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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Why is an action built from superfields guaranteed to be supersymmetric?

Given a superfield (in 0+1 spacetime + 2 superspace coordinates) $$X(t,\theta_1,\theta_2) = x(t) + \theta_i \psi_i(t) + \theta_1 \theta_2 F_{12}(t)\tag{1}$$ and given the standard supercharges ...
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Is there any reason that Landau and Lifshitz don't discuss Noether's theorem in their mechanics book? [on hold]

I'm currently working my way through the Course volume one. Unless I've completely missed it, the authors omit any discussion of Noether's theorem, instead deriving various conservation laws on a case ...
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Energy-Momentum tensor in Yukawa theory

Given the following Lagrangian: $$\mathcal{L}=\frac 1 2\left(\partial_{\mu} \phi \partial^{\mu} \phi-M^{2} \phi^{2}\right)+\overline{\psi}(i \not \partial-m) \psi+g \phi \overline{\psi} \psi$$ How ...
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Solving the Euler-Lagrange equation with the Axion Lagrangian

I am trying to show that for a constant axion field $\theta(\textbf{x},t)=const.$ the axion Lagrangian $\mathcal{L}_\theta=-\frac{\kappa\theta}{4\mu_0}F_{\mu\nu}\tilde{F}^{\mu\nu}$ does not lead to a ...
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Question on the $1/N$ expansion

My question is from Coleman's Aspect of Symmetry, on the large $N$ approximation. We will consider the $O(N)$ version of the $\phi^4$ theory. Its Lagrangian density is given by: $$ \mathcal{L}=\...
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Why is effective mass a tensor?

So I came across the effective mass concept for solids the other day. It was mentioned that the effective mass is a tensor and may have different values in different directions. However, this is stark ...
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Pendulum attached to Oscillating Fulcrum

In the specific scheme, I would expect that by pulling the spring to the right, the pendulum due to inertia would have to move to the left quartile. In order to find the r vector: $$r = R + l$$ and ...
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Quantum corrections in path integral

I am working the following exercise: Calculate the generating functional $$Z[j]=\int \mathcal{D}\Phi \exp\left(\frac{i}{\hbar}S[\Phi,j]\right),\quad S[\Phi,j]=\int d^4x(\mathcal{L}(\Phi)+j\Phi),$$ $...
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Constraining a Double Pendulum

My question is, how do I apply boundary constraints to a Lagrangian such as: $90 > \theta_1 > -45$ and $\theta_2 > 0$ I am trying to use a constrained double pendulum to simulate an ...
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33 views

EM Dual Lagrangian

I am working on the dual Lagrangian as given by $$\mathcal{L}_D=F_{\mu\nu}\tilde{F}^{\mu\nu}$$ In literature, this term is often written as $$\boxed{\mathcal{L}_D=2\partial^\mu(\varepsilon_{\mu\nu\rho\...
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Deriving the stress-energy tensor from the Einstein-Hilbert action

I'm a mathematician who knows very little physics and is trying to learn relativity theory from a mathematical perspective. Let $M$ be a compact, orientable manifold. In the vacuum, the Einstein-...
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Independents fields and the Lagrange Density of Schrodinger equation [duplicate]

I have a doubt about the lagrangian of the Schrodinger equation. If we consider the wave function $\psi(\textbf{x},t)$ that satisfy the Schrodinger equation as a field, one way of construct the ...
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Tensor Notation - David Tongs Notes [duplicate]

I'm trying to understand the Maxwell's Equation example from David Tongs QFT notes. He uses the Lagrangian: $$ L = -\frac{1}{2}(\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu})+\frac{1}{2}(\partial_{\mu}...
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1answer
39 views

Free fermion Lagrangian invariance under chiral symmetry

I want to apply this transformation to a free-fermion lagrangian: $$ L=\bar{\psi}(\gamma^\mu{\partial_\mu \,- m)\,\psi}$$ $$ \psi ' =\psi\; e^{i \alpha \gamma_5} $$ $$ \bar{\psi}'=\bar{\psi} \;e^{-i \...
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Gravitons and self-interaction

In the book quantum field theory and standard model by Schwartz, there is a problem 9.4 that says by considering lorentz invariency of compton scattering, you can prove that for spin 1 field there ...
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The Lagrangian for gravitational potential energy in a double pendulum

For a double pendulum what would be the gravitational energy. I am trying to work out the Lagrangian for the double pendulum. I got the kinetic energy but I am struggling on the gravitational ...
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Experimental methods to identify C.O.G of a highly heterogeneous cube

While taking to a college about calculating the centre of gravity of multibody basic objects, the question was raised on how one would determine the C.O.G of a highly heterogeneous object of a given ...
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Acceleration due to central forces in the Lagrangian

On Wikiversity it states that for central forces: Wouldn't $\ddot{\vec{r}}_1$ be the same as $\vec{g}$?
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Does it make sense to say that the action is even or odd under time reversal?

