Vol 1 - L D Landau, E.M. Lifshitz - Mechanics · Problem 8.1

The action $S$ is a fundamental scalar in relativistic mechanics, but it transforms in a subtle way between frames because it's built from the Lagrangian, which itself depends on velocities. The key insight is that $S$ is actually a Lorentz invariant—the same in all inertial frames—because it's the integral of the Lagrangian over time, and the combination $L \, dt$ transforms exactly like a four-scalar. This means you're really discovering why the action is frame-independent, which is the deep reason the equations of motion have the same form everywhere.

The action $S$ is a fundamental scalar in relativistic mechanics, but it transforms in a subtle way between frames because it's built from the Lagrangian, which itself depends on velocities. The key insight is that $S$ is actually a Lorentz invariant—the same in all inertial frames—because it's the integral of the Lagrangian over time, and the combination $L \, dt$ transforms exactly like a four-scalar. This means you're really discovering why the action is frame-independent, which is the deep reason the equations of motion have the same form everywhere.