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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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Non-quadratic kinetic energy [on hold]

Do you have examples of Lagrangians/Hamiltonians used in physics with non-quadratic kinetic terms? e.g. $\dot{x}^4$ What is the origin and the interpretation of such terms?
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Energy momentum tensor “mixed derivative” [on hold]

I'm supposed to determine the energy-momentum tensor given by: $$T^\mu_\nu = \delta^\mu_\nu\mathscr{L}-\frac{\partial \mathscr{L}}{\partial(\partial_\nu A^\mu)}(\partial_\mu A^\mu)$$ where $$\...
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Interaction Lagrangian up to quartic order

I have a Klein-Gordon Lagrangian of scalar fields and I add an interaction term that depends only on the fields (not their derivatives). The free Lagrangian is invariant under some infinitesial ...
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Why do we demand that the counterterms in $\varphi^3$ theory be $O(g^2)$?

In Srednicki's QFT book, section 9, he introduces the $\varphi^3$ lagrangian: $$\mathcal{L}= -\frac{1}{2}Z_\varphi(\partial_\mu\varphi)(\partial^\mu\varphi) -\frac{1}{2}Z_mm^2\varphi^2 +\frac{1}{6}...
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Independence of position and velocity vector [duplicate]

Hi I am a mathematics student with an interest in Physics. In our Physics elective our prof. said if $\vec r$ denotes the position vector then the velocity vector $\vec v = \vec {\dot r} $ is ...
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Origin of $\sqrt{-g}$ in the integral of action $S$

I have a question that might (and probably will) be stupid: I do not understand where does the factor $\sqrt{-g}$ (i.e. $\sqrt{-\det\left(g_{\mu\nu}\right)}$) come from in the action integral S when ...
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Are Instantons Massless?

That is, are the only field configurations which give a non-zero winding number ones in which the Fourier transform includes a factor like $\theta(k^0)\hat{D}\delta(k^2)$, where $\hat{D}$ is some ...
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How could I derive the Noether charge for a real scalar field?

I know for a (free) complex scalar field $\psi$ the Lagrangian is: $$ \mathcal{L} = \partial^\mu \psi^\ast\partial_\mu \psi$$ and that Noether's theorem from the $U(1)$ symmetry of the system gives a ...
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Can all phase space conserving dynamics be described by a Lagrangian system? [duplicate]

Given a system described by a set of ODE's that can be shown to conserve phase space, does there necessarily exist a Lagrangian (or Action) formulation that describes my system? I'm comfortable ...
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Constants of motion of an electron in a harmonic electromagnetic field in free space

I have encountered a question in Classical Electrodynamics, as below: In free space, an electron, initially at rest at $z=0$, is subjected to an intense laser field $\vec E=\hat x A \cos(\omega t-...
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What is a coordinate-free version of Noethers theorem? [closed]

What are some examples and derivations of some basic symmetries (not coordinate symmetries)? For example I remember a sufficient condition for being a symmetry of the lagrangian system is being an ...
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Variation in field theory with respect to one quantity

In my QFT course we are supposed to vary the action of a for a scalar field coupled to an electromagnetic field with the following Lagrangian density: $$\mathcal{L} = [D_\mu\phi(x)]^*D^\mu\phi(x)-m^2\...
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How to see linearity of an interaction if it's lagrangian density is known?

The Lagrangian of electrodynamics is $-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+A_\mu J^\mu$ we know that electrodynamics is linear in special relativity but when we go to general relativity it becomes non-...
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1answer
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Infinitesimal transformations that leave the action invariant

I have the Klein-Gordon Lagrangian for three scalar fields and I want to find three independent infinitesimal transformations that leave the action invariant. I suppose that these three ...
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1answer
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Why is the Lagrangian for space-like geodesics equal to 1?

