Monotonicity of a constrained minimum under added constraints ($A\subseteq B⟹{\inf }_{A}f\ge {\inf }_{B}f$)

Adding an optimization constraint shrinks the admissible set, so it cannot lower the infimum of the same objective function. Every value attained on the smaller set was already available on the larger set. If both minima exist and the larger problem has a unique minimizer absent from the smaller set, the inequality is strict. This compares beam energies for free, pinned, and clamped endpoints without evaluating the energy integrals.