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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Understanding the Leibniz rule and dynamical variables

On the following question Derivation of Maxwell's equations from field tensor lagrangian We try to calculate the equation of motion of $(1)$ where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu ...
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Coupling of gauge bosons to fermion current in Lagrangian

I am working on breaking the $SO(10)$ group to Pati-Salam. I don't want to go into detail (tell me if you need some) but basically there are a bunch of generators that are broken; the 24 generators of ...
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External force dependent on velocity on a Lagrangian

The question I have is, what conditions must satisfy an external force dependent on velocity so it can be a part of the lagrangian and Euler-Lagrange equations are still true.
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Is the numerical value of the Lagrangian conserved, when moving between inertial reference frames?

I am doing a course on Lagrangian mechanics and the instructor mentioned that the numerical value of the Lagrangian is conserved when I shift between two inertial reference frames, even though their ...
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Deriving the $0$-component of 4-momentum using the relativistic Lagrangian

My question arises from Susskinds book on Special Relativity and Classical Field Theory. (page 102 equation 3.29 to 3.30 and page 105 equation 3.34 to 3.36.) The relativistic Lagrangian for a free ...
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74 views

Is there anything natural about the principle of “stationary action”?

In Taylor's classical mechanics, he derived Lagrange equations and showed that they are equivalent to Newton's second law. Then, it was obvious that Lagrange equations are similar to the Euler-...
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Solution to any classical mechanics problem [duplicate]

Can Lagrangian Mechanics solve any classical mechanics problem, ie in non-inertial frame, involving non-conservative forces etc.
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Lagrange's Equation for Dependent Generalized Coordinates

In learning about Lagrange's Equations, I had always used generalized coordinates that are independent from each other. However, in this post it was mentioned that generalized coordinates can be ...
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$d$-dimensional Einstein equations

I am reading this paper https://arxiv.org/abs/2002.02577. On page 13, it is written that the $d$-dimensional Einstein equations are $$G_{\mu\nu}+\frac{(d-1)(d-2)}{6}\Lambda g_{\mu\nu}=8\pi T_{\mu\nu}...
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Derivation of Euler-Lagrange Equations

I am studying the Euler Lagrange equations and have some problems understanding its derivation. Consider a path $y(x)$ where a slight deviation from the path is given by $$Y(x,\epsilon) = y(x) + \...
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What would be the trajectory of a particle in a constant radial gravitational field? [closed]

For a constant field ($ \vec{E} = g \hat{r}$) We get the Lagrangian as, $$ \mathcal{L} = \frac{1}{2}m\dot{r}^2 + \frac{1}{2}mr^2 \dot{\theta}^2 + mgr.$$ If I put this in the Euler-Lagrange equation ...
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The generic form of instanton (antiinstanton) in Kahler $\sigma$-model

Suppose we have some compact Riemann surface $\Sigma$ , and scalar field $\phi$, which takes values in some Kahler manifold (target space) $M$. In other words, we have a map: $$ \phi : \Sigma \...
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Noether's Theorem and conservation of momentum

So as we all know for a system that has translational symmetry Noether's Theorem states that momentum is conserved, more precisely the theorem states that the quantity: $$\frac{\partial L}{\partial \...
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Disclosing the indeterminacy of a system of equations derived by Lagrangian dynamic formulation

I used Lagrangian dynamic formulation for calculation of torques and reaction forces of the robot with the four 2-DoF legs. As result, I got 9 equations with 12 unknowns (6 servomotors torque and 6 ...
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Problem solving Euler-Lagrange equations of a particle constrained to a spherical spiral

Problem I want to calculate the time it takes for a particle living in a spherical spiral to fall under de force of gravity down to the bottom. So far I've sketched the procedure but when I tried to ...
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Can the Brachistochrone problem be solved using Lagrangian? [closed]

I was thinking about the Brachistochrone problem while reviewing Calculus of Variations and I am perfectly clear with the "textbook" approach, in which we minimize the following integral: $$ f = \int_{...
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Symmetries in quantum field theory and anomalies

Suppose we have a lagrangian quantum field theory, thus a theory where we can write an action in the form \begin{equation} S = \displaystyle \int d^4 x \; \mathcal L \, \left( \partial_{\mu} \phi , \...
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Generalized coordinates of two unequal masses attached to a mass-less rigid rod

