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.
5,420
questions
1
vote
0
answers
27
views
Schwinger approach using the heat kernel [closed]
I am having trouble calculating the effective action for a constant magnetic background. The calculations I am trying to replicate are from this paper: https://arxiv.org/abs/hep-th/9807031 . The ...
- 11
0
votes
0
answers
40
views
Equation of motion for $X^{\mu}$ (geodesic equation)
The action for a relativistic particle of mass $m$ in a curved $D$-dimensional is $$\tilde{S}_0=\frac{1}{2}\int d\tau (\dot{X}^2-m^2)$$ for particular gauge and $\dot{X}^2=g_{\mu\nu}(X)\dot{X}^{\mu}\...
- 410
2
votes
1
answer
139
views
Proving that the Lagrangian of a free particle depends only on $|\boldsymbol{v}|^2$
The question is NOT answered by
Deriving the Lagrangian for a free particle,
as the answers therein assume the quadratic dependence, which is what
I am trying to prove. Additionally, while one of the ...
- 23
0
votes
0
answers
18
views
Partial differentiation assumption in development of equation of motion for Lagrangian [duplicate]
In the book "Quantum Field Theory Demystified", David McMahon derives the equation of motion for the Lagrangian:
$$
L=\frac{1}{2}(\{\partial{_u\phi})^2-m^2\phi^2\}
$$
where $ \phi $ is the ...
- 1
1
vote
1
answer
50
views
Lagrangian Formulation in Non-Conservative Systems
I am working in a non-conservative system. Would it make a difference if I
Formulate the Lagrange Equation with an additional term on the right hand side of the equation to account for the Rayleigh ...
- 11
0
votes
2
answers
44
views
Minimizing the action - particle in a potential well
Consider a particle starting at rest in a potential well, the problem is restricted to one dimension.
The lagrangian is $L=T-V$, set $V$ to 0 in the starting location, we get $L(t=0)=0$.
If the ...
- 17
0
votes
0
answers
24
views
Derivative returning zero in dynamic model (SymPy) [closed]
I'm modeling the dynamics of a beam coupled with an oscillator excited by an external force, but when I try to formulate the Euler-Lagrange equation for the beam, the code returns zero for the ...
1
vote
1
answer
50
views
Invariance of the action under a symmetry of 2D isotropic harmonic oscillator
I have a question on the invariance of the action under symmetry transformation.
As the simplest example, here I consider two dimensional Harmonic oscillator. After some rescaling, the Hamiltonian can ...
- 155
2
votes
0
answers
45
views
Propagator for Schwarzian action perturbation
I am reading a paper arxiv:1606.01857. This paper expand the Schwarzian action in equation (4.26)
$$
\tau=u+\epsilon(u).\tag{4.26}
$$
Then the action becomes equation (4.27)
$$
I=\frac{C}{2}\int du[\...
- 71
3
votes
1
answer
86
views
Where this definition $T_{\alpha\beta}=-\frac{2}{T}\frac{1}{\sqrt{-h}}\frac{\delta S}{\delta h^{\alpha \beta}}$ come from?
In the book "String theory and M-theory" by Becker, Becker and Schwarz, the author says that the Nambu-Goto action $$S_{NG}=-T\int d\sigma\, \tau \sqrt{(\dot{X}\cdot X')^2 -\dot{X}^2X'^2}$$ ...
- 410
1
vote
1
answer
72
views
Why $SU(N)$ and not $U(N)$?
I am going through one example where they introduce the lagrangian density
\begin{equation}
L(x) = \sum_{i = 1}^N \partial^\mu \phi_i^* \partial_\mu \phi_i - m^2 \phi_i^* \phi_i = \partial^\mu \Phi^\...
- 21
2
votes
1
answer
61
views
Diagrammatic computations in the theory with spontaneous symmetry breaking
I am looking for an reasonably detailed and instructive exposition of some computations with Feynman diagrams of a theory with spontaneous symmetry breaking (but with no Higgs mechanism). I wonder how ...