The action of a system in mechanics is an integral over time defined as $$S[x(t)]=\int\limits_{t_1}^{t_2}L(x,\dot{x},t)dt.$$ Here, the time $t$ is integrated making the left hand side depend only on ...
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1answer
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Swinging Atwood and Hoop And Pulley Lagrangian

The picture is showing the swinging atwood and a hoop and pulley. I know the lagrangian for both two, I have no problem with the kinetic energy of both but i couldn't convince myself that for the ...
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1answer
36 views

Generalized Coordinates Property for a System of Particles

I"m looking at "Principles of Dynamics: Second Edition" by Donald T Greenwood. I'm trying to figure out how he obtains Eq. (6-64) $$\frac{\partial\dot x_j}{\partial\dot q_i} = \frac{\partial x_j}{\...
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Why are these two variables being treated differently in the action?

I'm trying to understand the derivation provided in the section 2.4 of this paper. I have modified the notation and cut out the unimportant parts of the equations for clarity purposes, but for ...
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1answer
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Rotation as an example of symmetry in classical mechanics

I modified the question because it was confused. On my book there is this mathematical definition of symmetry transformation: "The equations of motion have a symmetry, if the solutions of the ...
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Solving a system of three masses and two springs

Let's say $m_1$ is attached to $m_3$ via a spring of constant $k_1$ and $m_3$ is attached to $m_2$ via a spring of constant $k_2$. Just to simplify the problem we can make $m_1=m_2=m_3$ and $k_1=k_2$. ...
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Dimensional reduction of higher-dimensional Einstein-Hilbert action

I take a spacetime of the form $\mathcal{M}_{d+1}\times \mathbb{S}^n$, with $\mathcal{M}_{d+1}$ some generic non-compact $(d+1)$-dimensional spacetime and $\mathbb{S}^n$ an $n$-dimensional sphere, so ...
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1answer
116 views

Lagrangian and equations of motion in a time-varying coordinate system

I am assuming a very simple case, where there is only a mass $m$ with position $x$ under an external force $F$. we know that the Lagrangian takes the form $L = (1/2) m \dot{x}^2$ from which equations ...
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1answer
47 views

How to check for invariance in Lagrangian after gauge transformation?

If I have the Lagrangian density: $$\mathcal{L}=\left(\partial_{\mu} \phi^{*}\right)\left(\partial^{\mu} \phi\right)-m^{2} \phi^{*} \phi$$ How can I show it is invariant under the following gauge ...
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Derivation of Equation of motion from Euler-Lagrange Equation [duplicate]

Hello I am new to Lagrange's dynamics and have some doubts regarding derivation of equation of motion given in a text : Deriving Equation of motion for a free body(no-external forces) ,in one ...
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A simple question about equation of motion in polchinski's String theory?

In page 14 to get the equation of motion, it takes the variation of the action $$ S_P[X,\gamma]=-\frac{1}{4\pi\alpha'}\int_Md\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}\partial_a X^\mu\partial_b X_\mu $$ ...
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Compute the Legendre transform for a singular Lagrangian

I'm given the lagrangian: $$ L(q,\dot{q}) = \frac{1}{2}(\dot{q_1}^2+\dot{q_2}^2+2\dot{q_1}\dot{q_2})-\frac{k}{2}(q_1^4+q_2^4). $$ I have to compute the Legendre transformation associated to it. The ...
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Rewriting a lagrangian in terms of Hodge duals?

Spinors have been found to have some interesting applications in general relativity (such as Wittens positive energy proof). Recently I'd come across a series of papers 1 2 3 (there are many more ...
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Fixed coordinate frame as limit of rotating coordinate frame

I have a question about a fixed coordinate system as limit of rotating system. Consider for example a pendulum. The Lagrangian in the rotating frame is given by \begin{equation} L(\mathbf{r}, \mathbf{\...
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3answers
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Motivation behind action when deriving ''Strings as Harmonic oscillators" in Zwiebach's book on String theory

