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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I have a fairly interesting problem concerning rotating reference frames and I need help setting up the required Lagrangian

So, in this question, what coordinate systems should I use to set up the initial position vectors as required? And from there, how can I proceed towards finding the Lagrangian?
MegaBytes's user avatar
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Ward identity in scalar QED; gauge transformations & plane wave solutions for polarization

I am prepping for my QFT2 exam tomorrow, and in one of the mock exams I found the following question (and I'm not quite sure how to go about this). Given the following Lagrangian: $$ L = -\frac{1}{4}...
Sophie Schot's user avatar
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Why are these terms not present in the QED Lagrangian?

I am working though some questions for my QFT/ QED exam and i am having trouble with the following question: Explain why the following terms cannot be part of the Lagrangian of QED: $-g(\bar{\psi}\...
ugur's user avatar
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Classification of equilibrium configurations for particles subject to elastic force constrained on a circle

I am interested in classifying all the possible equilibrium configurations for an arrangement of $l$ equal point particles $P_1, P_2, . . . , P_l$ $(l > 2)$ on a circle of radius $R$ and centre $O$....
ebenezer's user avatar
2 votes
1 answer
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Definition of generalized force in Lagrangian formalism

In some texts (e.g. Taylor's Classical Mechanics), the generalized force is defined to be (I'll simplify to one particle in one dimension for ease of notation): $Q \equiv \frac{\partial{L}}{\partial{q}...
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Connection between Quantum fluctuations and loops in the Feynman diagrams [closed]

I have a request. Please clarify these doubts for me: In the loops in quantum field theory there is a momentum $k$ which is integrated over. In a lecture, Professor Hong Liu says that this free $k$ ...
SX849's user avatar
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Where does the baryon number appear in the Lagrangian of the standard model?

In the standard model Lagrangian, the electric charges of the particles are the coefficients of the interaction terms (e.g. $(-2/3e)u'Au$ for the up quark shows it's charge is $(2/3)e $) How can we ...
KaraboMadisa's user avatar
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How to treat the processes involving propagators of two particles that mix with each other?

Consider a Lagrangian with two scalar particles $V,A$: $$ L =L_{\text{kin}}(V)+L_{\text{kin}}(A)+g_{VA}VA. $$ It looks to me that I can treat the $VA$ term either as a mixing term, diagonalizing the ...
Name YYY's user avatar
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Understanding 2nd-order solution for Euler-Lagrange equation [closed]

I'm currently working on a practice problem involving functional minimization: Given a path $y(x)$ passing through $(0,0)$ & $(1,1)$, I'm trying to minimize the functional $$ I(y)=\int\limits_0^1\...
OldWorldBlues's user avatar
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Residual Symmetry Group after Spontaneous Symmetry Breaking

I am seeking a proof of the following: Suppose we have a theory with $n$ scalar fields $(\phi_1,...,\phi_n)$ such that the Lagrangian $L$ is invariant under the action of some group $G$. However, $G$ ...
Mishary Al Rashed's user avatar
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Equation of Motion Invariance in Galilean Mechanics

Consider a particle moving freely, where $\vec{r}(t)$ is the position of the particle. Suppose I move into a frame with $$\vec{r}' =\vec{r} + \epsilon \vec{F}(\vec{r}, t)\tag{1},$$ where $\epsilon$ ...
CosminA's user avatar
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Derivation of lagrange equation in classical mechanics

I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
Jan Oreel's user avatar
1 vote
1 answer
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Mass dimension of ghost Lagrangian in BRST quantization

It seems from the BRST transformation rules that the ghost fields should be dimensionless: For eg. in the Abelian case in 4D: $$A_{\mu} \to A_{\mu} + d_{\mu}c.$$ Then the ghost Lagrangian density $\...
vvs's user avatar
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Mysterious Action Optimization [closed]

I've been running some simple computational experiments to better understand the principle of stationary action, and have encountered some mysterious examples. Specifically, I'm running an ...
William Lewis's user avatar
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Derivation of Dirac equation in curved spacetime by varying the action

I want to derive the Dirac massless equation in curved spacetime from the action. I have the symmetric form of the Dirac action: $$S = \frac{1}{2} \int \bigg[i\bar{\psi} \gamma^\mu D_\mu \psi - i D_\...
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Low-energy string effective action valid for large dilaton field?

The low-energy effective action of the bosonic string in the critical dimension $D=26$ is given by: $$S=\frac{1}{2\kappa_0^2}\int d^{26}x\sqrt{-G} \left[ \phi^2\left( R-\frac{1}{12}H_{\mu\nu\lambda}H^{...
John Eastmond's user avatar
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How did Fermi arrive at this particular expression of canonical conjugate?

