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So in the familiar case of non-relativistic particle Lagrangians/actions, we know the equations of motions are given by [tex]\frac{\partial \mathcal L}{\partial x^i} = \frac{\mathrm d }{\mathrm dt} \left( \frac{\partial \mathcal L}{\partial \dot x^i} \right)[/tex]

In the familiar case of relativistic field theory Lagrangians/actions, we have

[tex]\frac{\delta \mathcal L}{\delta \phi} = \partial_\mu \left( \frac{\delta \mathcal L}{\delta ( \partial_\mu \phi )} \right)[/tex]

However, it seems that if we now choose a time-splitting, like for example in ADM where the essence is to rewrite [itex]S = \int \mathrm d^4 x \; \mathcal L(g_{\mu \nu}, \partial_\rho g_{\mu \nu})[/itex] as [itex]\boxed{ S = \int \mathrm d t \; \mathrm d^3 x \; \mathcal L(g_{i j}, \partial_k g_{i j}, \dot g_{ij}, N, N^i)}[/itex]

In this case it seems the equation of motion is given by

[tex]\frac{\delta \mathcal L}{\delta g_{ij}} = \frac{\mathrm d }{\mathrm dt} \left( \frac{\delta \mathcal L}{\delta ( \dot g_{ij} )} \right)[/tex]

This seems a bit weird. Is it obvious the latter two equations of motions are compatible?

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# What happens to Euler-Lagrange in field theories? (ADM?)

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