\]

    This is Euler’s integral variational equation. Note that the shortest distance between two points, the minimum surface of rotation, and the brachistochrone, described earlier, all are examples where \(\frac{\partial f }{\partial x}=0\) and thus the integral form of Euler’s equation is useful for solving these cases.


    This page titled 5.6: Euler’s Integral Equation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.