Questions tagged [lagrangian-formalism]
For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.
4,734
questions
2
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1
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Calculations with co- and contravariant formalism in QFT
i have another question regarding calculations with the co- and contravariant formalism in QFT. It is not that i don't understand all of this, but most of the time i'm missing some "middle" ...
-3
votes
0
answers
53
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Is the angular momentum conserved? 2 [closed]
I have a Lagrangian equation and an expression for the generalized momenta, if I put the generalized momenta into the Euler-Lagrange equation and I get a differential equation as result, does that ...
- 11
0
votes
1
answer
149
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What is the full QED Lagrangian with physics units written out?
I wonder what the QED Lagrangian would look like if you carefully write out all units of the terms and make sure they are consistent. I think there is something missing about Coulomb.
Can you write ...
- 1,397
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vote
1
answer
36
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Peskin and Schroeder, Linear sigma model, renormalized perturbation theory
On Peskin & Schroeder's QFT pages 353-355, the book uses the Linear sigma model to illustrate the renormalization and symmetry.
We can write the Lagrangian of Linear sigma model with
$$
\begin{...
- 1,121
2
votes
0
answers
60
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Semi-classical limit of Feynman path integral
I am reading Blau's note on The Path Integral Approach to Quantum Mechanics. I am troubled for the derivations of semi-classical limit of Feynman path integral, which is located on Page.50 of Blau's ...
- 1,121
-1
votes
0
answers
23
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For the Yukawa theory of massless fermion with vector and tensor couplings, find the beta functions of the couplings using renormalization [closed]
Here is the Lagrangian density given. I have to use renormalization to find the beta functions of the vector and tensor couplings of the given Lagrangian density.
$$\mathcal{L} = \frac{1}{2}(\partial_\...
2
votes
0
answers
27
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Fourier expansion of positive and negative fields in In-In formalism
Recently, I am reading articles regarding In-In formalism, Schwinger-Keldysh formalism.
One advantage of this formalism is it is easy to construct the expectation values of operators in-state without ...
- 3,288
2
votes
4
answers
195
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Why do we put factors of zero in a Lagrangian that is to be extremized?
According to the Wikipedia page on Lagrange multipliers under the section - Example 3: Entropy, it is written that:
$$f(p_1,p_2,\ldots,p_n) = -\sum_{j=1}^n p_j\log_2 p_j$$
For this to be a ...
- 609
0
votes
1
answer
37
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Two constraints of $\bar\psi$ from equations of motion for Free Dirac Field Lagrangian
$$\mathcal{L}=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi,$$
taking Euler-Lagrange equation on $\bar\psi$ gives the more familiar Dirac equation
$$(i\gamma^\mu\partial_\mu-m)\psi=0$$ and its adjoint ...
- 523
1
vote
0
answers
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Second-order perturbation in Brans-Dicke gravity
Let be $g_{\mu \nu} = \eta_{\mu\nu}+h_{\mu\nu}$ the perturbation of the metric and $\phi=\phi_0 + \varphi$ the perturbation of a field.
The lagrangian of a scalar-tensor theory of gravity is:
\...
- 21
0
votes
0
answers
14
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Hoop and Pulley lagrange [closed]
With respect to the problem raised, I have doubts if obtaining the characteristic equations are appropriate, or I made a mistake in some step to obtain them, since I have had problems all week to be ...
-1
votes
0
answers
15
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Equation of motion for scalar field in Brans-Dicke Theory [closed]
The action is given by
S=∫▒〖d^4 x〗 √(-g) {(F(φ))/2 R-1/2 ∇_c φ∇^c φ-V(φ)}+S_m
I am trying to vary with respect to ϕ
using Euler - Lagrange equations in curved spacetime,
to get this
▢φ+3φ ̇H+ V_φ=1/2 ...
1
vote
0
answers
38
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How do equations of motion in BF theory imply triviality of powers of observables?
