Lagrangian Density

#Physics

$\displaystyle \mathscr{L}=\mathscr{K}-\mathscr{U}=-\frac{1}{2}\Box \phi$

• $\displaystyle \mathscr{L}$ is the Lagrangian density for a spin 0 particle
• $\displaystyle \mathscr{K}$ is the kinetic energy density
• $\displaystyle \mathscr{U}$ is the potential energy density
• $\displaystyle \Box$ is the d'Alembert operator
• $\displaystyle \phi$ is a field

$\displaystyle \mathscr{L}=i\bar{\Psi}\gamma^{\mu}\partial_{X^{\mu}}\Psi-m\bar{\Psi}\Psi$

• Dirac Lagrangian
• Leads to the Dirac equation

$\displaystyle \mathscr{L}=-\frac{1}{4}F^{\mu \nu}F_{\mu \nu}$

• Maxwell Lagrangian

$\displaystyle \mathscr{L}=\frac{1}{16\pi G}g^{\mu \nu}R_{\mu \nu}$

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