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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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When and how was the need for symmetry in the stress-energy tensor first realized

This question is somewhat historic. Let $\Theta_{\mu\nu}$ denote the canonical stress-energy tensor of some matter field $\psi$ in special relativity. It is often stated that the reason why we want ...
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Computing the spin degrees of freedom for a massless particle in $D$ dimensions

According to the paper A Lagrangian formulation of the classical and quantum dynamics of spinning particles, a relativistic spinless particle in $D$ spacetime dimensions can be described by the ...
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Varying a scalar field Lagrangian density

I was varying a scalar field density and I look at this term $${\cal L}~=~-\frac{1}{2}\partial _\mu\phi\partial^\mu\phi.$$ The result that I need to come is $$-\frac{1}{2}\delta(\partial _\mu\phi\...
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1answer
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Physical explanation of Dirac-Born-Infeld (DBI) inflation

I am studying the Dirac-Born-Infeld (DBI) inflation model and came across this question in a past exam paper from Cambridge that considered the following Lagrangian: $$\mathcal{L}=\sqrt{-g}A(X,\phi)$$...
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Superparticle in Background Gauge Fields

A simple model for a superparticle is $$L=m\int dt\left(\dot{x}^{2}-\frac{i}{2}\psi\dot{\psi}\right)$$ with SUSY algebra $\delta x=-i\epsilon\psi$ and $\delta\psi=-\epsilon\dot{x}$, where $\...
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D'Alembert's principle and equation of motion

Is obtaining proper equation of motion from D'Alembert's principle a mere coincidence or there is some logic behind this? This is asked because while we are finding the equations in a regime where ...
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What is the physical content of the principle of least action?

Say the world is governed by the Principle of Least Action (or Hamiltonian mechanics) and let's not worry about quantum mechanics too much. Independently of any Lagrangian or Hamiltonian, does that ...
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Lagrangian Mechanics - Charged particle in magnetic field

A particle of mass $m$ and charge $Q$ moves in the equatorial plane ($θ=π/2$) of a magnetic dipole, where the vector potential of the dipole is given by $$\mathbf{A} = \dfrac{\mu \sin \theta }{4 \...
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Simple Lagrangian Example [on hold]

Consider a mass $m$ placed at the topmost point of a wedge of mass $M$. The wedge is free to move on the smooth floor. What is the acceleration of the wedge and the block when the system is released ...
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1answer
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$\partial^{\nu} \partial_{\nu}$ vs. $\partial_{\nu} \partial^{\nu}$

I was doing a problem regarding field theory. I am given the following lagrangian density: $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi_i\partial^\mu\phi_i-\frac{m^2}{2}\phi_i\phi_i$$ for three scalar ...
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Branes without Fayet-Iliopoulos term

Often D-brane effective actions are given as (let's take D3-branes) $$ S=-T\int d^4x \sqrt{-det(g_{mn}+2\pi\alpha'F_{mn})}. $$ After expansion in $2\pi\alpha'$ there is a constant Fayet-Iliopoulos ...
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Lagrangian of a Heavy Symmetrical Top - Inertial or Non-inertial Frame?

I'm having some confusion with the analysis of a symmetrical top (specifically, a heavy top, but this is not very important for the question). Following Landau and Lifshitz's Mechanics, on page 110 ...
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Deriving kinematic equations for the slingshot maneuver using Lagrangian mechanics

I have an assessment/investigation coming up in my math class for which I plan to derive the equations of motion for the slingshot maneuver/gravity assist using Lagrangian mechanics. So far, I am ...
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2answers
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Exercise about symmetry in the Lagrange equations [closed]

This question was asked during a classical mechanics exam (no solutions were given afterwards). Suppose a free particle in $\mathbb{R}^n$ with the following Lagrangian: $$L = \frac{m}{2}\sum_{i=...
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2PI-effective action and functional derivatives

I'm trying to work out the 2PI-effective action for complex scalar fields. Introducing a multi field index $(a,b,c...)$ the complex conjugation and all other degrees of freedoms are suppressed, and ...
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SUSY variation Wess-Zumino

I'm following John Terning book on Supersymmetry and in particular I'm trying to check the susy variations of the Wess-Zumino model given by $\mathcal{L}_s = \partial^\mu \phi^* \partial_\mu \psi \,...
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Lagrangian Mechanics - Normal Modes

