古典電磁気学の共変定式
δ S int [ X , A ] δ X i μ ( λ ) = ∂ L int ∂ X i μ − d d λ ∂ L int ∂ X ̇ i μ = q i X ̇ i ν ( λ ) ∂ μ A ν ( X i ) − q i d d λ A μ ( X i ) = q i X ̇ i ν ( λ ) ∂ μ A ν ( X i ) − q i X ̇ i ν ∂ ν A μ ( X i ) = − q i X ̇ i ν ( λ ) F ν μ ( X i ) {\displaystyle {\begin{aligned}{\frac {\delta S_{\text{int}}[X,A]}{\delta X_{i}^{\mu }(\lambda )}}&={\frac {\partial L_{\text{int}}}{\partial X_{i}^{\mu }}}-{\frac {d}{d\lambda }}{\frac {\partial L_{\text{int}}}{\partial {\dot {X}}_{i}^{\mu }}}\\&=q_{i}{\dot {X}}_{i}^{\nu }(\lambda )\,\partial _{\mu }A_{\nu }(X_{i})-q_{i}{\frac {d}{d\lambda }}A_{\mu }(X_{i})\\&=q_{i}{\dot {X}}_{i}^{\nu }(\lambda )\,\partial _{\mu }A_{\nu }(X_{i})-q_{i}{\dot {X}}_{i}^{\nu }\partial _{\nu }A_{\mu }(X_{i})\\&=-q_{i}{\dot {X}}_{i}^{\nu }(\lambda )\,F_{\nu \mu }(X_{i})\\\end{aligned}}}