Variation Formulations of Physical Laws: The Correct Formulation

[JTVT]

Source of Question
J.N. Reddy states in Finite Element Method: "Variational forms of the laws of continuum physics may be the only natural and rigorously correct way to think of them. While all sufficiently smooth fields lead to meaningful variational forms, the converse is not true: There exists physical phenomena which can be adequately modeled mathematically only in a variational setting; they are nonsensical when viewed locally"

My Question
What are some examples of physical phenomena that can be adequately modeled mathematically only in a variational setting and why are they nonsensical when viewed locally?

 

[UltrafastPED]

Some approaches are "mean field theories" which result in very smooth fields - Maxwell's equations in matter are an example. But when you look at the microscopic sources of the fields (atoms and molecules, etc) their appear to be discontinuities ... hence you cannot meet the usual requirements for differentiability so that you can carry out the variational principle.