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Suppose that you want to calculate the variation $\delta S$ of an action induced by some arbitrary variation $\delta g_{\mu \nu}$ of the spacetime metric : \begin{equation} S = \int_{\Omega} \mathscr{L …

In general relativity and for its Einstein-Hilbert action, we usually ask that the metric variations $\delta g_{\mu \nu}$ cancel on the boundary $\partial \, \Omega$ of some region $\Omega$ of the spacetime … EDIT 2 : Under the action integral variation, is the perturbed field $\phi'(x) = \phi(x) + \delta \phi(x)$ "on-shell" or "off-shell" ? …

I'm wondering about a similar (inverted) interpretation for the action (1) (this is my own interpretation): Action $A$ is a measure of the mechanical information that you already have on the state … , but all of the answers are systematically refering to the extremal action principle (or stationary action principle), and they aren't answering the question about the action itself. …

Suppose that $L(q^i, \dot{q}^i)$ is a standard and well behaved lagrangian associated to some Dirichlet boundary conditions : $q^i(t_1) = q_1^i$ and $q^i(t_2) = q_2^i$. Now I have this new lagrangian …

Consider the standard Einstein-Hilbert action in $D \ne 2$ dimensions spacetimes : \begin{equation} S_{EH} = \frac{1}{2 \kappa} \int_{\Omega} R \; \sqrt{- g} \; d^D x, \end{equation} where $\Omega$ is … Usually, this forces us to introduce the Gibbons-Hawking-York surface integral into the gravitational action to remove that variational issue. I don't like that. …

We could suppose that since $\mathcal{V}_4$ should be very large and the action $S$ "reasonable", then $\Lambda$ should be small. … Does it make sense to interpret $\Lambda$ as a Lagrange multiplier associated to a constrained 4-volume introduced into the action ? …

application of the stationary action principle ? … Very important: Take note that I may be using the "Nature" Hamilton-Jacobi action and not the "observer" Euler-Lagrange action (I'm not sure yet), as defined in this paper : https://arxiv.org/abs/1203.2736 …