Lagrangian and action
$$L(t)=\int d^3x\,\mathcal{L}(\phi_k,\partial_{\mu}\phi_k),\quad S=\int_{t_1}^{t_2}dt\,L(t)$$
\(\delta\phi_k(\vec{x},t_1)=\delta\phi_k(\vec{x},t_2)=0\)
\(\delta\left(\partial_{\mu}\phi_k\right)=\partial_{\mu}\left(\delta\phi_k\right)\)
$$\delta S=\int d^4x\,\left(\frac{\partial\mathcal{L}}{\partial\phi_k}-\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\phi_k\right)}\right)\right)\delta\phi_k+\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\phi_k\right)}\delta\phi_k\right)=0$$
Lagrange's equations
$$\frac{\partial\mathcal{L}}{\partial\phi_k}-\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\phi_k\right)}\right)=0$$
Conserved current
$$\mathcal{J}^{\mu}=\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\phi_k\right)}\delta\phi_k-\mathcal{F}^{\mu},\quad\delta\mathcal{L}=\partial_{\mu}\mathcal{F}^{\mu}$$
$$\partial_{\mu}\mathcal{J}^{\mu}=0,\quad\frac{d}{dt}\int d^3x\,\mathcal{J}^{0}=0$$
Momentum density conjugate to \({\phi_k}\)
\({\phi_k}\)
$$\pi_k=\frac{\partial\mathcal{L}}{\partial\dot{\phi}_k}$$
Hamiltonian density:
$$\mathcal{H}=\frac{\partial\mathcal{L}}{\partial\dot{\phi}_k}\,\dot{\phi}_k-\mathcal{L}$$
The Lagrangian density of a scalar complex field
$$\mathcal{L}(\phi)=\partial_{\mu}\phi\,\partial^{\mu}\phi^{*}-m^2\left|\phi\right|^2$$
Klein-Gordonequation
$$\left(\partial_{\mu}\partial^{\mu}+m^2\right)\,\phi=0$$
and Hamiltonian density
$$\mathcal{H}=\left|\partial_{0}\phi\right|^2+\left|\nabla\phi\right|^2+m^2\left|\phi\right|^2$$
Symmetry:\(\phi\rightarrow e^{i\alpha}\phi\),\(\phi^{*}\rightarrow e^{-i\alpha}\phi^{*}\),\(\alpha\in R\)
$$\delta\mathcal{L}=0,\quad\delta\phi=i\alpha\phi,\quad\delta\phi^{*}=-i\alpha\phi^{*}$$
Conserved current:\(\mathcal{J}^{\mu}=i\left(\phi\partial^{\mu}\phi^{*}-\phi^{*}\partial^{\mu}\phi\right)\)