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Learning some field theory, and many authors just claim "it is nice to express the Lagrangian as an integral $L = \int\mathcal{L}\,\mathrm{d}^3x$." Now I understand when dealing with fields, the sum's over all the particles turn into integrals and hence $L$ is an integral over space, but is that ALWAYS the case? Could I possibly have a Lagrangian who wasn't the integral of anything?

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    $\begingroup$ A single-particle Lagrangian is not typically the integral of anything. I'm not sure I understand the question. $\endgroup$ – ACuriousMind Nov 3 '16 at 22:38
  • $\begingroup$ does the OP maybe mean that you always define the action $S$ as an integral over time / integral over time and space of a lagrangian density? $\endgroup$ – Quantumwhisp Nov 3 '16 at 23:22
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In fundamental physics, we traditionally assume that the action functional is local, cf. e.g. this Phys.SE post.

However, there exist ideas that a local Lagrangian formulation is not always the most efficient description, cf. e.g. the amplituhedron formulation of planar pure ${\cal N}=4$ SYM by Arkani-Hamed et al.

Other theories are bi-local, see e.g. this Phys.SE post, or even more non-local.

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Ultimately, we want to work with relativistic field theories. To achieve that, the usual Lagrangian $L=L(\mathbf{q}(t), \dot{\mathbf{q}} (t), t) $ is not good enough. The Euler-Lagrange equations are: $$ \frac{\partial L}{\partial q_j} - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_j} \right) = 0$$ You could also write it as $$ \frac{\partial L}{\partial \mathbf{q}} - \frac{d}{dt} \left( \frac{\partial L}{\partial \left( \frac{d\mathbf{q}}{dt} \right) } \right) = 0 $$ This is obviously not Lorentz invariant. We need some Lorentz-scalar $\mathcal{L} = \mathcal{L} (\phi (x), \partial_\mu \phi (x))$ which satisfies: $$ \frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) = 0$$

This $\mathcal{L}$ is usually called the Lagrangian density and it satisfies: $$L = \int d^3 x \mathcal{L} $$

That's basically it, in simple terms.

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