Following the lectures of Nathan Seiberg at PiTP in 2015 https://www.youtube.com/watch?v=pqgNrVTQ4yM&t=666s, consider $U(1)$ BF theory in 2D
$$S(B,A)=\frac{n}{2\pi}\int_\Sigma B\text{d}A,$$
and ...
- 3,665
-1
votes
0
answers
25
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About the Classical Scalar Field Lagrangian in Flat Space FLRW Spacetime
So the action for a scalar field in spacetime is typically given as:
$S[\phi]=\int dx^4 (\frac{1}{2}\partial^\mu\phi \partial_\mu \phi - V(\phi))$, thus
$\mathcal{L}[\phi] = \frac{1}{2}\partial^\mu\...
- 79
2
votes
1
answer
142
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Geometrical intuition for Noether's Theorem
I have been reading some questions about the relation between Noether's Theorem and Lie Algebras and I wanted to get some intuition on it, but I didn't find what I really wanted. Also, the majority of ...
- 154
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votes
0
answers
31
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Effective action of interacting electron gas (Altland & Simons derivation)
I have a question re: equation (6.6) in Altland and Simons, which claims that the effective action of an interacting electron gas takes the form
$$S\propto \sum_q \phi_q\left( \frac{\mathbf{q}^2}{4\pi}...
- 227
1
vote
1
answer
30
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How to find Belinfante-Rosenfeld SEM tensor?
Using definition of SE tensor as a response to the infinitesimal coordinate change
$$\delta_{\epsilon} S=\int\partial_{\mu}\epsilon_{\nu}T_{\mu\nu}d^Dx;\quad \partial_{\mu}T_{\mu\nu}=0;\quad (7)$$
We ...
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2
votes
1
answer
58
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Lorentz/rotational invariance parameter doesn't vanish on boundaries
As I know Stress-Energy tensor is defined as Noether current under arbitrary coordinate transformations $\boldsymbol{x} \rightarrow \boldsymbol{x}+\epsilon(\boldsymbol{x})$.
$$
\delta_\epsilon S=\int_{...
- 93
1
vote
1
answer
55
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Virtual displacement in semi-holonomic constraints
I am currently studying Lagrangian Mechanics for systems whose constraints equations have the form $$\sum_{k=1}^na_{\ell k}(q,t)\dot{q}_k+a_{\ell t}(q,t)=0\tag{1}$$
or, equivalently
$$\sum_{k=1}^na_{\...
- 154
2
votes
2
answers
60
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Why rescale the kinetic term in Wilsonian renormalization?
I have been doing some reading on Wilsonian renormalization and also Effective Field Theories.
It's my understanding, and I could be wrong, that part of the process is to continually rescale the ...
- 121
1
vote
1
answer
36
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Is the divergence of the energy tensor related to the equations of motion?
Given a Lagrangian $L[g,\phi]$ we can define its energy tensor as $T=\frac {\delta L}{\delta g}$ and ihe equations of motion for the field $\phi$ are $\frac{\delta L}{\delta \phi} =0$. For the wave ...
- 166
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votes
0
answers
29
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What is wrong with my single Nose-Hoover thermostat?
I am trying to implement a single Nose-Hoover thermostat inside of my leapfrog velocity verlet algorithm in Python. This is what I have so far:
...
- 57
0
votes
1
answer
38
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How Feynman's path integral lead to least action principle? Math proof needed [duplicate]
I have read about Feynman path integral which leads to classical limit.
It said that because $\hbar \rightarrow 0$ in classical view. The function of path integral $\int e^{\frac{1}{\hbar}f(x)} dx$ ...
3
votes
2
answers
250
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Does there exist a square root of Euler-Lagrange equations of a field? (Factorization)
Does there exist a square root of Euler-Lagrange equations $\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0$ in the sense that $(x+...
- 31
0
votes
1
answer
48
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Interpretation of $\phi^n$ terms in Lagrangian density
Why in QFT are $\phi^n$, where $n>2 $, terms in your lagrangian density interpreted as interaction terms?
so $\phi^4$ is considered a self-interaction term.
