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Before going to field theory, it seems instructive to first ask the same questions in point mechanics: Can the Lagrangian $L(q,v,t)$ depend on time explicitly? Yes. The Lagrangian $L(q,v,t)$ can depend explicitly on time. E.g. there could be external sources. On the other hand, if the Lagrangian does not depend on time explicitly, then the ...

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We have such equation: $$H = \frac{\partial L}{\partial \dot{q}} \dot{q} - L$$ You can show by calculation, that it holds in your special case, too. Now we use chain rule, and Euler-Lagrange equation: \frac{dL}{dt} = \frac{\partial L}{\partial q} \dot{q} + \frac{\partial L}{\partial \dot{q}} \ddot{q} =\frac{d}{dt}\left(\frac{\partial L}{\partial ...

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The mathematical concept that I was searching for in this question is the following: http://en.wikipedia.org/wiki/Hodge_dual I will not elaborate more, but except to say that, the Hodge dual allows you to define a conserved current corresponding to any choice of the "time axis". (Sorry for the vagueness).

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Comments to the question (v2): It is often is possible to formulate (discrete) time-evolution equations/equations of motion (eoms) in a discretized theory. This is of course useful in computational physics. However OP asks for a variational action principle for a fully discretized theory. Hence we will not discuss further the case where eoms (without a ...

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Not all conserved charges are obtained by integrating the time component of some conserved current. For example, momentum and angular momentum are conserved charges and are obtained by integrating a spatial component of a conserved current. So the equations and interpretation for conserved charges in a Euclidean theory are the same as in the ...

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Assuming no quantum gravity, $\eta^{\mu\nu}$ is a constant and can be pulled out of the derivative and what remains looks like a $\delta^k_{\mu}$ or $\delta^k_{\nu}$-type expression (in the sense of a Kronecker $\delta$), pulling the $k$ into the $\partial^\mu$ or $\partial^\nu$ respectively. If you are confused about where the minus sign comes from, I ...

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