System is as in the diagram shown with the hoop having mass $M$, bead having mass $m$ and the moment of inertia about the pivot point for the hoop is given by $I = 2MR^2 $. For the bead, we have $$ ...
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How do I calculate the Lagrangian for a moving System that has moving parts in its own? [on hold]

I want to calculate the Lagrangian for a ball that is actuated with wheels above it. Now since the wheels are attached to the ball, should I a) calculate the kinetic energy with the velocities with ...
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1answer
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Derivatives with Two Indices in Electromagnetic Lagrangian [duplicate]

I was reading about the derivation of Maxwell's equations from an electromagnetic Lagrangian density from Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity. The Lagrangian ...
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1answer
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What is the dimensionality of each part of a covariant derivative?

In the standard model, we have the following covariant derivative: $$D_\mu = \partial_\mu - ig_sG_\mu^a\lambda_a-igW_\mu^a\frac{\sigma^a}{2}-ig'B_\mu\frac{Y}{2}$$ If we let this work in on e.g. the ...
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1answer
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Noether's theorem for scale invariance

When we have the Lagrangian $$\mathcal{L} = \frac{1}{2} \partial _\mu \phi\partial^\mu \phi \tag{1} $$ We have a symmetry given by $$x^\mu\mapsto e^\alpha x^\mu, \qquad\phi\mapsto e^{-\alpha} \phi.\...
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Why can we add counterterms?

I'm having a hard time understanding why renormalized perturbation theory works. Why is it permissible to add counterterms to the Lagrangian? Terms which are often divergent themselves and carry ...
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Mass of an anti-symmetric spin-1 field

I was wondering how does one calculate the mass of an anti-symmetric spin-1 field. For a vector field one writes $m^{2}A_{\mu}A^{\mu}$ for mass term. How does one write the mass for "any" anti-...
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1answer
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Non-conservative forces in Lagrangian mechanics

In the Lagrangian formalism with a dissipative frictional force $F$, we can write $$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{q}_{k}}-\frac{\partial\mathcal{L}}{\partial q_{k}}=Q^{(nc)}_{k}...
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Are Maxwell's equations “physical”?

The canonical Maxwell's equations are derivable from the Lagrangian $${\cal L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$ by solving the Euler-Lagrange equations. However: The Lagrangian above is ...
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Variation of action for massive point particle (pp)

So I'm pretty sure I'm missing something obvious, but for the life of me I cannot replicate the step between 1.2.2 and 1.2.3 in Polchinski Vol 1. Basically, I'm trying to find the variation of: $$S_{...
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Noether's conserved charges: why are there not “spatially” conserved charges? [duplicate]

Noether theorem implies that there is a conserved current $j^\mu$ for every continuous symmetry of the action, i.e. $$\partial_\mu j^\mu=0 $$ to each conserved current we can associate a conserved ...
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1answer
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Why does the Lagrangian Density have to be a polynomial of the field?

In a lecture, a professor appeared to have said that the Lagrangian can only contain terms that have powers of $\phi$ and a term with $\partial_\mu \partial^\mu \phi$ . I imagine this would make any ...
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1answer
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Constrained Hamiltonian systems: spin 1/2 particle

I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action \begin{equation} S=-m\int ...
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Euler-Lagrange eqs. use in QFT [duplicate]

It is known that in QFT the Euler-Lagrange equations are used to obtain the equations of the quantum fields. Nevertheless, from the path integral's point of view (where you integrate over all $\it{...
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QED lagrangian: gauge fixing term

I have a question about the structure of the QED lagrangian, in particular the free photon lagrangian which is contained in it. My premise is: I only know how to exploit canonical quantization in ...
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Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5)

In Weinberg's QFT textbook, using a gauge transformation $$A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu}\epsilon(x)\tag{8.1.3},$$ it has: $$\delta I_{M} = \int d^4 x \frac{\delta I_{M}}{\delta A_{...
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Question about Lagrange method and line element

Consider the following line element: $$ds^{2} = K(x,y,z,t)(-dt^2+dx^2)+M(x,y,z,t)dxdt+dy^2+dz^2$$ Then the lagragian method give to us the lagrangian from line element: $$\mathcal{L}^2 = K(x,y,z,t)(...
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On the “Derivation of the Electromagnetic Lagrangian density”