- 2,271
2
votes
1
answer
81
views
Rewriting the Euclidean on-shell action of Schwarzschild-AdS
The Schwarzschild-AdS solution is given by
\begin{align}
\label{eq: Schwarzschild-AdS metric}
ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2\,,
\end{align}
where
\begin{align}
f(r)=1-\frac{2MG}{r}-\...
- 329
1
vote
0
answers
36
views
Renormalization on External Field in Effective Potential Formalism
Effective potential formalism provides an intuitive definition of stabe states in quantum field theory, see chapter 11 in Peskin and Schroeder, that is defined at the minimum of effective potential. ...
- 811
0
votes
0
answers
38
views
Calculations Regarding Neutrino Self-Interaction
I have recently been looking at papers talking about neutrino self-interactions and have seen quite different results among papers. One considers the addition of ${\mathcal{L}} \supset \frac{\lambda_{\...
- 181
0
votes
0
answers
45
views
Inverse propagator matrix from Lagrangian
Given a Lagrangian, how do I find the inverse propagator matrix?
For example,
$$
\begin{aligned}
&\mathscr{L}_\pi=\mathrm{D}_0 \pi^* \mathrm{D}_0 \pi-\nabla \pi \nabla \pi^*-\mathrm{b}^2 \mu^2 \pi^...
- 367
1
vote
0
answers
48
views
1PI quantum effective action for free scalar theory [duplicate]
I have a very basic question on quantum effective action. Consider the free scalar field with the action
$$I[\phi]=\int d^4x \frac{1}{2}((\partial\phi)^2-m^2\phi^2).$$
How to compute explicitly the ...
- 2,271
-3
votes
1
answer
71
views
Deriving Euler-Lagrange Equation [closed]
The author is demonstrating how you might derive the Euler-Lagrange equation by minimising the action at a certain point. He substitutes a point $x_8$ into the lagrangian and then differentiates the ...
2
votes
0
answers
50
views
Ambiguity for boundary conditions after conformal transformation
Abbreviations
EOM = Equations of motion
BCs = Boundary conditions
CT = conformal transformation
Intro
I was playing around a bit with EOMs, action principle, CTs and BCs. There, I met a problem. ...
- 877
1
vote
1
answer
50
views
How to calculate the functional derivative of a product?
Let $\omega_{J}^{I}=\omega^{I}_{\nu~J}dx^{\nu}$ be a one-form connection with values in the Lorentz group $SO(3,1)$ and $$B_{J}^{I}=B^{I}_{\mu\nu~J}dx^{\mu}\wedge dx^{\nu}$$ a two-form with values in ...
- 166
1
vote
2
answers
77
views
Identifying Lagrangian as the solution to the variational problem from the Hamilton's principle
We understand that we derive Lagrange equations,
$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\right)-\frac{\partial L}{\partial q_j}=0, \tag{1}$$
starting from d'Alembert's principle of ...
- 341
2
votes
0
answers
83
views
1-loop renormalization for interacting scalar theory - choice of the subtraction point
I would like to clarify some things about interacting scalar fields. In particlar I am having trouble with subtraction points in 1 loop renormalization of the theory.
The lagrangian for 2 interacting ...
3
votes
2
answers
264
views
Why half in Lagrangian density of Klein-Gordon field? [duplicate]
The Lagrangian density for the real Klein-Gordon field, which describes a real scalar field $\phi$ with mass $m$ is given by
\begin{equation}
\mathcal{L} = \frac{1}{2} \partial^\mu \phi \, \partial_\...
- 51
3
votes
2
answers
259
views
Does nature maximize or minimize spacetime curvature?
This question is based on the Einstein-Hilbert action, which states $S=\frac{1}{2\kappa}\int_{ }^{ }R\sqrt{-g}d^{4}x$, where $R$, the Ricci Scalar, is some measure of the spacetime curvature.
The ...
- 396
0
votes
0
answers
40
views
Question about the action function of the electromagnetic field (Landau & Lifshitz)
In Landau's Classical Field Theory, the action function of the electromagnetic field is constructed based on the following facts (Chapter 4,$\S 27$):
$S_f$ is the integration of sum scalar function ...
- 57
0
votes
1
answer
63
views
Does anyone know if a lagrangian term of the form $\mathcal{L}=\frac{g}{4}\bar{\psi}\sigma^{\mu\nu}F_{\mu \nu}\psi$ has a name?
