First variation

The first variation is a generalized directional derivative of a functional . Their properties are relevant in applied mathematics and theoretical physics . The first variation plays a central role in the calculus of variations and is used in analytical mechanics . A related concept is functional derivation .

definition

Be a function space; a functional with or ; Functions and . Then the first variation of the functional after is defined as ${\displaystyle X}$ ${\displaystyle J:X\rightarrow \mathbb {K} }$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle y,h\in X}$ ${\displaystyle \varepsilon \in \mathbb {R} }$ ${\displaystyle J}$ ${\displaystyle y}$

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

This corresponds to the Gâteaux differential of the functional at the point in direction . ${\displaystyle J}$ ${\displaystyle y}$ ${\displaystyle h}$

properties

1. The first variation is a linear map :

   ${\displaystyle \delta (F(y)+\alpha G(y))(h)=\delta F(y)(h)+\alpha \delta G(y)(h)\quad \forall \alpha \in \mathbb {K} ,\,\forall F,G\in {\mathcal {D}}^{\prime }(\Omega )}$

2. The product rule applies to a functional product: ${\displaystyle F(y)=G(y)H(y)}$

   ${\displaystyle \delta F(y)(h)=\delta G(y)(h)\ H(y)+G(y)\ \delta H(y)(h)}$

example

The first variation of

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

is according to the above definition

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

See also

• variation
• Hamilton's principle
• Euler-Lagrange equation

Web links

• Exampleproblems.com has more examples.