I've been implementing lately few algorithms based on energy functions of the form.
$$ E = \int_\Omega \mathcal{L}(x,f,f')dx $$
In the above the cost function $E$ defines an actual energy that we want to minimize, you can assume in the integral above that $x$ is not a temporal variable.
Now with that in mind... Inspired by physics (I'm not an expert so forgive my terminology) I've designed an energy function which is summation of kinetic energy $T$ and potential energy $-V$, these two energies don't depend on time but from the generalized coordinates $q$ and $q'=p$, by the d'Alambert principle the associated cost function is given by
$$ C = \int_{t_0}^{t_1} \mathcal{L}(t,q,q')dt $$
in the integral above the lagrangian $\mathcal{L}$ is the energy at time $t$, and indeed $t$ is a temporal variable.
Total variation of $E$ would provide me a $PDE$ to be solved, while the total variation of $C$ would provide a ODE, I have the feeling there's some kind of relationship between $E$ and $C$ that I'm missing, but I really can't explain why, is there any formal argument or research that relates the two functional above?
Probably the observation might sound silly, however my hunch is... if we relate somehow $C$ and $E$ and differentiate $C$, and discretize the ODE associated we might be able to find some optimal sequence that can converge to the minimum of $E$, in this way we can solve the PDE of $E$ by a sequence.