In Schwarzschild spacetime, the Lagrangian can be defined as $$ L = -\left( 1 - \frac{2M}{r} \right) \dot{t}^2 + \left( 1- \frac{2M}{r} \right)^{-1} \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2\theta ...
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1answer
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Free boson Equation motion from action

So in David tongs notes we have $$S=\frac{m}{8\pi}\int d^2x\partial_i\varphi\partial^i\varphi$$ and he finds that the equation of motion is $$[\partial_{t}^2-v^2\partial_{x}^2]\varphi=0$$ now my ...
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Considerations for formulating Lagrangians in general

My question albeit trivial is something I wish to understand better. Given a system either say 3 masses, 2 masses and a pulley (with mass), and 2 masses attached via a string with mass $m$ (and ...
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Quantum field theory, Dirac field interaction Yukawa theory

From this Phys.SE question: Please can someone answer me to get the scattering amplitude and the cross section
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2answers
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Conserved currents in quantum electrodynamics

A general Noether theorem in fields theory says that an infinitesimal symmetry of the action leads to a conserved current $j^\mu$, i.e. $\partial_\mu j^\mu=0$. Below I would like to consider a minor ...
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1answer
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Clarifications regarding the Maupertuis/Jacobi principle

I'm slightly confused regarding the Maupertuis' Principle. I have read the Wikipedia page but the confusion is even in that derivation. So, say we have a Lagrangian described by $\textbf{q}=(q^1,...q^...
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$\int (f(x+\delta x) - f(x)) dx = \int \left ( \frac{df(x)}{dx} \delta x \right) dx$

From Landau and Lifshitz's Mechanics Vol: 1 $$ \delta S= \int \limits_{t_1}^{t_2} L(q + \delta q, \dot q + \delta \dot q, t)dt - \int \limits_{t_1}^{t_2} L(q, \dot q, t)dt \tag{2.3b}$$ $$\Rightarrow ...
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Potential energy requirements for the Lagrangian to work

It's clear to me that in acquiring the equations of motion using the Euler-Lagrange equations and the Lagrangian, defined as $$L \equiv T - V,$$ the potential energy may have a time dependence if the ...
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The Lagrangian and inertial reference frames [duplicate]

From my understanding, my instructor told me that in order to use the Lagrangian, defined as $$L \equiv T - V,$$ to find the equations of motion via the Euler-Lagrange equations, the generalized ...
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1answer
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Confusing with the equation $(2.4)$ and $(2.5)$ of Landau and Lifshitz, Mechanics, Chapter 1, The principle of Least Action

I'm a 12th Grader and I'm interested in Lagrangian Mechanics and having a bit of knowledge about the Newtonian Mechanics. So, I found a book of Landau and Lifshitz's Mechanics and started reading from ...
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Non-relativistic quantum electrodynamic lagrangian: number of dynamical variables greater than 6

There is an argument I do not understand given in "Introduction to quantum electrodynamics" by Cohen Tanoudji (page 111 for the french version of the book). We are dealing with the non-relativistic ...
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1answer
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Testing the second type of equations of Lagrangian Mechanics

I'm new to classical mechanics and learned a new cool thing, second type equations of Lagrangian Mechanics. So, I was just testing if it really works or not. So, I made a question myself to test it, ...
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Field degrees of freedom from equations of motion and higher spin

It is my understanding that we compute the number of degrees of freedom of a quantum field as the number of its components minus the number of non trivial equations we get by taking the divergence of ...
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Derivation of Coulomb's law from classical field theory

In the section on Coulomb's law in QFT by Schwartz, he expands $-\frac{1}{4}F_{\mu\nu}^{2}$ to get $-\frac{1}{2}(\partial_{\mu}A_{\nu})^{2} + \frac{1}{2}(\partial_{\mu}A_{\mu})^{2}$, can someone ...
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2answers
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How does $r$ depend on $\varphi$ in the Schwarzschild metric?

I am confused about the Wikipedia derivation of the equation for geodesic motion in the Schwarzschild spacetime. The derivation of this equation involves variation with respect to the longitude $\...
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1answer
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Equation of constraint - Falling disc unrolling from an attached string

Where does the equation of constraint below come from? I've tried to rationalize it, but the angle will be 0 more than one time as the string unrolls, even though y will keep going down (right?), not ...
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1answer
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Getting a Conserved Quantity from a Lagrangian [duplicate]

So I've been messing around with the implications of Noether's theorem, and though I conceptually get what it's saying, I'm having a hard time actually using it to retrieve a conserved quantity from a ...
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Hamiltonian directly expressed in $(q,\dot{q})$ : how to find what is $p$?

