Order the logic of solving a variational problem.

1. Write the functional $J[x] = \int F(t, x, \dot{x})\,dt$
2. Demand that no small variation improves $J$
3. Derive the Euler–Lagrange equation
4. Solve the differential equation for $x(t)$
5. Fix constants with the boundary conditions

Model → optimality principle → equation → solution family → pin down endpoints. The differential equation in step 4 is why the next lessons build integration and ODEs.