I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form $$\ddot{u}(t) = \nabla F(t,u(t))$$ where $F:[0,T] \times \mathbb{R}^n \to \mathbb{R}$ satisfies various assumptions.
I understand that equations of the above form arise from various dynamical systems, especially from the Lagrangian formulation of classical mechanics.
The papers that I read usually only begin by saying that the above equation arises from Lagrangian mechanics, or that they are important in Physics, etc. They then went on to prove various results on the existence of periodic solutions satisfying certain properties (with prescribed minimal period, prescribed energy, etc.)
There are also many criteria and many questions in this forum that were asking about whether such and such equation has (or doesn't have) periodic solution.
My question is: Why do we care that the above equation has periodic solution say for every period $T>0$ ? What is its relevance to say classical mechanics, or its place in the context of the study of differential equations and dynamical systems in general?
As a beginning grad student in Math, I am interested to know about the bigger picture and the relevance to Physics.
Thank you.
