In classical mechanics the Lagrangian \(L=T-V\) (kinetic minus potential energy) is the function whose time integral, the action \(S=\int L\,dt\), is stationary along the physical trajectory. In field theory the corresponding object is the Lagrangian density \(\mathcal L\), and the two are related by a spatial integral \(L=\int d^3x\,\mathcal L\). In particle-physics usage "the Lagrangian" almost always means the Lagrangian density \(\mathcal L\).

The dynamics follow from demanding \(\delta S=0\). For a field \(\phi\) with

\[ S = \int d^4x\ \mathcal{L}(\phi,\partial_\mu\phi), \]

the stationarity condition gives the Euler–Lagrange equations of motion

\[ \partial_\mu\!\left(\frac{\partial \mathcal L}{\partial(\partial_\mu\phi)}\right) - \frac{\partial \mathcal L}{\partial\phi} = 0 . \]

• Symmetries made manifest: invariances of \(\mathcal L\) map, via Noether's theorem, onto conserved currents (e.g. the color current of QCD). This is why building a Lagrangian invariant under local gauge invariance fixes the interactions.
• Lorentz covariance: writing \(\mathcal L\) as a Lorentz scalar guarantees relativistic invariance of the theory — see Poincare invariant Lagrangian.
• Quantization: the same \(\mathcal L\) feeds directly into the path integral formulation, \(Z=\int[D\phi]\,e^{iS}\), providing the bridge from classical to quantum field theory.

For a single fermion the Lagrangian is

\[ \mathcal{L} = \overline\psi\,(i\gamma^\mu\partial_\mu - m)\,\psi , \]

whose Euler–Lagrange equation is the Dirac equation. Promoting \(\partial_\mu\to D_\mu\) turns this free Lagrangian into the interacting QCD Lagrangian.