Page 248 gives us this action and he simply says that we will assume it correct. $$ S=\int d \tau d \sigma ~\mathcal{L}=\frac{1}{4 \pi \alpha^{\prime}} \int d \tau \int_{0}^{\pi} d \sigma\left(\dot{X}...
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1answer
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Kinetic energy always time independent?! Where is my mistake? [closed]

I have some problems understanding the Lagrangian and the Hamiltonian formalism. Those can be condensed in the following "derivation" of $\frac{\partial T}{\partial t} = 0$ from the equation $\frac{\...
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Asymmetry in Hamilton Equations

I noticed that in deriving Hamilton equations from the total deriveative of the Hamiltonian with respect to time, for the first equation $$\frac{dx_k}{dt}=\partial_{p_k}H$$ we do not need Lagrange's ...
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Most general form of Lagrangian only with respect to Galilean invariance

Let us assume we are doing classical one point particle mechanics. Assume that the least action principle holds. Also, assume that Lagrangian $L$ is a function only of coordinate $x$, its derivative $\...
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Rotating coordinate frame

Hello I have a question about rotating coordinate frames. Following the book of Brizard the Lagrangian is given by \begin{equation} L(\mathbf{r}, \mathbf{\dot{r}}) = \frac{m}{2} \vert \mathbf{\dot{r}} ...
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Goldstein expression for the Lagrangian

I was looking for help in order to proove 2 relations that Goldstein has put in his book. $$ L(q, \dot{q}, t)=L_{0}(q, t)+\tilde{\mathbf{\dot{q}}} \mathbf{a}+\frac{1}{2} \tilde{\boldsymbol{\dot{q}}} \...
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Doubts in an introduction to classical field theory

I started to study classical field theory using the book "Field Quantization" of Greiner and Reinhardt, and I have some doubts. First, the book write the Lagrangian $L(t)$ as a functional of a field $\...
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Anderson-Higgs Mechanism

Consider an abelian gauge field coupled with a complex field: $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D\varphi)^\dagger D\varphi+\mu^2 \varphi^\dagger\varphi-\lambda(\varphi^\dagger\varphi)^2.$...
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107 views

Explicit counting of gauge field degrees of freedom

Consider a connection on a principal $U(1)$-bundle $A_\mu$ over the flat base manifold $M_4$. The action of the theory is described in terms of the curvatures of such connection coupled to some source ...
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28 views

How do interaction terms appear in the Lagrangian?

How does forcing the Lagrangian to be invariant under $U(1)$ group give rise to the electromagnetic interaction term?
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Velocity of particle in non-inertial frame [closed]

Can velocity of the free particle remain constant in non-inertial frame as contrast to free particle in an inertial frame? I know the answer is straightforward yes but taking a different perspective ...
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Noether current Lorentz rotation massive vector field

I'm considering a massive vector field in classical field theory. With the Lagrangian density $$\mathscr{L}=-\frac{1}{4}V^{\mu\nu}V_{\mu\nu}+\frac{1}{2}m^2V^{\mu}V_{\mu}.$$ I want to prove from the ...
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Why is there a Lagrangian? [duplicate]

In all discussions regarding the Lagrangian formulation it has always been said that $L = T - V $, only is a correct guess that when operated via through the Euler -Lagrange equation yields something ...
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1answer
32 views

On the use of Lagrange multipliers in deriving the Lagrange eqn. in classical mechanics

Can one derive the Lagrange eqn based on the methods of Lagrange multipliers? That is, we need to minimize the action with respect to the trajectory keeping the net energy of the body in motion ...
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1answer
79 views

Is it there any relation between an action and entropy?

I've found papers that seem to suggest that these concepts are the same, like this one: https://arxiv.org/abs/1005.3854 But I've found answers in Physics Stack Exchange that say that both are ...
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1answer
52 views

How to calculate the conserved energy $E$ from the Lagrangian?

I am reading a PhD thesis that considers the Lagrangian $$\mathcal{L}=\partial_\mu\phi\partial^\mu\phi^\star-U(|\phi^2|)$$ where $\phi$ is a complex scalar field and $U(|\phi|^2)$ is an arbitrary ...
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Why are only the Lagrangian and Hamiltonian used in mechanics? [duplicate]

Why is it that we have a closed set of four functions, connected by Legendre transforms, in thermodynamics but nobody ever mentions but two of the corresponding functions in mechanics? I've read that ...
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How can dissipative/friction terms be incorporated into a Lagrangian?

I'm trying to find a suitable Lagrangian for a damped harmonic oscillator, a system that satisfies the following equation of motion: $$m \ddot{x} + \gamma \dot{x} + \frac{d\phi}{dx} = 0.$$ What I ...