Euler-Lagrange Equations for Fields

Derives field dynamics from a local Lagrangian density. Theory, formulas, worked examples and interactive graphs.

Conceptual context

In field theory, dynamics is not assigned point by point through an external force. Instead, it emerges from a global rule: the physical configuration makes the $S$ stationary. The $𝓛$ locally summarizes the contributions of the $\phi$, the $∂_\mu \phi$, and interactions. Varying the full configuration through a $\delta \phi$ produces two local responses that must balance. This condition turns a global principle into a differential equation valid at each $x^\mu$. The aim is not merely to memorize a formula, but to understand how the architecture of the $𝓛$ selects allowed evolutions and how the $𝓔(\phi )$ diagnoses whether a proposed configuration truly satisfies the dynamics.

🟢 Essential level

The central idea is to compare two ways in which the $𝓛$ changes. The first comes from modifying the $\phi$ directly at a point.

The second appears because local slopes and rates, represented by the $∂_\mu \phi$, also change. The Euler-Lagrange equation requires these contributions to balance after accounting for how the response to gradients varies across spacetime. If they do not balance, a $𝓔(\phi )$ remains, showing that the chosen configuration is not an exact physical evolution. The $S$ provides a global criterion, while the resulting equation acts locally. This is why a single $𝓛$ can generate an entire field law. Boundary conditions matter: an admissible variation must respect them, otherwise the $B$ contains information that cannot be discarded. Boundary data therefore participate in the definition of the physical problem rather than acting as an afterthought.

Joseph-Louis Lagrange was only nineteen when, in 1755, he sent Euler a letter describing a new algebraic method of variations; Euler, who had already founded the calculus of variations in 1744, immediately adopted Lagrange's notation and named the new discipline in his honor.

🔵 Formal level

Basic formulas

The $S$ is built by integrating the $𝓛$ with the $d^4x$:

Field action

An arbitrary interior $\delta \phi$ is introduced and the $\delta S$ is computed:

Stationarity condition of the action

Both relations are the mandatory starting point of the variational principle: the first defines the quantity to extremize and the second formalizes how the field is perturbed.

Real student questions

Why vary the whole field instead of one point? Because the $S$ depends on the complete configuration. The variation explores possible directions in configuration space.

Does zero residual guarantee a unique physical solution? No. A vanishing $𝓔(\phi )$ guarantees the local equation, but initial conditions, boundary conditions, and stability criteria are still needed.

What if the boundary term does not vanish? The $B$ may impose natural conditions or require additional contributions to the $S$.

Why do signs matter so much? Because the $□\phi$ depends on the metric signature. Mixing conventions changes the resulting equation.

Is a stationary action always stable? No. Stability requires second-order variations, not only the first-order $\delta S$.

Final synthesis

The $𝓛$ determines evolution because its variation produces a local condition. The $∂𝓛/∂\phi$ and $∂_\mu [∂𝓛/∂(∂_\mu \phi )]$ must balance, canceling the $𝓔(\phi )$. The method turns symmetries, interactions, and geometry into coherent field equations.