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.
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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 ...
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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" ...
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Lagrangian of a Josephson junction
The lagrangian density of the system, as given in Squeezed-state generation using a Josephson parametric amplifier, B Yurke, is given by
$$ L(x) = \delta(x)\ \frac{\hbar}{2e} [I_c Cos(\phi) + \phi \...
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1
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Lagrangian in a system with a specific velocity dependent potential
I have a system of a particle moving under the generalized central potential
$$
V= \frac{1}{r}(1+\dot{r}^2) \tag{1}
$$
The general Euler-Lagrange equations for such type of potentials are:
$$
\frac{...
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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 ...
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1
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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
$$
\begin{...
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1
answer
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Determine canonical fields of action
I'm working on an exercise which asks me to determine the canonical fields, and their equations of motion, of this invariant action:
$$
S = \int d\tau \sqrt{g_{\tau\tau}}\left( \frac{\tilde m}{2} g^{\...
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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 ...
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4
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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 ...
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Derive Linearized Einstein's equation from Lagrangian approach
Given the Hilbert action:
$$
S_{H}=\int \sqrt{|g|}R d^{n}x
$$
and the metric written in terms of Minkowski and perturbed metric:
$$
g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}.
$$
I am able to derived the ...
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Noether's theorem derivation by Greiner
I'm reading Quantum Mechanics (Symmetries) by Greiner, in the topic of Noether's theorem (pp. 6-7) there are points where it is a little bit confusing. I'll add a link to the google book version so as ...
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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.
$$\mathcal{L} = \frac{1}{2}(\partial_\...
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Second-order perturbation in Brans-Dicke gravity
Let be $g_{\mu \nu} = \eta_{\mu\nu}+h_{\mu\nu}$ the perturbation of the metric and $\phi=\phi_0 + \varphi$ the perturbation of a field.
The lagrangian of a scalar-tensor theory of gravity is:
\...
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Can you rewrite the QCD lagrangian in terms of hadron?
Is it possible to (exactly) rewrite the QCD lagrangian in terms of hadrons?
I get that it's probably practically too difficult to do, but would it be possible in principle?
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Fourier expansion of positive and negative fields in In-In formalism
Recently, I am reading articles regarding In-In formalism, Schwinger-Keldysh formalism.
One advantage of this formalism is it is easy to construct the expectation values of operators in-state without ...
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2
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What properties make the Legendre transform so useful in physics?
The Legendre transform plays a pivotal role in physics in its connecting Lagrangian and Hamiltonian formalisms. This is well-known and has been discussed at length in this site (related threads are e....
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Two constraints of $\bar\psi$ from equations of motion for Free Dirac Field Lagrangian
$$\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$$ and its adjoint ...
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2
answers
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Transformation of coordinates in Noether's Theorem
I am confused, in the proof of Noether's theorem, by the change of boundary in the action integral during the transformation of coordinates. I have seen on Wikipedia
that along with the change of ...
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answer
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Index position when varying an action with respect to the metric
I'm confused about where we should put tensor indices when we vary an action wrt the metric. For example, if I have in the Lagrangian a term such as
$$
A_{\mu\nu}B^{\mu\nu},
$$
do I necessarily have ...
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Effective action of interacting electron gas (Altland & Simons derivation)
I have a question re: equation (6.6) in Altland and Simons, which claims that the effective action of an interacting electron gas takes the form
$$S\propto \sum_q \phi_q\left( \frac{\mathbf{q}^2}{4\pi}...
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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 ...
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2
answers
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Wilsonian RG and Effective Field Theory
I'm having trouble reconciling the discussions of the Wilsonian RG that appear in the texts of Peskin and Schroeder and Zee on the one hand, and those of Schwartz, Srednicki, and Weinberg on the other....
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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 ...
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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 ...
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0
answers
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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\...
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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 ...
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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 ...
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2
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Lorentz invariance of the Klein-Gordon equation action
What I will say is not exclusively true for the KG equation, but let's take it as a simple example.
When proving the invariance of its action under a Lorentz transformation, it suffices to show that ...
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Ball rolling in a cylindrical trough
I am trying to understand an interesting effect I observed while playing with my kids' toys (video). The energy in this system seems to slosh back and forth between the trough and the ball, with the ...
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Local Lorentz invariance of General Relativity
Consider the gravitational action as an integral over a differential 4-form $$ S = \int_{\mathcal{M}} \star F_{ab}\wedge e^a \wedge e^b$$ where $\star F_{ab} = \epsilon_{abcd} F^{cd}$ and $F$ is the ...
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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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Lorentz/rotational invariance parameter doesn't vanish on boundaries
As I know Stress-Energy tensor is defined as Noether current under arbitrary coordinate transformations $\boldsymbol{x} \rightarrow \boldsymbol{x}+\epsilon(\boldsymbol{x})$.
$$
\delta_\epsilon S=\int_{...
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On a trick to derive the Noether current
Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.
The trick to get the Noether current consists in making the ...
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2
answers
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Does there exist a square root of Euler-Lagrange equations of a field? (Factorization)
Does there exist a square root of Euler-Lagrange equations $\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0$ in the sense that $(x+...
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1
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Virtual displacement in semi-holonomic constraints
I am currently studying Lagrangian Mechanics for systems whose constraints equations have the form $$\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_{\...
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2
answers
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About virtual displacement
Thornton Marion
The varied path represented by $\delta y$ can be thought of physically as a virtual displacement from the actual path consistent with all the forces and constraints (see Figure above)....
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Angular momentum conservation reduces degree of freedom
In 2 dimention dynamics, if angular momentum is conserved: mr^2(theta dot)=constant, does that mean degree of freedom is reduced from 2 to 1?
I think it should since r and theta(although theta dot ...
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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 ...
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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:
...
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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 ...
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1
answer
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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 $\int e^{\frac{1}{\hbar}f(x)} dx$ ...
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1
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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,...
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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}=\...
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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-...
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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.
$$
\begin{aligned}
\...
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What is the physical significance of the vector field term $X_{\nu}$ in the improved Noether current $T^{\mu\nu}X_{\nu}$?
In Pedro Lauridsen Ribeiro's answer to deriving the improved stress-energy tensor using the improved Noether current, the variational equation for the improved stress-energy tensor is given by:
\begin{...
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Feynman rules for Majorana fermions?
I want to deduce the Feynman rules for neutrino magnetic moment, and there are Dirac and Majorana terms for this
$$\mathcal{L}_D=\mu_{ij}\bar{\nu}_{iL}\sigma_{\alpha\beta}\ \nu_{jR}F^{\alpha\beta}$$
$$...
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1
answer
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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}
\...
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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\...
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2
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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 ...
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