Lagrangian
$ L(q(t),\dot q(t),t)
作用 (action)
$ S=\int_{t_i}^{t_F}L dt
運動方程式
Newton の運動方程式$ {\bf F}=m\ddot{\bf x},$ \frac{d^2{\bf r}(t)}{dt^2}=F
$ {\bf r}=\braket{\psi(x,t)|x|\psi(x,t)}
Euler-Lagrange 方程式$ \frac d{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i}=0 $ {\cal L}=T-V
$ {\cal L}_X(\dot X,e)=\frac e 2(e^{-2}\dot X^\mu\dot X_\mu(\lambda)-m^2c^2)
Hamiltonian の正準方程式$ \dot q_i(t)=\frac{\partial H}{\partial p_i},$ \dot p_i(t)=-\frac{\partial H}{\partial q_i} Lagrangian 密度 (場の理論)
Maxwell の方程式
$ {\cal L}=-\frac 1 4 F_{\mu\nu}F^{\mu\nu}
電氣・磁氣
$ {\cal L}_{\rm QED}={\cal L}_d+{\cal L}_{\rm em}+{\cal L}_{\rm int}=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi-\frac 1 4 F_{\mu\nu}F^{\mu\nu}-gA^\mu\bar\psi\gamma_\mu\psi
$ {\cal L}_{\rm GWS}={\cal L}_{\rm YM}+{\cal L}_\psi+{\cal L}_\Phi+{\cal L}_{湯川}=-\frac 1 4 F^{a\mu\nu}F^a_{\mu\nu}+\sum_\psi i\bar\psi\bar\sigma^\mu{\cal D}_\mu\psi+\left(({\cal D}^\mu\Phi)^\dag{\cal D}_\mu\Phi-\lambda\left(\Phi^\dag\Phi-\frac{v^2}2\right)^2\right)+\left(-\sum_{i,j}y_{i,j}\psi_i\psi_j\Phi+{\rm h.c.}\right)
$ {\cal L}_{\rm QCD}=\sum_\psi(i\bar\psi^j\gamma^\mu({\cal D}_\mu\psi)_j-m_\psi\bar\psi^j\psi_j))-\frac 1 4 G^a_{\mu\nu}G^{a\mu\nu}