Fermi (Quantum Theory of Radiation 1932), using the electromagnetic energy expression $W_e$, a new variable $v_s$ is derived in equation: $$v_s=\frac{\partial W_e}{\partial \dot{u}_s}$$ which is ...
Jyotishraj Thoudam's user avatar
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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-...
DrD's user avatar
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Conserved current from a symmetry

Good morning. I was reading Tong's Quantum Field Theory course and got stuck on a somewhat stupid step. Essentially, considering the Lagrangian density $$ L = - F_{\mu \nu}F^{\mu \nu} + i \bar{\psi} \...
Gorga's user avatar
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Extending relativistic action for a particle to a relativistic field [duplicate]

The action for a relativistic particle is given by $$S_R = -m\int\sqrt{dx^\mu dx_\mu} = -m \int_{t_1}^{t_2} \sqrt{1-\dot{\mathbf{x}}} dt$$ where I have parametrized using time in the last equality. ...
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Does every field correspond to a particle?

I know that particles in QFT are just excitations of its corresponding field. But is it possible to have a field which cannot generate particles? If yes, what terms must be added to the Lagrangian so ...
Gabriel Ybarra Marcaida's user avatar
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Why impose constraints in (Path Integral) Quantization of Proca action?

I was reading the Wikipedia page on Proca Action. To summarize, it is almost like Maxwell action, but with a mass term because of which Proca action does NOT have gauge invariance. From the equation ...
baba26's user avatar
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Is Principle of Least Action a first principle? [closed]

It is on the basis of Principle of Least Action, that Lagrangian mechanics is built upon, and is responsible for light travelling in a straight line. Is its the classical equivalent of Schrodinger's ...
megamonster68's user avatar
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What is the Lagrangian for the interaction of graviphoton with matter?

There are some models that postulate the existence of graviphoton. What is the Lagrangian for the interaction of graviphoton with matter?
physics_2015's user avatar
5 votes
1 answer
394 views

Dirac Lagrangian in Classical Field Theory with Grassmann numbers

The concept of the Grassmann number makes me confused. It is used to describe fermionic fields, especially path integral quantization. Also, it is used to deal with the classical field theory of ...
Jaeok Yi's user avatar
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1 answer
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Trouble with Lagrangian and Newtonian mechanics [closed]

I'm a pure mathematician and I was doing some physics for fun. I was trying to obtain the equations of motion of a particle moving along a curve $y(x)$ under the effect of gravitational force which ...
Guillermo García Sáez's user avatar
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Correspondence between Feynman diagrams in the $n$-point correlation function expansions for 2 different cutoffs

My understanding of QFT is quite elementary. I'm reading through Kevin Costello's book on Renormalization and effective field theory, which is based on Wilsonian low energy theory. The integral for an ...
Yashasvi Aulak's user avatar
4 votes
1 answer
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Doubts about "whether a given system has a Lagrangian" and "inverse problem of the calculus of variations"

There has been extensive discussion in the literature and on this forum regarding the question of "whether a given system has a Lagrangian" (e.g. post1, post2, post3, and paper1, paper2). ...
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Interpreting $4D$ massive scalar momentum space action as a gauge-field action in 1D?

Consider the following action for massive scalar as follows $$S = \int d^4x \left(-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}m^2\phi^2\right) \tag{1}$$ with Minkowski signature $(-,+,+...
Dr. user44690's user avatar
-3 votes
1 answer
56 views

A least principle action and human behaviour [closed]

Does anyone have an idea how the authors of this paper https://www.nature.com/articles/s41598-021-81722-6#additional-information solved the equation 10? Thank you in advance
physics22's user avatar
1 vote
1 answer
80 views

Schrödinger picture formulation of a velocity-dependent potential of the form $V(x,\dot{x}) = a + bx + cx^{2} + d\dot{x} + ex\dot{x}$

In Shankar Chapter 8, there is a section at the end of the chapter on the path integral formulation for a potential of the form $$V(x,\dot{x}) = a + bx + cx^{2} + d\dot{x} + ex\dot{x}.$$ I follow the ...
Jack's user avatar
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2 votes
1 answer
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Deriving Feynman rules for scalar QED

I am a bit confused about Matthew D. Schwartz's statement of the Feynman rules in scalar QED (chapter 9, section 9.2 titled Feynman rules for scalar QED. The Lagrangian is \begin{equation} \mathcal{L} ...
QFTheorist's user avatar
2 votes
0 answers
36 views

Sigma-Omega model on curved space-time

I'm trying to get the equations of motion for the scalar meson $\sigma$, vector meson field $\omega_{\mu}$ and finally for the nucleons $\Psi=(\Psi_n,\Psi_p)^{T}$ in the sigma omega model on a curved ...
martín canullán's user avatar
1 vote
1 answer
80 views

Hamiltonian density of EM fields

I'm learning about the field theory of electromagnetism. The Lagrangian density for an electromagnetic field can be taken to be $$ \mathcal{L} = -\frac{1}{4} F^{\mu\nu} F_{\mu\nu} + \mu_0 A^\mu J_\mu $...
Bio's user avatar
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General matrix element in weak interaction

During my studies on weak interactions and $\beta$-decay, I've to study how theory and experimental results bring us to achieve the V-A structure of weak vertices. My doubts regard the theory. Suppose ...
Matteo Brini's user avatar
1 vote
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Reparametrization invariance of Einbein action [closed]

I'm going through David Tong's online lecture notes on String theory. At the end of section 1.1.2, where he introduces the einbein action $$S=\frac{1}{2} \int d\tau (e^{-1}\dot{X}^2-em^2),\tag{1.8}$$ ...
Learner667's user avatar
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Is there a Lorentz invariant action for a free multi-particle system?

