# Why is the functional of an elastic system equal to it's total potential energy and why is it stationary at it's minimum?

Okay this was even quite difficult to formulate but let's start with some statement's in my FEM book:

1. Elastic continuum has a functional which is equal to it's total potential energy
2. Elastic system is in balance when the stationary value of it's functional equals the minimum of potential energy

I understand nothing about it but I will try to form some more exact questions:

a) Is there any physical interpretation of a functional? All I can find is that a functional it's an integral of a function and the arguments of that function are also functions. But that doesn't tell me how it is derived and what is it's meaning

b) Connected with a), why is this functional equal to total potential energy in elastic systems?

c) Why is the elastic system in balance when the stationary value of the functional is equal to the minimum of potential energy? Why not maximum or any other state?

• Have taken any courses in the calculation of variations that included Lagrange methods of multivaiable optimization? This stuff took the world's best mathematicians about 100 years to figure out, and even the basic Rienmannian surface maths aren't commonly taught any more. Jun 4 '20 at 9:52
• I haven't taken any such courses. We have used some Lagrange and Euler-Lagrange methods but again with no background. I understand it's difficult to figure out, but explaining it (at least in Layman terms) might be easier, no? It is interesting to me how difficult is it to find anything about it on the internet Jun 4 '20 at 12:31