There are many problems in physics whose solution (a function on some space) can be obtained by a variational principle, namely, the solution $\phi$ makes some functional $S$ extremal: \begin{align} \frac{\delta S[f]}{\delta f}|_{\phi} = 0 \end{align}
Now I'd like to know: If a function is a solution to a variational calculation / if a function minimizes some functional, does it have properties that other functions don't have? For example uniqueness of the solution with respect to special boundaries of a certain type of equations, or something like that.
The reason I'm asking is that the following observation from physics: Not all problems do have a variational formulation, but most of them do, and when they do, we usually have some kind of information conservation. I will use the point particle as an example. Subject to conservative forces, it does have a Lagrangian formulation, and the motion will be deterministic for all times. Subject to velocity dependent (Lorentz-force aside) forces like a velocity dependent friction, there won't be an formulation of the problem as a variational problem anymore. At the same time, the problems are not deterministic anymore, we can't reconstruct a future state from the present one.
That's why I'd like to know what the properties are that we in general can assign to solutions of variational calculus. I know that being the solution to a variational principle is equivalent to the function satisfying some kind of lagrangian equation, but I'd like to know wether there are other equivalent properties that this function does have as well.