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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36 views

Noether's theorem and energy in four-momentum

In Newtonian physics, momentum and energy are often treated as distinct entities, which happen to be separately conserved. In relativity, energy is regarded as the "time" component of the ...
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Lorentz-invariant Lagrangian for spinor field

Schwartz book on QFT (page 167), Zee book on group (page 461) and Maggiore book on QFT (page 55), prove that $\psi_R^{\dagger}\sigma^{\mu}\psi_R$ is a 4-vector, where $\psi_R$ is a right-handed spinor ...
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67 views

Intuition about non-invariance of the Hamiltonian in canonical transformation

Suppose $q={\{q_i\}}_{i=1}^n$ is the set of generalized coordinates of a dynamical system. $L(q,\dot q,t)$ is the Lagrangian of the system. Now we make coordinate transformation $Q_i=Q_i(q,t)$. Then ...
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Torsion-free Einstein-Cartan action and Hilbert-Einstein action equations of motion equivalence

Let $e^\alpha_{\ \mu}$ be the tetrad i.e. \begin{equation} g_{\mu\nu} = \eta_{\alpha\beta}e^{\alpha}_{\ \mu}e^{\beta}_{\ \nu} \end{equation} I'm denoting the internal indices using greek letters $\...
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84 views

Higher-order derivatives than second-order differential equations

From https://doi.org/10.1063/1.2155755 he limited himself to second-order differential equations. Our experience in elementary-particle physics has taught us that any term in the field equations of ...
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1answer
60 views

Noether's Theorem and Poynting Theorem

For simplicity, let's assume the Lagrangian formulation of Noether's theorem, that is, our equations of motion can be derived from the Euler-Lagrange equations, or, simply, that we can use a ...
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1answer
104 views

Action perturbation vs Equation of motion perturbation

I have a simple question which has been in my mind for some time and I would be thankful if anyone help me to fix it. Consider the following action : \begin{equation} S=\int\!dt\,({\textstyle\frac12}\...
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37 views

Relative signs in interaction terms of the Standard Model Lagrangian

I'm trying to understand how to decide if a minus sign is to be put in front of a vertex of interaction when dealing with Feynman diagrams at lowest order. At first I thought that you just take the ...
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1answer
46 views

Why is the "$\phi$" equation not differentiated with the affine parameter in Schwarzschild case?

The equation for motion determined by the Lagrangian formulation takes the form: $$L^2 = g_{ab}\frac{dx^a}{dm}\frac{dx^b}{dm}$$ We arrive, for the $\phi$ coordinate in $(t, r, \theta , \phi)$ system ...
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84 views

Connection between the Least Action principle and the Schwarzschild's metric in General Relativity

I'm looking for some simple paper/textbook explaining the fact that time dilation in the Schwarzschild's metric can be translated into the classic equation of motion in a gravitational field. That is: ...
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What are the physical reasons for considering the Maxwell invariant ${\cal G}=E.B$ in the action?

As I know, the Maxwell invariant ${\cal G}=E.B$ is the fundamental invariant such as ${\cal F}=\frac{1}{2}\left( {{{\bf{B}}^2} - {{\bf{E}}^2}} \right)$ that can be used to construct all possible ...
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Making point transformations of the coordinates of the whole universe [closed]

In theoretical mechanics, we make a transformation from regular coordinates to general coordinates, and then we make a coordinate space, using these independent general coordinates on the ...
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Decomposing Lagrangian into CM and relative parts with presence of uniform gravitational field

Most problems concerning two-body motion (using Lagrangian methods) often only consider the motion of two particles subject to no external forces. However, the Lagrangian should be decomposable into ...
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49 views

Metric variation and functional chain rule

I want to take the metric variation of a functional which depends explicitly and implicitly on the metric \begin{equation} F[g,G[g]]. \end{equation} How will the corresponding chain rule be defined ? ...
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Variational derivative of $\Phi_a(-\partial^2 - m_0^2 - \Sigma)\Phi_a$

Let me refer to the below link http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-17-phi4on.pdf In Eq: 18.40, $\Gamma[\Phi_a, \Sigma]$ is given as, $\Gamma[\Phi_a,\Sigma] = NA_{coll}[\Sigma]...
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How did Lagrange derive this Hamilton-like equation in Mechanique analytique

I was trying to understand how Hamiltonian formulation was derived in the history, and read Mechanicque Analytique, and found these pages from section V of this book, which derived the Hamiltonian-...
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97 views

Why do we use two ways to write the kinetic term in a Lagrangian?

