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* Page found: Euler-Lagrange-Gleichungen (eq math.1118.14)

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\begin{align}
  & \delta S\left[ q \right]=S\left[ {{q}_{0}} \right]-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{L\left( q+\delta q,\dot{q}+\delta \dot{q},t \right)dt} \\
 & =S\left[ {{q}_{0}} \right]-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( \underbrace{L}_{=S\left[ {{q}_{0}} \right]}+{{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt} \\
 & =-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left( {{\partial }_{q}}L\delta q+{{\partial }_{{\dot{q}}}}L\delta \dot{q} \right)dt}
\end{align}

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δ ⁢ S ⁢ [ q ] = S ⁢ [ q 0 ] - ∫ t 1 t 2 L ⁢ ( q + δ ⁢ q , q ˙ + δ ⁢ q ˙ , t ) ⁢ 𝑑 t = S ⁢ [ q 0 ] - ∫ t 1 t 2 ( L ⏟ = S ⁢ [ q 0 ] + ∂ q ⁡ L ⁢ δ ⁢ q + ∂ q ˙ ⁡ L ⁢ δ ⁢ q ˙ ) ⁢ 𝑑 t = - ∫ t 1 t 2 ( ∂ q ⁡ L ⁢ δ ⁢ q + ∂ q ˙ ⁡ L ⁢ δ ⁢ q ˙ ) ⁢ 𝑑 t missing-subexpression 𝛿 𝑆 delimited-[] 𝑞 𝑆 delimited-[] subscript 𝑞 0 superscript subscript subscript 𝑡 1 subscript 𝑡 2 𝐿 𝑞 𝛿 𝑞 ˙ 𝑞 𝛿 ˙ 𝑞 𝑡 differential-d 𝑡 missing-subexpression absent 𝑆 delimited-[] subscript 𝑞 0 superscript subscript subscript 𝑡 1 subscript 𝑡 2 subscript ⏟ 𝐿 absent 𝑆 delimited-[] subscript 𝑞 0 subscript 𝑞 𝐿 𝛿 𝑞 subscript ˙ 𝑞 𝐿 𝛿 ˙ 𝑞 differential-d 𝑡 missing-subexpression absent superscript subscript subscript 𝑡 1 subscript 𝑡 2 subscript 𝑞 𝐿 𝛿 𝑞 subscript ˙ 𝑞 𝐿 𝛿 ˙ 𝑞 differential-d 𝑡 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\delta S\left[q% \right]=S\left[{{q}_{0}}\right]-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{L\left(q+% \delta q,\dot{q}+\delta\dot{q},t\right)dt}\\ &\displaystyle=S\left[{{q}_{0}}\right]-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{% \left(\underbrace{L}_{=S\left[{{q}_{0}}\right]}+{{\partial}_{q}}L\delta q+{{% \partial}_{{\dot{q}}}}L\delta\dot{q}\right)dt}\\ &\displaystyle=-\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left({{\partial}_{q}}L% \delta q+{{\partial}_{{\dot{q}}}}L\delta\dot{q}\right)dt}\end{aligned}}}

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δS[q]=S[q0]−∫t1t2L(q+δq,q˙+δq˙,t)dt=S[q0]−∫t1t2(L⏟=S[q0]+∂qLδq+∂q˙Lδq˙)dt=−∫t1t2(∂qLδq+∂q˙Lδq˙)dt

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