Energy functional and Euler Lagrange equation

Question

We know that for potential energy functional, its derivative is called the Euler Lagrange equation and physically, it means that at the given point there is a force balance. Now if the energy functional is not convex, it could also have local maxima, which also indicates force balance. But physically, we want to minimize the energy, so what do the maxima mean for the energy functional? It seems that the maxima also solve the Euler Lagrange equation, which means that is a force balance. But why is it physically not correct?

Answer 1

As A.G. stated, the (strict) local maxima of potential energy are unstable equilibria. E.g., a ball at rest at the top of a sphere, or a pendulum balanced at its highest position. These can't be sustained in reality, due to inevitable perturbations (air motion, etc).

A non-strict local maximum may happen to be neutrally stable if the potential energy is constant in its neighborhood (i.e., this local maximum is also a local minimum). This is an idealized scenario, like a ball resting on a perfectly flat and perfectly horizontal surface.