Questions tagged [boundary-terms]
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Minimum or Stationary Value of a Mixed Boundary Problem
Take the volume integral of the dissipated DC current in a finite volume $\mathcal V$ of conductivity $\sigma$ and stationary potential distribution $\phi$ while assuming charge conservation $\nabla \...
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Robin conditions from action principle
Consider the Lagrangian density
$$L(\tilde{\phi}, \nabla \tilde{\phi}, \tilde{g}) = \tilde{g}^{\mu \nu} \nabla_{\mu} \tilde{\phi} \nabla_{\nu} \tilde{\phi} + \xi \tilde{R} \tilde{\phi}^2$$
with $\...
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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
$$ \delta I = \int_V ...
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Variation of the Lagrangian expressed as a time derivative of a function
In chapter 4.5 of Jakob Schwichtenberg's Physics from Symmetry, he expresses the variation of the Lagrangian $L = L\left ( q, \dot{q}, t \right )$ with respect to the generalized coordinate $q$ as
$$\...
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Dirac "GTR" Eq. 27.11 -- how to show that a boundary term vanishes?
In Dirac's "General Theory of Relativity", p. 53, eq. (27.11), Dirac is deriving Einstein's field equations and the geodesic equation from the variation $\delta(I_g+I_m)=0$ of the actions ...
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Dirac field coupling to gauge fields
I've seen in couple sources that the gauge invariant Lagrangian for the Dirac field being written as follows:
$$\mathcal{L} = \frac{i}{2}[\bar{\psi}\gamma^{\mu}D_{\mu}\psi-(\bar{D}_{\mu}\bar{\psi})\...
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Boundary terms in the gravitational action
A correct derivation of Einstein's field equations from an action principle $\delta I=0$ is given in
Poisson, E. (2004). A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge ...
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Partial integration of the Gibbons-Hawking-York boundary term
In https://arxiv.org/abs/1402.6334 on page 16 in their Eq. (5.21), they go from the equation
$$S_G+S_\chi\approx\int d^2x\sqrt{-g}XR+\int dt\sqrt{-\gamma}XK$$
to the equation
$$=\int d^2x \sqrt{-g}\...
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How did the boundary term vanish in deriving equation of motion from Lagrangian? [closed]
I was deriving the equation of motion from Lagrangian, by using the principle of least action. Usually, at this point in derivation,
$$\int dt \frac{\partial L}{\partial \dot{q}} \frac{\partial}{\...
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Boundary conditions in $\delta I=0$ to derive Einstein's equations -- why the derivatives of $g_{\mu\nu}$ are held constant?
Dirac derives Einstein's field equations from the action principle $\delta I=0$ where $$I=\int R\sqrt{-g} \, d^4x$$ ($R$ is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-...
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Generating function condition not satisfied?
We want to find a generating function $S(q_i,P_i,t)$ such that we get the best possible canonical transformations. So it must satisfy the Hamilton-Jacobi equation:
$$H(q_i,\frac{\partial S}{\partial ...
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Something fishy with canonical momentum fixed at boundary in classical action
There's something fishy that I don't get clearly with the action principle of classical mechanics, and the endpoints that need to be fixed (boundary conditions). Please, take note that I'm not ...
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What is the boundary action need for topological massive gravity (TMG)?
For pure Einstein gravity with Dirichilet boundary conditions, Gibbons-Hawking-York boundary action is needed to make the variational principle well defined. I am considering the case for topological ...
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Does the term $d ( \omega_{ab} \wedge \theta^a \wedge \theta^b )$ have any significance?
If $\omega_{ab}$ is the spin connection 1-form, and $\theta^a$ are the tetrad 1-forms, then one has the equality
\begin{equation}
\int \, d ( \epsilon_{abcd} \omega^{ab} \wedge \theta^c \wedge \theta^...
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Why do the boundary terms vanish for a function in Hilbert Space?
Below I have attached a solution to a problem from a quantum mechanics textbook, and I'd simply like someone to explain why the boundary terms vanish in Hilbert Space for the functions $f(x)$ and $g(x)...
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