 Solution

For an admissible variation $\eta$ with $\eta (0)=\eta (L)=0$,Stationarity for every variation therefore gives the Euler-Lagrange equationThe first energy term is bending energy: the curvature of an inextensible planar beam is ${\theta }_s$. The shortening relative to the straight reference is $\int (1-\cos \theta )ds$. A prescribed compressive dead load does positive work on that shortening, so its potential-energy contribution is minus $P$ times that integral. A minimizer must satisfy the stationary equation, but a stationary solution need not be a minimizer.