Let $f$ be a smooth function $[0, L] \times \mathbb{R} \to \mathbb{R}$, and consider an initial value problem (IVP) of the form $$ \begin{cases} x' = f(t, x),\\ x(0) = x_{0} \in [0, 2\pi). \end{cases} $$ In particular, suppose that the problem is globally solvable, meaning that, for any $L$, a solution $[0, L] \to \mathbb{R}$ exists and is unique.
Now, consider a strictly positive function of the form $$ E(x_{0}) = \int_{0}^{L} F(t, x(t), x'(t))\, dt, $$ where $F(t, x(t), x'(t)) >0$, and where $x$ is a solution of the IVP defined by the initial condition $x_{0}$. The function $E$ may be thought of as assigning an energy to each solution of the IVP (i.e., to the physical system that the IVP represents).
I am wondering if anything interesting can be said about the dependence of the energy $E$ on the initial condition.
It seems plausible to me that, as $L \to \infty$, the dependence of $E$ on $x_{0}$ should become less and less important. In other words, I would expect that, for any $x_{0}, x_{1} \in [0, 2\pi)$, $$ \lim _{L \to 0} E(x_{0})/E(x_{1}) =1. $$
Question. Is my expectation justifiable from a mathematical perspective?
Edit: As the answer appears to be no in general, I would like to specialize the question to a function $F$ of the form
$$F(t, x(t), x'(t)) = \cos(x(t))^{2} g(t)^{2}.$$