I am currently working in a problem that involes an interaction lagrangian of the form $\mathcal{L}=\frac{g}{4}\bar{\psi}\sigma^{\mu\nu}F_{\mu \nu}\psi$ and i would like to know if it has a name so ...
- 3
0
votes
0
answers
61
views
Why is the gauge field $A_\mu$ real for $\mathrm{U}(N)$ symmetry? [duplicate]
This question is a duplicate of this question asked on Maths S-E
From my notes I have that
The transformation law, $$A_\mu\to MA_\mu M+\frac{i}{g}\left(\partial_\mu M\right)M^\dagger\tag{1}$$ can be ...
- 198
4
votes
1
answer
128
views
In which dimensions is it possible to define supersymmetric actions?
Susy gauge theories only exist in dimensions 3,4,6,10. This is stated by Baez and Huerta as:
Nonabelian Yang-Mills fields minimally coupled to massless spinors are
supersymmetric if and only if the ...
- 1,861
0
votes
0
answers
47
views
Renormalization verse dimensional regularization [duplicate]
I am trying to learn QFT for fun, well past college age and have never sat in a college classroom. I understand counter terms, and how they re-normalize the Lagrangian fairly well. I understand a ...
0
votes
0
answers
34
views
How to compute the derivative of action with respect to magnetic field?
In Magnetohydrodynamics, the Lagrangian density can be written as [1]:
$$\mathcal{L}=\mathcal{L}_0-\frac{|\mathbf{B}|^2}{2\mu_0}+\boldsymbol\Gamma\cdot\left(
\frac{\partial\mathbf{B}}{\partial ...
- 81
4
votes
0
answers
66
views
Why do we treat quadratic counterterms as interactions rather than free terms? [duplicate]
For example, in QED (or any other theory containing particles with spin 1/2) the renormalized countains
$$
\mathcal{L} \subset \bar{\psi} (i \displaystyle{\not}\partial -m)\psi + \bar{\psi} (\delta_2 ...
- 1,925
0
votes
1
answer
86
views
Definition of global supersymmetry on curved spacetimes and use of constant spinor fields
According to "Quantum Fields and Strings: A Course for Mathematicians" (Deligne et al.), we call theories supersymmetric if their action functional is invariant under the action of a super ...
- 359
1
vote
0
answers
33
views
Problem with Dirac-Bergmann algorithm [closed]
This is from Lemos, Analytical Mechanics, Problem 8.24.
Consider the Lagrangian
$$L = (\dot{y}- z)(\dot{x} - y).$$
I found the Lagrangian equations of motion to be
$$\ddot{y} = \dot{z},$$
$$\ddot{x} = ...
4
votes
1
answer
116
views
Boundary conditions on Schwarzschild event horizon
Consider the variational problem for a scalar field in Schwarzschild spacetime $M$ with respect to Eddington-Finkelstein coordinates $(v,r, \theta, \varphi)$, i.e.
$$\delta I(\phi) = \int_M dV \ \Big(...
- 877
1
vote
0
answers
42
views
About Yo-yo motion and forces constraint
The purpose of the Euler-Lagrange equation, is supposed to enable us to describe a system with the fewest possible coordinates, using generalized coordinates instead of traditional ones.
However, in ...
2
votes
1
answer
136
views
On the Background Independence condition
In General Relativity, one has that the equations of motion for any matter distribution are given by extremizing the following action:
\begin{equation}
S[g] = \int\left[\frac{1}{8\pi}(R - 2\Lambda) + ...
- 477
1
vote
0
answers
35
views
How to derive D brane effective action? The result is super Yang-Mills action
It is said that the effective action on $N$ coincident D branes is $U(N)$ super Yang-Mills (SYM) action with 16 super charges.
On D3 branes it is $4d,\mathcal{N}=4$ SYM
$$
L = -\int d^4x. Tr[ \frac{1}{...
- 128
2
votes
2
answers
84
views
Relativistic Lagrangian for a System of Massive Particles
The standard Lagrangian $L$ written in local coordinates for a free, relativistic particle of mass $m > 0$ is given by
$$L(q, \dot{q}) = -m \sqrt{-g_{\mu\nu}(q) \dot{q}^\mu \dot{q}^\nu}\tag{1}$$
...
