I recently asked a related question and was interested in why/or why we cannot use convolutional variational principles in practice or in theory.
Summarizing the points I made in the earlier post:
Generally, we can only formulate the Lagrangian (and Hamiltonian) for "potential" systems, where in the dynamics satisfy the condition that: $$ m\ddot{\mathbf{x}}=-\nabla V $$ If this is true, we can formulate a functional which is stationary with respect to the system as: $$ F[\mathbf{x}]=\int^{t}_0\left(\frac{1}{2}m\dot{\mathbf{x}}(\tau)^2-V(\mathbf{x}(\tau))\right)\,\text{d}\tau $$
Taking the first variation of this functional yields the dynamics of the system, along with a condition that effectively states that the initial configuration should be similar to the final configuration (variation at the boundaries is zero).
The functional above is an inner product based formulation.
Now, given the functional: $$ F[\mathbf{x}]=\frac{1}{2}[\mathbf{x}^{\text{T}} * D(\mathbf{x})]+\frac{1}{2}[\mathbf{x}^{\text{T}} * \mathbf{Ax}]-\frac{1}{2}\mathbf{x}'(0)\mathbf{x}(t) $$ With $\mathbf{A}$ symmetric and $\mathbf{x}(0)$ being the initial condition, and: $$ [\mathbf{f}^{\text{T}} * \mathbf{g}]=\int^{t}_0 \mathbf{f}^{\text{T}}(t-\tau)\mathbf{g}(\tau)\,\text{d}\tau $$
If we take the first variation and assume only that the initial variation is zero, the functional is stationary with respect to: $$ \frac{d\mathbf{x}(t)}{dt}= \mathbf{Ax}(t) $$
This is a functional derived by Tonti and Gurtin, it represents a variational principle for linear initial value problems with symmetric state matrices and shows, as a proof of concept, that functionals can be derived for non-potential systems, initial value or dissipative systems.
This functional is a convolution based formulation.
Now, one might make the argument that these functionals should somehow be related to the "energy" of the system.
We can use Parseval's theorem to come up with such a relationship.
For some function $\mathbf{x}(t)$, we have: $$ \int^{\infty}_{-\infty}\left|\mathbf{x}(\tau)\right|^2\,\text{d}\tau = \int^{\infty}_{-\infty}\left|\mathbf{X}(\xi)\right|^2\,\text{d}\xi $$
Where $\mathbf{X}(f)=\mathcal{F}\!\left\lbrace \mathbf{x}(t)\right\rbrace$, the Fourier transform of the signal.
Either of the sides can be said to be the "energy" content of the signal and the left-hand side is the result of an inner product, which implies some relationship between inner products and "energy" content.
One can similarly relate the total frequency content of a real function $\mathbf{x}(t)$ (righ-hand side above) to a convolution using Plancherel's theorem, this shows that convolutions are related to energy as well.
Now, are there any other arguments as to why convolutions shouldn't be used in a variantional principle context from a physical standpoint?