I am reading a book about non relativistic quantization of E.M field. But first we do classical field theory. We directly wrote the Hamiltonian of our study, and a part of our Hamiltonian is the ...
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Size of quantum corrections at infinity

Suppose we have a one dimensional field theory for the field $\phi(r)\;r\in[0,\infty]$ and that the solution for the background (Euler Lagrange equations) give a function $\phi_0$ that goes to a ...
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Piecewise solution to Euler-Lagrange equations

I would like to consider a background for a quantum field theory made up by connecting continuously two different solutions of the Euler Lagrange equations. The problem is one dimensional (let's call ...
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1answer
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Galilei Invariance and Newton Third Law

Let's say we have a system of two point particles that can interact with each other by forces that are position and velocity dependent. The forces might or might not be derivable from a generalized ...
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How do fermions explicitly interact with curvature via the tetrad?

I am aware of the basics of the tetrad formalism and am clear on why bosonic fields do not have couplings to curvature via their covariant derivatives in a curved space Lagrangian i.e. why $\nabla_\mu\...
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Field momentum of Klein-Gordon Lagrangian

Given the Lagrangian $L$ of the field $\phi$ the field momentum $\Pi$ reads: $$L_{KG}=-\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m^2\phi^2$$ $$\Pi=\frac{\partial L}{\partial(\partial_\...
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1answer
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What objective function is Lagrange's equation of the first kind based on?

In Lagrangian mechanics, Lagrange's equation of the first kind states that $$ \frac{\partial L}{\partial r_k} - \frac{d}{dt}\frac{\partial L}{\partial \dot{r_k}} + \sum_{i=1}^C \lambda_i \frac{\...
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1answer
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Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary

Suppose we want to make an integral $S$ of the form $$S = \int_{x_1}^{x_2} f\left[y_1(x), \dots, y_n(x), y'_1(x), \dots, y'_n(x), x\right]dx$$ stationary with the constraint $y_1\left(x_1\right) = \...
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1answer
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$\phi R$ term for scalar field in a curved background

Condider the following action for a free scalar field $\phi$ in a curved background $$S=\int dx\Big( \frac12g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+\gamma \phi R\Big)$$ Here $g_{\mu\nu}$ is a ...
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Trying to prove the Wess Zumino invariance under a SUSY transformation

I have the Lagrangian density $$L=-\partial_\mu \phi^\star \partial^\mu \phi - \bar{\chi}_R \gamma^\mu \partial_\mu \chi_L - \bar{\chi}_L\gamma^\mu\partial_\mu \chi_R.$$ where $\epsilon$ is the ...
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Questions about Euler-Lagrange derivation in Classical Field Theory

I'm new to classical field theory, so I have a few basic questions: From the derivation of the Euler-Lagrange equations, we have the following: \begin{align} \delta S[\phi]&=\int d^4x\delta L(...
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Why is the generating function in the Hamilton–Jacobi equation equal to the action? [duplicate]

The aim in Hamilton Jacobi formalism is to find a canonical transformation that generates a new Hamiltonian $H'$ which is equal to $0$. Therefor we find the equation: $$H(q_1,...,q_n,\frac{\partial F}{...
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2answers
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Different definitions of Functional Derivative

In studying QFT and General Relativity, I came across two different definitions of Functional Derivative, and I'd like to know if they are equivalent. Firstly, in Wald's book General Relativity, as ...
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Is there a physical connection between Lagrangian mechanics and Hamilton mechanics

I know that the Hamilton's equations results from the Legendre transformation of the Lagrange equation. It's also well-known that the Hamiltonian is equal to the energy of a system if the system is ...
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1answer
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Difference between kinematic momentum and conjugated momentum in purely mechanical setup

I don't know much about physics, but I wanted to understand what was the difference between the "kinematic momentum" and the conjugated momentum. As I understand it, kinematic momentum is mass times ...
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1answer
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On the definition of Lagrange's equation

Is it really Newton's third law eq (1.5) ? Isn't the 2nd? I found it on "Abers Ernest Quantum mechanics".
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Time-independent Schrödinger equation Lagrangian derivation

Recently I was taking a calculus of variations class and our professor casually obtained the time-independent Schrödinger equation for a free particle from the integral (constants dropped) and it's ...
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1answer
362 views

Applying the Euler-Lagrange equations to Maxwell's Theory

In Prof. David Tong's notes, specifically on page 10, he gives the Lagrangian of Maxwell's theory to be $$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\...