All Questions
Tagged with conventions lagrangian-formalism
75
questions
3
votes
1
answer
89
views
Equation for real/complex $\phi^4$ theory
On wikipedia (see this link), the Lagrangians of the $\phi^4$ equation for real AND complex scalar fields are given. One may derive the Klein-Gordon equation by inserting into the Euler-Lagrange-...
- 743
2
votes
3
answers
110
views
Sign conventions for the Lagrangian from the EM Lagrangian density
In Chapter 13.6 of the 3rd edition of Goldstein's Classical Mechanics, Goldstein proposes the Lagrangian density of the electromagnetic field as:
$$\mathcal{L} = -\frac{F_{\lambda \rho} F^{\lambda \...
- 45
-1
votes
2
answers
126
views
Why do we multiply the Euler-Lagrange equations by negative one?
As I've learned classical mechanics from different sources, I've seen both
$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_k} \right) - \frac{\partial L}{\partial q_k} = 0,$$
and
$$\frac{\...
- 183
0
votes
1
answer
90
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The Klein-Gordon equation and the sign of the mass term
A derivation of the Klein-Gordon equation starts with the following lagrangian for a scalar field ϕ:
$$
L=\frac{1}{2}g^{ab}(∇_a\phi)(∇_b\phi)-V(\phi)
$$
If we plug this lagrangian in the Euler-...
- 141
1
vote
0
answers
82
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Diagrammatic derivation of the Dyson–Schwinger equation
I am following chapter 9 of the Rammer's book on Field Theory (which you can find here: https://www-thphys.physics.ox.ac.uk/talks/CMTjournalclub/sources/Rammer.pdf). I am referring to section 9.2.2, ...
- 143
1
vote
0
answers
23
views
Definition of energy for a Lagrangian with constraints [duplicate]
Suppose I have the Lagrangian in spherical coordinates:
$$\mathcal{L} = T=\frac{1}{2}m (\dot{r}^2 + r^2 \sin^2 \theta \dot{\phi}^2 + r^2 \dot{\theta}^2)$$
Where $T$ is the kinetic energy.
Now suppose ...
- 2,599
1
vote
1
answer
69
views
Sign convention for the Lagrangian of a free massive point particle in general relativity
As far as I understand, the Lagrangian of a massive free particle in the context of general relativity is the following:
$$L=-mc\sqrt{g_{\mu\nu}\dfrac{dx^\mu}{dt}\dfrac{dx^\nu}{dt}}.$$
But is this the ...
- 293
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votes
0
answers
61
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Combinatorics in $\lambda \phi^4$ theory and critical exponents
I am trying to understand critical phenomena from the perspective of Statistical Mechanics.
The interacting term in the $\lambda \phi^4$ scalar theory is usually (but not always) multiplied by $\frac{...
- 143
2
votes
2
answers
273
views
Does the Lagrangian have to have units of energy?
I've read that the standard formulation of the lagrangian is $L = T-V$ which has energy units (Joules).
But when I read a bit about electromagnetism and relativity I've seen that the lagrangian is ...
- 25
1
vote
0
answers
46
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On the metric signature and the energy-momentum tensor [duplicate]
Given a Lagrangian $$\mathscr{L} = -\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - V(\phi),\tag{1}$$ is the metric always with signature $(-, +, +, +)$? It seems to me that This post Sign Convention ...
- 298
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0
answers
31
views
Sign of Lagrangian constraint function in rotations
How can I foolproof decide the correct sign of the constraint function in Lagrangian mechanics, particularly in rotations?
In a simple example, lets consider a cylinder rotating around its main axis ...
- 51
1
vote
1
answer
50
views
Convention when considering a mathematical pendulum attached to an object with a spring [closed]
Sorry for the horrible picture, it was the best I could do.
I am trying to find the lagrangian for this set up, but I have problem with the convention of the potential energy, whether it should be ...
- 1,398
1
vote
3
answers
791
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How did we find the Noether current $j^\mu(x) = \bar\psi(x)\gamma^\mu\psi(x)$ for Dirac equation?
The Dirac Lagrangian reads:
\begin{equation*}
\mathcal{L} = \bar\psi(i\not\partial-m)\psi.\tag{1}
\end{equation*}
It's invariant under the transformation $\psi(x)\rightarrow e^{i\alpha}\psi(x)$. Now ...
- 1,783
1
vote
1
answer
89
views
Why do we multiply $(1+\delta)$, but just add $\delta$ to construct counterterms for a Lagrangian?
Consider the Lagrangian
$$
L = \frac{1}{2}(\partial^\mu\phi\partial_\mu\phi-m^2\phi^2)+\bar\psi(i\not\partial-m)\psi-g\phi\bar\psi\psi.
$$
I was told when we include the counterterms, it becomes
$$
L =...
- 1,783
0
votes
1
answer
279
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Lagrangian formalism for boson mass
Why for charged boson, it's mass term is $m^2 W^+_{\mu}W^{-\mu}$? While for neutral boson, it's mass term is $\frac{1}{2}m^2 Z_{\mu}Z^{\mu}$. (Is their a mathematical reason that charged boson mass ...
- 1,421