I have just started reading Schwartz's book on QFT and I see from the first few chapters that he writes the kinetic part of the Lagrangian in a way I find strange. As an example, for the massless ...
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Lagrangian in parabolic co ordinates given by $x= u*v$ and $y= (u^2+v^2)/2$

I found 2 equations of motion in $v$ and $u$ for $L=m/2*[x^2+a*y^2+w^2(x^2+y^2)]$ where $a,w$ are constants so the phase space has the generalized axes $u, v $. How does the parabolic coordinate work?...
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High wire artist

We want to derive a Lagrangian equations for a high wire artist which uses a balancing rod for stability purposes. We know the masses and the polar moment of inertia of the artist and the rod.
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49 views

Kerr metric geodesics using EL equations

Let there exist a Kerr metric $ds^2$ in Boyer–Lindquist coordinates such that $$ds^2=-\left(1-\frac{r_sr}{\Sigma}\right)dt^2+\left(\frac{\Sigma}{\Delta}\right)dr^2+\Sigma d \theta^2+\left(r^2+a^2+\...
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How do I expand the Hermitian conjugate in the Lagrangian?

I am trying to expand the following equation: $$\mathcal{L} = \partial_\mu \phi^\dagger_1\partial^\mu \phi_1 - m^2_1 \phi_1^\dagger \phi_1 - \lambda (\phi^\dagger _1 \phi_1)^2\tag{1}$$ using the ...
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1answer
54 views

Factor $\frac{1}{2}$ in scalar kinetic Lagrangian in QFT [duplicate]

Why is it that sometimes I see kinetic term of scalar Lagrangians written like this $$\mathcal{L}=\partial_\mu\phi^\dagger\partial^\mu\phi+\dots$$ like for example in scalar electrodynamics, while ...
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113 views

Can I write the Hamiltonian $H$ in the standard way $p\dot{q}-L$ for a general QFT?

I have read some questions (and the Wikipedia article) about the hamiltonian formulation of a QFT, but the only example that seems to be brought up is the scalar case, saying that $$\mathcal{H}_S=\Pi\...
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Anharmonic terms of Lagrangian of spring pendulum with free support

I am trying to find the normal modes of a spring pendulum with moving support. The spring has spring constant $k$ and unstretched length $l_0$. Sorry for my bad paint skills. The problem was stated ...
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27 views

How to solve lagrangian equations of motion in momentum space?

How do you find the equations of motion for a field given the Lagrangian by working in momentum space?
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51 views

Some index operation confusion about Maxwell field's Euler-Lagrange e.q

Consider this Lagrangian density $$\mathcal{L}(x)=-\frac{1}{2}[a\partial_{\mu}A^{\nu}\partial^{\mu}A_{\nu}+b\partial_{\mu}A^{\nu}\partial_{\nu}A^{\mu}+c(\partial_{\mu}A^{\mu})^2+dA_{\mu}A^{\mu}]$$ The ...
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113 views

Is nonlocality consistent with scale invariance?

For sure I'm excluding gravity at first step, the question is that if nonlocality is compatible with scale invariance. At the classical and quantum levels for field theory in Minkowski spacetime. Then ...
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1answer
39 views

Newtonian limit of geodesic equation and Euler-Lagrange equations

As far as I know the Euler-Lagrange (EL) equations $$\frac{\partial L}{\partial q^m}-\frac{d }{dt}\frac{\partial L}{\partial \dot{q}^m}=0 $$ are covariant time dependent coordinate transformations, $$...
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34 views

Lagrangian for a Pendulum with Free Support

In David Morin's Introduction to Classical Mechanics, he poses the following problem: a pendulum of mass $m$ is fixed onto a block of mass $M$ with a rod of length $l$. The block of mass $M$ is free ...
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Boundary terms of action in AdS

Let's consider the action of a free scalar field in a space-time with metric $g_{\mu \nu}$ $$ S = -\frac{\eta}{2} \int d^{d+1}x \, \sqrt{g} \{g^{AB} \partial_{A} \phi \partial_{B} \phi + \frac{1}{2} m^...
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Calculating $Φ(t)$ during inflation

I am trying to solve for $\Phi(t)$ during inflation for a given metric, $$ ds^2 = dt^2 - a^2dx^2 - b^2dy^2 - c^2dz^2. $$ I just wanted to know how should I proceed while taking Lagrangian $L$ as $ L =...
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Is $T^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g_{\mu\nu}}$ a true tensor or a density?

