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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?

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  • $\begingroup$ This post (v1) seems like a list question. $\endgroup$ – Qmechanic Jul 24 '16 at 15:42
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    $\begingroup$ @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. $\endgroup$ – Javier Jul 24 '16 at 15:58
  • $\begingroup$ I think the form of the action in field theory is necessary for the E.O.M. to be local. $\endgroup$ – Jahan Claes Jul 24 '16 at 21:50
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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.

The advantage of such models emerges after quantization. With discrete lattice, the path integral becomes an ordinary integral although on a vector space of very large dimension, and that means one can study the theory numerically.

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Usually what we are interested in is not the solving the variational problem itself. We merely want to have neat object which allows us to derive equations of motion, conserved quantities etc. This object is action integral. Action not given by an integral will generally not lead to differential equations so it is just not interesting.

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