- 348
0
votes
0
answers
32
views
Proca lagrangian equations of motion [duplicate]
In his book Quantum Field Theory and the Standard Model, Matthew D. Schwartz postulates the Lagrangian for a massive spin 1 field (eq. 8.23):
\begin{equation}
\mathcal L=\frac{a}{2}A_\nu \partial_\mu\...
- 1,268
2
votes
1
answer
60
views
Deriving an explicit form of the boundary term in the symmetry variation of action
In many treatments of Noether theorem [e.g. Srednicki's book eq. (22.27), Banados' review eq. (2.65), David Tong's notes on QFT eq. (1.38), Peskin & Schroeder eq. (2.12), etc.] an almost explicit ...
- 170
6
votes
3
answers
914
views
Conservation of energy in a mechanical system with a discontinuous potential function
I've started reading Landau-Lifshitz Mechanics, and I'm having trouble with the problem at the end of section 7.
A particle of mass $m$ moving with velocity $\mathbf{v}_1$ leaves a half-space in ...
- 63
1
vote
1
answer
69
views
Building fields from the Kallen-Lehmann representation formula
The standard QFT textbook presentation is usually such that the Lagrangian/theory is given first and then things like the 2-point function, propagator, and Kallen-Lehmann representation are derived as ...
- 3,542
0
votes
2
answers
53
views
Doubt when considering the signs for the Lagrangian for a charged particle in an electromagnetic field
This is a trivial question probably, i have a doubt when considering the signs for the Lagrangian for a charged particle in an electromagnetic field.
Considering that the Lagrangian for a charged ...
- 1
2
votes
1
answer
70
views
Deriving Beltrami identity when there are multiple dependent functions
I'm trying to understand this solution to the brachistochrone problem inside a uniform sphere. Going from equation (19) to (20) and (21), the integrand of the functional we're trying to minimize is of ...
- 23
-2
votes
3
answers
121
views
When computing the Euler–Lagrange equations, why do we assume the coordinates do not depend on time?
I've just started to learn lagrangians through this video and I'm a bit confused. The setup has that $L = T-V$. With $T=\tfrac{1}{2}mv^2$ and $V=mgx$. So, $L= \tfrac{1}{2}m(dx/dt)^2-mgx$. This is all ...
- 123
4
votes
1
answer
117
views
Commutation of the functional derivative for the 1PI effective action and off-diagonal propagators in field space
Given a scalar real Yukawa theory,
\begin{equation}
L = \varphi_1G_0^{-1}\varphi_1 + \varphi_2 G_0^{-1} \varphi_2 - \frac{\lambda}{2}\varphi_1 \varphi_2^2\tag{1}
\end{equation}
of the two fields $\...
- 113
3
votes
4
answers
678
views
The proof of conservation of momentum in Mechanics by Landau and Lifshitz
In reading the first chapter of Mechanics by Landau and Lifshitz, there is one point on which I consistently get stuck. This regards the proof in $\S 7$ that space homogeneity implies conservation ...
- 33
4
votes
1
answer
59
views
Need help understanding lagrangian applied to mass on spring problem [closed]
I am interested in learning physics, and I was trying to learn about the Lagrangian method for classical mechanics through https://scholar.harvard.edu/files/david-morin/files/cmchap6.pdf, but I ...
- 43
1
vote
1
answer
79
views
How can you have a potential in a theory without any forces?
Consider the typical Lagrangian:
$$L=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) - V(\phi).$$
I interpret the above (please correct me) as a theory consisting of a field which can move through ...
- 31
-2
votes
1
answer
88
views
Deriving Feynman rules for QED using the path integral
The Lagrangian for QED is
$$\mathcal{L} = \bar{\psi}(i\displaystyle{\not}\partial -m)\psi - \frac14(F_{\mu\nu})^2 - e\bar{\psi}\gamma^\mu\psi A_\mu = \mathcal{L}_0 - e\bar{\psi}\gamma^\mu\psi A_\mu.$$
...
- 3,542