The energy-momentum tensor is defined by $$T^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g_{\mu\nu}}$$ where $S_m$ is the matter action $$S_m =\int d^4x\sqrt{-g}\mathcal{L}_m$$ and $\mathcal{...
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A strange term in the variation of an action

I am varying an action to compute some equations of motion. The starting point is \begin{equation} \delta S \sim \int d^4x\, \log\Box\delta \phi \end{equation} where $\Box$ is the d'Alembertian ...
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1answer
113 views

Lagrangians for Non-Interacting Scalar Fields in QFT

I am currently taking a QFT class and we are using both canonical and path integral quantization to solve non-interacting scalar fields. We have seen the real scalar field with Lagrangian $$\mathcal{L}...
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Intuitive explanation on why velocity = 0 for a inverted pendulum on a wheel system

I believe I have solved below problem. I am not looking for help on problem-solving per se. I am just looking for an intuitive explanation. Problem statement: wheel mass = $m_1$, even mass rod BC mass ...
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47 views

Help solving one dimensional third order Euler-Lagrange condition

I have the following Lagrangian: $$ \mathcal{L}(t, x, \dot{x}) = \frac{df}{dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \dot{x} + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} b(...
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What is the reason behind the stationarity of action? [duplicate]

I am reading Goldstein right now to understand the least action principle. I understood that the action needs to be stationary under small variation and this specifies the equation of motion, but do ...
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1answer
48 views

Propagator/phase factor of a free particle

I am reading a literature review on modelling standard model particles wavefunctions. I am struggling with deriving the result Ive attached as a photo: From a quantum mechanics course this year i ...
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1answer
108 views

How can we represent thermal energy and heat diffusion in the Lagrangian?

My question has two parts. But let's first introduce the problem: In Lagrangian mechanics, a central part is the Lagrangian $$ \mathcal L\left(t, q,\dot{q}\right) = T\left(t, q,\dot{q}\right) - V\left(...
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Lagrangian for Gauge theory of gravity

There are a number of questions here discussing gravity as a gauge theory of the Lorentz group. I am trying to find the Lagrangian this gauge produces, and the other discussions stop just short of ...
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1answer
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Replacing $U(1)$ covariant derivative with $GL(4,\mathbb{R})$ covariant derivative... does it give quantum gravity?

I realize that many questions about deriving quantum gravity have been asked multiple times before on this forum, but it hasn't been asked exactly like I am doing here. I would like to know what ...
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Action of scalar field with quartic interaction, but no mass

I am working with an action of this form: $$ S=\int d^4 x [ g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi +\lambda \phi^4 ] $$ Typically, an action would contain a mass term $m^2 \phi ^2$. But here ...
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1answer
89 views

Wald's approach to deriving the Einstein field equations and the Levi-Civita connection through Palatini's action

I'm reading Appendix E of Wald's General Relativity book and I'm a bit confused in how he derives the Einstein field equations and the Levi-Civita connection through Palatini's action. The Palatini ...
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1answer
59 views

Geometric meaning of conjugate momentum

Let's say I have a free particle moving in an $n$-dimensional Manifold $M$. There is a tangent space $TM$ associated with all possible infinitesimal motions of a particle at each point in this ...
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3answers
112 views

Can Lagrangians be not related to energy?

I understand that in Lagrangian mechanics, a Lagrangian can be written as $L=T-V$ where $T$ and $V$ are the kinetic and potential energy of the system, respectively. However, in this paper, it ...
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35 views

Does the Harmonic conjugate of the Hamiltonian have to do with the Lagrangian?

Say, we have a Hamiltonian $H(x,p)$. We find a function $G(x,p)$ such that the function $H(x,p)+iG(x,p)$ has a complex derivative. $G$ is then the harmonic conjugate of $H$. Since the change in $H(x,p)...
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1answer
32 views

Differentiation of tensors in lagrangian formalism: two questions

I think I have this right, but I have no way to check it and would appreciate a second opinion. I want to calculate the following: $$ \frac{\partial}{\partial\rho_\nu}\left[\left(\partial_\mu\rho_\nu-\...
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1answer
70 views

$\phi^4$ theory in 5 dimensions

$\phi^4$ theory is not perturbatively renormalizable in 5 dimensions. I have come across literature where renormalizability is discussed w.r.t $N$, for fields obeying $O(N)$ symmetry. But it is not ...
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60 views

Symmetry of a time-dependent Lagrangian

How do I get the group of symmetries and the constant of motion of $L=\frac{\dot{x}^2}{2}m+V(x+ct)$ where c is a constant? When I tried to solve it, it was to look for shifts in $x$ and $ct$ under ...
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Mechanical gauge transformation and energy conservation

I'm studying lagrangian mechanics, and there's a property where you could obtain an equivalent lagrangian $\mathcal{L'}$ from $\mathcal{L}$ by adding a function which satisfies: $$ \mathcal{L'}\...

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