Similarly for two different fields $\phi,...
- 370
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vote
0
answers
90
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Can we call it "quantization" when we specify Hilbert space and operators to write a classical field theory into a quantum theory?
Can we call it quantization when we specify Hilbert space and operators to write a classical field theory into a quantum theory? Suppose there is a single spin 1/2 system with Hamiltonian $\hat{H}=\...
- 21
1
vote
2
answers
82
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Photon propagator and the Fermi Lagrangian density
I'm stuck with the photon propagator, at chapter 5 of Mandl and Shaw QFT book.
They say that since the Maxwell Lagrangian density for the free Electromagnetic field has a conjugate momenta to the ...
0
votes
1
answer
84
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Understanding this abstract Lagrangian of effective field theory
I'm learning Wilson's approach to renormalization and the Effective Field Theory. Typically, the theory is defined by a Lagrangian valid up to some scale $Λ$. I saw these two definitions for 4-...
- 1,305
0
votes
0
answers
47
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Feynman rules for Majorana fermions?
I want to deduce the Feynman rules for neutrino magnetic moment, and there are Dirac and Majorana terms for this
$$\mathcal{L}_D=\mu_{ij}\bar{\nu}_{iL}\sigma_{\alpha\beta}\ \nu_{jR}F^{\alpha\beta}$$
$$...
0
votes
1
answer
63
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What is the physical significance of the vector field term $X_{\nu}$ in the improved Noether current $T^{\mu\nu}X_{\nu}$?
In Pedro Lauridsen Ribeiro's answer to deriving the improved stress-energy tensor using the improved Noether current, the variational equation for the improved stress-energy tensor is given by:
\begin{...
- 1
0
votes
1
answer
72
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Second-order Lagrangian of Einstein-Hilbert action
I'm having trouble deriving the equation (44) of https://arxiv.org/abs/1710.08863 .
The question is how to get the second-order lagrangian of the Einstein-Hilbert action, i.e.
\begin{equation}
\...
- 21
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votes
1
answer
37
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Proof that the axial current is conserved in classical QED
I am trying to use the Lagrangian of QED (without kinetic terms for photons) to prove that the axial current of QED satisfies $\partial_\mu j^\mu_5 = 2im\bar\psi\gamma^5\psi,$ where $j^\mu_5 = \bar\...
- 643
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1
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92
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Vertex factors for Feynman rules in QCD
I am stuck in deriving the vertex factor for the Feynman diagrams for the QCD Lagrangian. For the quantum Yang Mills theory we will have the following interacting Lagrangian.
$$
\begin{aligned}
\...
- 175
4
votes
2
answers
339
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How do I understand the Hodge $⋆$ operator in Yang-Mills Lagrangian?
The gauge-invariant part in Yang-Mills Lagrangian is
$$
\mathcal{L}_{\text{gauge}} = -\frac{1}{2}TrF_{\mu\nu}F^{\mu\nu} = -\frac{1}{4}F_{\mu\nu}^aF^{a, \mu\nu}.
$$
Sometimes I see the lagrangian ...
- 1,305
-1
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0
answers
49
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The Einstein varied action
Have I used the variation of the metric properly in the following? I've set the constant $k=1$. Using the product rule for variations on the action I get:
$$\delta A = \int d^4x\ \left[\delta\mathcal{...
-2
votes
1
answer
73
views
Finding equation of motion for given Lagrangian with respect to metric
Given the following action in $d$ dimensional $(0,1,...,d-1)$ curved spacetime:
$$ S= \int d^dx\sqrt{-g}\mathscr{L}[\chi,\Phi,g^{\mu\nu}] $$
Where:
$$\mathscr{L}=e^{-2\Phi} \left(-\frac{1}{2\kappa^2}[...
2
votes
1
answer
53
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Connection between definitions of "conjugate momentum density" as "generator of displacement of the field" and as "Lagrangian partial derivative"
I am reading Jakob Schwichtenberg Physics from Symmetry where in 5.2 conjugate momentum density $\pi(x)$ is defined as generator of displacement of the field itself (1):
$$
\pi(x) = −i\hbar\frac{\...
