Lagrangian Density

$\mathcal{L}=\mathcal{K}-\mathcal{U}=-\frac{1}{2}\square \varphi$

• $\mathcal{L}$ is the Lagrangian density for a spin 0 particle
• $\mathcal{K}$ is the kinetic energy density
• $\mathcal{U}$ is the potential energy density
• $\square$ is the d’Alembert operator
• $\varphi$ is a field

$\mathcal{L}=i\bar{\Psi }{\gamma }^{\mu }{\partial }_{X^{\mu }}\Psi -m\bar{\Psi }\Psi$

• Dirac Lagrangian
• Leads to the Dirac equation

$\mathcal{L}=-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }$

• Maxwell Lagrangian

$\mathcal{L}=\frac{1}{16\pi G}{g}^{\mu \nu }{R}_{\mu \nu }$

• Einstein-Hilbert Lagrangian
• Leads to Einstein’s equations