All Questions
Tagged with tensor-calculus lagrangian-formalism
63
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Notation for vector density in Lagrangian density
Consider a manifold $M$ and a Lagrangian density $\mathcal{L} \equiv \mathcal{L}(\phi, \nabla \phi)$. By varying the action, one obtains the equation
$$\int_M \, dV \; \Big( \frac{\partial \mathcal{L}}...
- 743
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Derivation of Einstein-Cartan (EC) action for parametrized connection $A$ & introduction of torsion
I have some trouble with one missing step when I want to get the teleparallel action from general EC theory, which I am not fully understanding.
The starting form of action (3-Dimensional) is:
$$
S_{...
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0
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Variation of nonminimal derivative coupling term
all. Can I request you assistance about the following problem?
How do I vary this action with respect to metric $\delta g_{ab}$
$$
\int d^4x \sqrt{-g} \Big[\kappa R+ G_{ab}\nabla^a \phi \nabla^b \phi \...
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Timelike normal vector becomes null
I have a metric given by
\begin{equation}
ds^2 = \frac{e^{2 A(z)}}{z^2} \left(-g(z) dt^2 + \frac{dz^2}{g(z)} + dx^2 + dx^2_1 + dx^2_2 \right)
\end{equation}
where $A(z) = -a \ln(b z^2 + 1)$ and $g(z)$ ...
- 404
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62
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Equation of motion for gravity in scalar-tensor theory
I'm trying to derive equation of motion in Higgs scalar-tensor theory with the Lagrangian given by
$$\mathcal{L}=[\frac{1}{16\pi}\alpha \phi^{\dagger}\phi R+ \frac{1}{2}\phi^{\dagger}_{;\mu}\phi^{;\mu}...
- 1
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44
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Derivative of Ricci tensor and Euler-Lagrange equations ambiguity
I'm currently working in a problem about formulating a Lagrangian for Newton-Cartan theory and i'm currently proving if it works.
In order to do this i'm required to compute the derivative of the ...
- 309
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46
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Lagrange multipliers for tensor properties
Suppose a tensor has to be numerically consistent throughout simulation however due to uncertainty in taking derivatives etc elements of the tensor drift away from having required properties. How ...
- 1
5
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3
answers
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Why do we need to make a tensor for the electromagnetic field?
I was wondering why we need the electromagnetic field tensor $F_{\mu\nu}$ to be a tensor and why can't we work with the electric and magnetic fields while dealing with the electromagnetic field ...
- 53
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100
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Variation vs. derivative wrt a symmetric and traceless tensor
Consider a Lagrangian, $L$, which is a function of, as well as other fields $\psi_i$, a traceless and symmetric tensor denoted by $f^{uv}$, so that $L=L(f^{uv})$, the associated action is $\int L(f^{...
- 11
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2
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211
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Can the Lagrangian density of vacuum Maxwell equation be written into tensor contraction without a basis?
The Lagrangian density of the Maxwell equations in vacuum is
$$
\mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} . \tag{1}
$$
My question is, $F$ is a tensor, namely
$$
F = \frac{1}{2}F_{\mu\nu} dx^{\...
- 77
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answers
55
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Rewriting Maxwell Lagrangian [duplicate]
I'm having some problems with rewriting the Maxwell Lagrangian. The text states, \begin{align}\mathcal{L}&=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-A_\mu J^\mu \\
&= -\dfrac{1}{2}(\partial_\mu A_\nu)^...
- 11
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2
answers
117
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How to understand the quadratic form of kinetic energy with $\dot{q}$ coefficients?
Kinetic energy can be written as:
$$ T=\frac{1}{2}\sum_{\alpha=1}^K\sum_{\beta=1}^K a_{\alpha \beta}(q)\dot{q}^\alpha \dot{q}^\beta$$
Where the object $a_{\alpha \beta}$ is a certain tensor. How to ...
- 2,035
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2
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74
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Swap index in Maxwell's Tensor multiplication
Why in
\begin{align} F_{\kappa\lambda}F_{\kappa\lambda} = \left( {\partial _\kappa A_\lambda \partial _\kappa A_\lambda - \partial _\kappa A_\lambda \partial _\lambda A_\kappa - \partial _\...
1
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Calculating conjugate momenta for a spin-2 field
Consider a symmetric spin-2 field $h_{\mu \nu}$. I have the following Lagrangian for this field:
$$\mathcal{L} = - \frac{1}{4}\left(\partial_{\lambda}h_{\mu \nu} \text{ } \partial_{\phi}h_{\alpha \...
- 593
4
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1
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192
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Kalb-Ramond current fall-offs at future null infinity
I can couple the electromagnetic field to a current generated by the complex scalar field for example:
$S=- \int d^4x \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + A_\mu J^\mu$
with $J_\mu = i(\partial_\mu \phi^...
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