auch Prinzip der kleinsten Wirkung genannt
δ S = ∫ t 1 t 2 ( δ T − δ A ) d t = 0 {\displaystyle \delta S=\int \limits _{{t}_{1}}^{{t}_{2}}{\left(\delta T-\delta A\right)dt}=0}
mit
δ A = ∑ i X _ i δ r i _ {\displaystyle \delta A=\sum \limits _{i}{{{\underline {X}}_{i}}\delta {\underline {{r}_{i}}}}}
führt zur Wirkung S [ q ] := ∫ t 1 t 2 L ( q , q ˙ , t ) d t {\displaystyle S\left[q\right]:=\int \limits _{{t}_{1}}^{{t}_{2}}{L\left(q,{\dot {q}},t\right)dt}}
FragenID::M1
δ S [ q ] = ∫ t 1 t 2 δ L ( q , q ˙ , t ) d t = ∫ t 1 t 2 ( ∂ q L δ q + ∂ q ˙ L δ q ˙ ) d t {\displaystyle {\begin{aligned}\delta S\left[q\right]&=\int \limits _{{t}_{1}}^{{t}_{2}}{\delta L\left(q,{\dot {q}},t\right)dt}\\&=\int \limits _{{t}_{1}}^{{t}_{2}}{\left({{\partial }_{q}}L\delta q+{{\partial }_{\dot {q}}}L\delta {\dot {q}}\right)dt}\end{aligned}}}
oder
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 {\displaystyle {\begin{aligned}\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{aligned}}}
mit partieller Integration ( ∫ u ′ v = u v − ∫ v ′ u {\displaystyle \int {u'v=uv-\int {v'u}}} ) mit u = δ q , v = ∂ q ˙ L {\displaystyle u=\delta q,v={{\partial }_{\dot {q}}}L}
∂ q ˙ L δ q ˙ = d t ( ∂ q ˙ L δ q ) − d t ( ∂ q ˙ L ) δ q {\displaystyle {{\partial }_{\dot {q}}}L\delta {\dot {q}}={{d}_{t}}\left({{\partial }_{\dot {q}}}L\delta q\right)-{{d}_{t}}\left({{\partial }_{\dot {q}}}L\right)\delta q}
δ S [ q ] = − [ ∂ q ˙ L δ q ] t 1 t 2 − ∫ t 1 t 2 ( ∂ q L δ q − d t ( ∂ q ˙ L ) δ q ) d t = ∫ t 1 t 2 ( d t ∂ q ˙ − ∂ q ) L δ q d t {\displaystyle {\begin{aligned}\delta S\left[q\right]=-{\cancel {\left[{{\partial }_{\dot {q}}}L\delta q\right]_{{t}_{1}}^{{t}_{2}}}}-\int \limits _{{t}_{1}}^{{t}_{2}}{\left({{\partial }_{q}}L\delta q-{{d}_{t}}\left({{\partial }_{\dot {q}}}L\right)\delta q\right)dt}\\&=\int \limits _{{t}_{1}}^{{t}_{2}}{\left({{d}_{t}}{{\partial }_{\dot {q}}}-{{\partial }_{q}}\right)L\delta qdt}\end{aligned}}}
( d t ∂ q ˙ − ∂ q ) L = 0 {\displaystyle \left({{d}_{t}}{{\partial }_{\dot {q}}}-{{\partial }_{q}}\right)L=0} FragenID:M2
![{\displaystyle S\left[q\right]:=\int \limits _{{t}_{1}}^{{t}_{2}}{L\left(q,{\dot {q}},t\right)dt}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d865adce5e3f364c1ad12e060497fca36db05743)
![{\displaystyle {\begin{aligned}\delta S\left[q\right]&=\int \limits _{{t}_{1}}^{{t}_{2}}{\delta L\left(q,{\dot {q}},t\right)dt}\\&=\int \limits _{{t}_{1}}^{{t}_{2}}{\left({{\partial }_{q}}L\delta q+{{\partial }_{\dot {q}}}L\delta {\dot {q}}\right)dt}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f88604fff7467be071d79286faaa701a7de874f)
![{\displaystyle {\begin{aligned}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33669a26cc4c75a555ceffd63580d80a63748ebf)
) mit
![{\displaystyle {\begin{aligned}\delta S\left[q\right]=-{\cancel {\left[{{\partial }_{\dot {q}}}L\delta q\right]_{{t}_{1}}^{{t}_{2}}}}-\int \limits _{{t}_{1}}^{{t}_{2}}{\left({{\partial }_{q}}L\delta q-{{d}_{t}}\left({{\partial }_{\dot {q}}}L\right)\delta q\right)dt}\\&=\int \limits _{{t}_{1}}^{{t}_{2}}{\left({{d}_{t}}{{\partial }_{\dot {q}}}-{{\partial }_{q}}\right)L\delta qdt}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/376d85394df80bb2533e1b79404a7afd6164401c)
FragenID:M2