In the most upvoted answer here : Deriving Lagrangian density for electromagnetic field, how do we know that equations (015) and (016) therein \begin{equation} \boxed{\: \dfrac{\partial }{\partial t}\...
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1answer
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How can the action can describe a movement? What is the argument behind? [duplicate]

We define the action of a system as $$S(q)=\int_{t_1}^{t_2}L(t,q(t),q'(t))dt,$$ where $q(t)$ is the evolution of the system and $L$ is the Lagrangien. How can a stationary point of $S$ can describe ...
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How to calculate Hamiltonian when Lagrangian has higher order derivatives? [duplicate]

If we have a Lagrangian density $\mathcal{L}$ for a scalar field $\phi$ depending on $\phi$, $\partial _{\mu} \phi$, and $\partial _{\mu} \partial _{\nu} \phi$, what is the Hamiltonian? Additionally, ...
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Classical method VS Hamiltonian method [duplicate]

I'm very confuse with the method using Hamiltonian to derive the equation of the movement. In example I have, it's easier to derive the equation of the movement using classical method (namely 2nd law ...
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Trying to make some sense of the cosmological constant

The question is simple: Does the naive reasoning below have some physical sense, or is it pure gibberish junk? (I use $c = 1$ and metric signature $\eta = (1, -1, -1, -1)$) I suppose that the ...
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Where do I go from here to show that linear momentum is conserved under all instances of translation symmetry?

I've worked through a simple derivation of symmetries implying conservation laws from an invariant Lagrangian. Namely a quantity $Q$ is conserved in the equation below, where $i$ is a degree of ...
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1answer
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Lagrangian Equation of Motion for Planetary Orbit in a Single Plane

Assume that a mass $m$ is in a gravitational orbit around a much larger mass $M$, as in the case of the earth revolving around the sun. Also, assume the motion is constrained to a single horizontal $...
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1answer
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Is the Dirac Lagrangian locally gauge invariant without gauge field $A$?

When it comes to the check of the invariance of the Lagrangian of the Dirac equation under local $U(1)$-transformations I have made the following observation: $$L = \bar{\psi} (i\gamma^{\mu}\...
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1answer
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Problem using Noether's theorem in time-dependent lagrangian

I have some problems calculating the conserved quantity for a lagrangian of the form $$ L = \frac{1}{2}m\dot{q}^2 - f(t) a q, $$ because I found the general problem too abstract, I tried at first ...
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Surface Tension and Potential Energy in a Lagrangian

It seems natural to define the potential energy due to surface tension to be $U = \int \sigma dA$. But then I have the following problem. I wanted to investigate surface tension in a simple ...
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Why can't we insert gravity in the special relativistic lagrangian?

I am a math student and I have taken four-five lessons about special relativity in a course about Lagrangian and Hamiltonian mechanics, so be patient with me if my question is stupid. My teacher says ...
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Intuitive explanation for the free field Lagrangian?

The free field Lagrangian is $$\mathcal{L}=\frac 1 2 \partial^\mu\phi\partial_\mu\phi-\frac 1 2m^2\phi^2$$ with sign convention $(+,-,-,-)$. Plugging this into the Euler-Lagrange equations gives the ...
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Equilibrium points of three masses on a rigid spring ring with gravity

I'm trying to find the equilibrium points of a given system using Lagrangian mechanics (the system is still not rotating at the beginning). should I find the diagonal matrix for the characteristic ...
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2answers
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$SU(2)$ Invariant Lagrangian

Consider two arbitrary scalar multiplets $\Phi$ and $\Psi$ invariant under $SU(2)\times U(1)$. When writing the potential for this model, in addition to usual terms like $\Phi^\dagger \Phi + (\Phi^\...
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Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\...
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Does Noether's theorem apply to constrained system?

The Lagrangian of a constrained system will be $$L-\lambda_1f_1-\lambda_2f_2-...\lambda_kf_k.$$ If a transformation will not affect the constrained Lagrangian, the there is some corresponding ...
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E.L. Equations in QFT

In QFT, we use the Lagrangian to construct the Hamiltonian, and in the Interaction Picture (with regards to the Free Field Hamiltonian) use the full Hamiltonian to calculate the changes in the field (...