PHY.7:4.4 - Derive the interior and boundary conditions together

Apply MATH.10 to the allowed family. Vary the complete action, including dependent quantities and boundary terms. For the smooth finite-dimensional form above, integration by parts gives

delta S = [L_qdot dot eta] at t0,t1 + integral (L_q-d/dt(L_qdot)) dot eta dt.

With fixed endpoints and otherwise arbitrary interior variations, stationarity gives d/dt(L_qdot)-L_q=0. With restricted variations, derive the condition supported by that restricted family. Retain force terms when the selected principle includes their virtual work.

For a field, perform the corresponding integration in space as well as time. First identify the boundary terms, then decide which vanish because the boundary value is prescribed and which produce a condition because its variation is free. An endpoint force or boundary energy can change the latter condition without changing the interior equation.

Differentiate symbolically or computationally when helpful, but retain the variables held fixed and the substitution rules. A result from varying only part of the action answers that smaller calculation. It does not justify omitting a physical term.

Stationarity alone does not establish a minimum. Use the stronger comparison only when required and supported. MATH.10 gives both a minimizing free-particle case and a stationary oscillator history with changes of either sign in the action.