- 21
0
votes
1
answer
58
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Writing the Euler-Lagrange equation for variation of an action with respect to metric using only the Lagrangian
Given Lagrangian which dependent on collection of fields ${\phi^a},a=1,...,N$
and on a tensor metric $g^{\mu\nu}$ such that the action in $d$ dimension which describes the system is $$S=\int d^dx \...
1
vote
1
answer
32
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Spherical Potential and Angular Momentum Conservation
I have always found it clear that since a spherical potential has all components of angular momentum conserved since the entire system is symmetric under any rotation. However, I was trying to prove ...
2
votes
1
answer
72
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Contribution of Counterterm Lagrangian to $n$-point ($n>4$) correlators
I am learning renormalisation in QFT and I had a question regarding the counter-terms we put in the Lagrangian. For the purpose of this question I shall consider a $\phi^4$ theory in 4D. The "...
- 672
-2
votes
1
answer
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Recovering the Newtonian lagrangian for a point particle from its stress-energy tensor in GR?
The stress energy tensor is related to the matter Lagrangian:
$$ T_{\mu \nu} = - 2 \frac{\partial (L_M \sqrt{-g})}{\partial g^{\mu \nu}} \frac{1}{\sqrt{-g}}.$$
Now, the stress energy tensor of a point ...
- 1,039
2
votes
1
answer
62
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Relationship between Lagrangians describing a particle interacting with a scalar field
In Susskind's Particles and Fields lecture, he considered the Lagrangian obtained by considering a particle and the effects of a scalar field $\phi(t, x)$ with coupling constant $g$ on the particle (...
1
vote
1
answer
69
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Deriving smoothing kernels
I'm watching a video on smoothed particle hydrodynamics it just blindly claims that these smoothing kernels are pretty good.
$$W(r-r_b,h)\equiv\dfrac{315}{64\pi h^9}\left(h^2-|r-r_b|^2\right)^3$$
$$\...
- 210
0
votes
1
answer
122
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Geodesics: Energy functional vs length functional
The Wikipedia article about geodesics talks about the equivalence of obtaining the geodesic by either minimizing the length functional $L$, or by minimizing the energy functional $L^2/2$, cf. the Phys....
1
vote
0
answers
57
views
Dirac's pseudo-energy tensor
Having trouble with Dirac's eq. (31.3) ("General Theory of Relativity"). Probably a simple math question, but I need to provide a little background on the symbols.
Dirac defines the pseudo-...
- 173
1
vote
0
answers
62
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Reference request for QFT $SO(3)$ non-linear sigma model
I was wondering if anyone has a reference that could help me understand quantum field theories that have a nonlinear configuration space. For example, from classical mechanics if we have a three-...
1
vote
2
answers
106
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Why does the Lagrangian not show particle-interaction? Why are normal/tension forces not considered?
(1) For formulating a lagrangian for a system of particles compared to one free particle, we start with the kinetic energy and formulate a potential energy term that is in terms of each of the radius ...
- 13
1
vote
0
answers
43
views
Coleman-Weinberg mechanism at two-loop
I'm trying to understand how to perform the CW mechanism (http://www.scholarpedia.org/article/Coleman-Weinberg_mechanism) to scalar theories at two-loop order. More specifically how to find the ...
- 371
1
vote
0
answers
57
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From EH action to Newtonian Mechanics action? [closed]
How does one start from the Einstein Hilbert action go to the action of a (point particles + some field) in special relativity and then Newtonian mechanics for a local neighborhood around a point for ...
- 1,039
0
votes
1
answer
56
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Calculating the Generalized force with and without the lagrangian
In my mechanics class, I learned that the components of the generalized force, $Q_i$, could be calculated using:
$$\begin{equation}\tag{1}Q_i = \sum_j \frac{\partial \mathbf{r}_j}{\partial q_i}\cdot \...
