Minimum total potential energy principle

Remarks

The functional $\Pi$ is a strictly convex quadratic functional on the Sobolev space $V$. Korn's inequality, together with $\abs{\Gamma_D} > 0$, shows that the strain energy controls the $H^1$ norm on $V$, so the Lax–Milgram theorem gives existence and uniqueness. For a convex quadratic functional, minimisers are exactly the points where the first variation vanishes, and the first variation of $\Pi$ is the weak form of equilibrium. Without $\abs{\Gamma_D} > 0$, the body can move rigidly: $\Pi$ is unchanged along rigid motions when the loads are balanced, so uniqueness fails, and $\Pi$ is unbounded below when they are not. Example in one dimension: a bar $[0, \ell]$ with stiffness $k > 0$, clamped at $0$ and pulled by a force $P$ at $\ell$, has $\Pi(v) = \tfrac{k}{2}\int_0^\ell v'^2\,dx - P v(\ell)$, whose minimiser is $u(x) = Px/k$. The same idea, that stable equilibria minimise the total potential energy, holds for conservative systems more generally. It is the basis of the finite element method in structural mechanics.

Going deeper

For a student

The principle is the variational form of linear elasticity, and it is a model case of the direct method of the calculus of variations. It replaces a system of partial differential equations by the minimisation of a convex functional. Restricting the minimisation to a finite-dimensional space of trial displacements gives the Ritz and finite element approximations, and the convexity gives their convergence.