Questions tagged [action]
The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.
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Weyl transformation of induced metric
Consider the Weyl/conformal transformation in four dimenions
$$\tilde{g} \enspace = \enspace \Omega^2 g \quad \Longrightarrow \quad \sqrt{-|\tilde{g}|} \enspace = \enspace \Omega^4 \sqrt{-|g|}$$
The ...
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Energy-momentum tensor and equation of motion in Einstein-Dilaton theory
I am following this paper (see eq. 19-22) and trying to derive the equation of corresponding to Einstein-Dilaton gravity (ignoring the Maxwell part for now)
\begin{align}
S_{\text{E-D}} = \int d^4 ...
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Independence of the equations resulting from the action principle $\delta (I_{\text{gravity}} + I_{\text{other fields}}) = 0$
In Dirac's "GTR", Chap. $30$, he discusses the "comprehensive action principle" and shows that variation of the combined action of the Hilbert-Einstein action plus all other matter-...
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Action principle dependent on spacetime-topology?
Consider the Lagrangian density
$$L(\phi, \nabla \phi, g) = g^{\mu \nu} \nabla_{\mu} \phi \nabla_{\nu} \phi$$
If one varies the action as usual, then one finds the equation
$$\delta S = \int_{\mathcal{...
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Weyl variation of a generic action
In this paper https://arxiv.org/abs/hep-th/9906127 (see eq. 15)
The following identity appears
$$ \delta_{W} \int d^d x \sqrt{-\gamma} \tilde{\mathcal{L}}^{(n)} = \int d^d x \sqrt{-\gamma} \sigma\left(...
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Possible boundary conditions in derivation of Euler-Lagrange equations
Given a Lagrange density
$$\mathcal{L} = g^{ij} \phi_{,i} \phi_{,j} - V(\phi)\tag{1}$$
I have read (e.g. here) that the boundary term that occurs through variation of the action
$$ \delta I = \int_V ...
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Plugging solution into Lagrangian [duplicate]
Consider a simple ODE in $x \equiv x(t)$, e.g.
$$\ddot{x} = -k x \quad \Longrightarrow \quad x(t) = e^{ik t}.$$
This system's Lagrangian is
$$L = \frac{1}{2} \dot{x}^2 - \frac{k}{2} x^2.$$
Knowing the ...
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Varying the Chern-Simons action
Summary/TL;DR
I want a detailed calculation of the derivation of classical equations of motion from the Chern-Simons action using differential forms, using variational derivatives. I mentioned "...
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How to transform coordinates in the Action equation?
For a particle moving in one dimension, and we have two people describing the motion of that particle one is stationary (let's call him Lenny) and the other (George) is moving relative to Lenny. Lenny'...
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