I want to write down a Lorentz-invariant action of free multi-particle systems. I know that a Lorentz-invariant action for each particle might be expressed as $$ S[\vec{r}]=\int dt L(\vec{r}(t),\dot{\...
watahoo's user avatar
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Equation of motion in conformal gravity theory?

In conformal gravity theory, the action is given by $$L=\int \sqrt{-g}C^{abcd} C_{abcd} d^4x=\int \sqrt{-g}(R^{ab}R_{ab}- \frac{1}{3}R^2)d^4 x.$$ However, the variation of the first term $\int \sqrt{-...
user392063's user avatar
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1 answer
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Variation of action under coordinate transformations

I am currently studying General Relativity from M.P. Hobson's "General Relativity: An Introduction for Physicists" and I had difficulty in understanding some concepts in variational field ...
Ethan's user avatar
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Do particles oscillating in coupled oscillation have only normal modes frequency?

If two bodies are coupled and they are performing oscillations, then do they have only two allowed frequencies (normal modes frequencies) with which they can oscillate or do they have a number of ...
Nikhilesh Singh's user avatar
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1 answer
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Relationship between Hamilton's principle and covariant derivative

The first time I was introduced to the covariant derivative I didn't even realise that was another "kind" of derivative. Following Hamilton's principle taking an action such that: $$ S=\int ...
Álvaro's user avatar
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Electromagnetic interaction of a scalar field before KK reduction

Consider the following gravitational action in $d+1$ accompanied with a complex scalar field $\chi$ with a mass $m$: $$S=\int d^{d+1}x\sqrt{-\tilde{g}}\left[\tilde{R}+\tilde{g}^{\mu\nu}\partial_{\mu}\...
Daniel Vainshtein's user avatar
1 vote
1 answer
50 views

Landau/Lifshitz action as a function of coordinates [duplicate]

In Landau/Lifshitz' "Mechanics", $\S43$, 3ed, the authors consider the action of a mechanical system as a function of its final time $t$ and its final position $q$. They consider paths ...
CW279's user avatar
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$B$-field reducion in the Kaluza-Klein mechanism

Given the following $d+1$ dimensional dilaton-gravity-Maxwell low-energy effective action in the target space of a bosonic string: $$S=\frac{1}{2\kappa^2}\int d^{d+1}x\sqrt{-\tilde{G}}e^{-2\tilde{\Phi}...
Daniel Vainshtein's user avatar
1 vote
0 answers
45 views

How scalar field couples to the $B$-field in Dilaton-Gravity-Maxwell action?

Given the following $d$ dimensional dilaton-gravity-Maxwell low-energy effective action in the target space of a bosonic string: $$S=\frac{1}{2\kappa^2}\int d^dx\sqrt{-G}e^{-2\Phi}\left[R-\frac{1}{12}...
Daniel Vainshtein's user avatar
1 vote
1 answer
62 views

Total derivative of Grassmann variables

From page 21 of "Conformal Field Theory" by Di Francesco, Mathieu, and Sénéchal, the free Fermion Lagrangian is given by: $$L=\frac{i}{2}\psi_i T_{ij}\dot{\psi}_j-V(\psi)$$ Where the $\psi$ ...
QPhysl's user avatar
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Motion around stable circular orbit

Hello I am to solve whether it is possible for body of mass $m$ to move around stable circular orbit in potentials: ${V_{1} = \large\frac{-|\kappa|}{r^5}}$ and ${V_{2} = \large\frac{-|\kappa|}{r^{\...
Optimammal's user avatar
1 vote
1 answer
42 views

Definition of generalized momenta in analytical mechanics

I've seen mainly two definitions of generalized momenta, $p_k$, and I wasn't sure which one is always true/ the correct one: $$p_k\equiv\dfrac{\partial\mathcal T}{\partial \dot q_k}\text{ and }p_k\...
Joan S. Guillamet F.'s user avatar
1 vote
0 answers
45 views

Self-energy of a spinon coupled to a $\rm U(1)$ gauge field

I would like to understand how to calculate the 1-loop self-energy for spinons coupled to a $\rm U(1)$ gauge field. For context, I am going through Nagaosa & Lee's "Gauge theory of the normal ...
dumbpotato's user avatar
1 vote
1 answer
70 views

Does the Lagrangian being invariant under substitution of variables imply a conserved quantity?

Consider the following Lagrangian: $$ \mathcal{L} = \frac{Ma^2\dot\theta^2}{6} +\frac{1}{2}ma^2\left(4\dot\theta^2 + \dot\phi^2 + 4\dot\theta\dot\phi\cos(\theta - \phi) \right) - \frac{a^2k}{2}\left( ...
sconsolato's user avatar

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