# Actions that are not integrals

So far every action I've seen in physics has been an integral of a Lagrangian, be it a point particle:

$$S = \int dt\ L$$

or fields (relativistic or not):

$$S = \int d^4x\ \mathcal{L}$$

and so forth. Authors don't usually justify this (and I'm not saying they should), so I wonder: are there applications of actions that aren't integrals of Lagrangians?

For example, we could have something like $S[x(t)] = \sup\{\dot{x}(t)^2\}$, or variations of the same theme (I haven't figured out how to find the extrema). Or can any functional be written as an integral?

• This post (v1) seems like a list question. – Qmechanic Jul 24 '16 at 15:42
• @Qmechanic: do you have a suggestion on how I could reword the question? I could ask something like "are these actions useful" but I feel like that's essentially the same question. – Javier Jul 24 '16 at 15:58
• I think the form of the action in field theory is necessary for the E.O.M. to be local. – Jahan Claes Jul 24 '16 at 21:50

In fact, one can consider discrete sums instead of integrals as the action. For example, one can consider an action of the form $$S=\sum_{i=0}^n\frac{1}{2}(\phi_{i+1}-\phi_i)^2+\frac{1}{2}\phi_i^2$$ with $\phi_{n+1}=\phi_0$,$\phi_i\in\mathbb{R}$.This will give a 1d field theory on a discrete circle. One can of course compute the equation of motion,which will be a recursive relation rather than a differential equation.