First variation

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In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional ${\displaystyle \delta J(y)}$ mapping the function h to

${\displaystyle \delta J(y,h)=\lim _{\varepsilon \to 0}{\frac {J(y+\varepsilon h)-J(y)}{\varepsilon }}=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0},}$

where y and h are functions, and ε is a scalar.^[1] This is recognizable as the Gateaux derivative of the functional.^[1]

Compute the first variation of

${\displaystyle J(y)=\int _{a}^{b}yy'\mathrm {d} x.}$

From the definition above:

${\displaystyle {\begin{aligned}\delta J(y,h)&=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}\\&=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\ \mathrm {d} x\right|_{\varepsilon =0}\\&=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ \mathrm {d} x\right|_{\varepsilon =0}\\&=\left.\int _{a}^{b}{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ \mathrm {d} x\right|_{\varepsilon =0}\\&=\left.\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime })\ \mathrm {d} x\right|_{\varepsilon =0}\\&=\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\ \mathrm {d} x\\\end{aligned}}}$

• Calculus of variations
• Functional derivative
• Second variation

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