Euler–Lagrange equations from variations

Choose fixed initial and final times and fixed endpoint configurations. A nearby path is the original path plus a small parameter multiplying a smooth variation function that vanishes at both endpoints. The velocity variation is the time derivative of that function. Position and velocity arguments are treated independently when taking partial derivatives of the Lagrangian; their variations are linked only after substituting the varied path. Confusing a partial derivative with a total time derivative is a common source of missing terms.

Differentiate the action with respect to the small parameter at zero. One term multiplies the position variation and another multiplies its derivative. Integration by parts transfers the time derivative onto the conjugate momentum, leaving an endpoint contribution. The endpoint contribution vanishes because we fixed configurations, not because momentum is zero. If endpoint conditions change, this term must be reconsidered; discarding it automatically can remove a genuine natural boundary condition.

For independent holonomic coordinates, each variation can be chosen freely inside the interval. If a continuous coefficient had a nonzero sign near some time, choosing a variation localized there would make the action change nonzero. Therefore every interior coefficient must vanish. This gives the Euler–Lagrange equations. A regular velocity Hessian allows these equations to be solved locally for accelerations; singular Lagrangians require additional constraint analysis rather than automatic inversion of a zero matrix.

For a mass on a spring, differentiating with respect to velocity gives mechanical momentum and differentiating with respect to position gives minus the spring force coefficient times displacement. The resulting equation is the familiar oscillator equation. Multiplying the whole Lagrangian by a nonzero constant changes the numerical action but not its stationary paths. Adding a total time derivative of a configuration function changes endpoint contributions while leaving these fixed-endpoint equations unchanged. Neither operation licenses changing individual kinetic or potential terms arbitrarily.

The first variation tests the term linear in the small path change. It says nothing by itself about the sign of the quadratic change. For a harmonic oscillator observed over a long enough interval, some fixed-endpoint variations decrease the action even though the actual solution remains stationary. The action principle is therefore not a mechanical search for the shortest visible curve or the least energy at every instant. Boundary-value problems can also have multiple stationary paths or special intervals with nonunique solutions.

The oscillator lab provides a separate check of a differential-equation solution, not a path-variation engine. Set damping ratio to zero, predict that changing natural frequency compresses the time scale, then play or step the normalized trajectory. The fixed initial data are displacement one and velocity zero. Introducing damping explores a different equation with a dissipative force; it does not show that the same autonomous T−V action suddenly produces damping. Keep the variational derivation and the illustrative initial-value model conceptually distinct.