Consider a system of two particles of masses $m_1$ and $m_2$ moving in a plane. Let the respective position vectors be $\mathbf{r_1}$ and $\mathbf{r_2}$. The particles are attached at the end of a ...
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Generalised coordinates

I am working on a scientific project for my university and I am reading a german paper (Karas: "Platten unter seitlichem Stoß") which makes use of generalised coordinates. It's about an analytical ...
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Different results for the Hamiltonian of a disc rolling on an inclined plane

$\hskip2in$ Starting from a Lagrangian of a disc rolling down on a inclined plane without slipping, given by: $$ \mathcal{L}=\frac{M}{2}\dot{x}^2+\frac{MR^2}{4}\dot{\theta}^2+Mg(x-L)\sin(\alpha) \...
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1answer
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C, P and T transformations of $\phi$ that preserves symmetry

I have a series of exercises regarding C, P and T symmetry but I am not really sure how to start with the problems. If anyone could help me with one of the problems, or show me a few example problems ...
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How are asymptotic states obtained from a Lagrangian?

I am trying to obtain the formula for a scattering process but I don't quite understand the concept of asymptotic states. I know that: For a Lagrangian, such as $(1)$, where the last term represents ...
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First order expansion of Euler-Lagrange equations

I know that in field theory Euler Lagrange equations are $p_i-d_\mu p^\mu_i=0$. (Classical notations, $p_i=\frac{\partial L}{\partial y^i}, p_i^\mu=\frac{\partial L}{\partial y^i_\mu}$). Being a ...
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Noether theorem in field theory [closed]

Text books define Symmetry of the Lagrangian as the Lagrangian being equal to a total derivative (or sum of derivatives over the manifold coordinates) under a continuous transformation. This can ...
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Setting $N=1$ and $N^a=0$ in the Einstein-Hilbert action

In the ADM formalism of general relativity, one obtains a $3+1$ split of spacetime by setting $$\mathrm d s^2=(-N^2+N_a N^a) \,\mathrm d t^2 + 2N_a\,\mathrm d t\,\mathrm d x^a + q_{ab} \,\mathrm d x^a\...
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Force of constraint for particle constrained to a surface

In Goldstein's Classical Mechanics, he wrote If a particle is constrained to move on a surface, the force of constraint is perpendicular to the surface. Why must the force of constraint be ...
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1answer
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Velocity-dependent potential under generalized coordinates transformation

If under some generalized coordinates the force can be written as: $$Q_j =-\frac{\partial U}{\partial q_j}+\frac{d}{dt}\left(\frac{\partial U}{\partial \dot{q}_j}\right).$$ Then can the force always ...
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Which components of the linear and angular momentum are conserved in the following situation?

A particle moves on a gravitational potential produced by the distribution of mass on a sphere of radius R. First I calculated the lagrangian: $T=\frac{m}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})$ $...
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How to prove time translation invariance of Lagrangian for a free particle?

In my textbook, the author deduce the expression of the lagrangian $L(q_i(t), \dot q_i(t), t )$ of a free particle only using classical physical symmetries where the $q_i(t)$ are independent ...
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1answer
46 views

Partial time derivative of the on-shell action

I have a few questions about differentiating the on-shell action. Here is what I currently understand (or think I do!): Given that a system with Lagrangian $\mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, ...
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1answer
55 views

What's the difference and connection between symmetry breaking and anomaly?

I'm just wondering what's the difference between symmetry breaking and anomaly. From my understanding, symmetry breaking means: there is a symmetry in the action, but in the ground state of the ...
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Boundary terms for stringy correction to GR

We know that there can be possible higher derivative corrections (stringy corrections) to the Einstein-Hilbert action. In GR, to ensure that we get the Einstein Field equations from varying the E-H ...
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Why does the path modification ($\delta$) commute with the time derivative in the derivation of the Euler-Lagrange equation?

When deriving the Euler-Lagrange equation my notes bring the $\delta$ into the action integral, which is fine, which gives $$\delta S=\int_{t_0}^{t_1}dt \frac{\partial \mathcal L}{\partial q_i}\delta ...
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Does modifying the geodesic Lagrangian $L$ with a smooth function $f(L)$ give the same geodesic curves as solutions?

Mathematical side of the problem Given the metric $$ds^2 = dr^2+r^2d\theta^2+r^2\sin^2\theta d\varphi^2$$ we can easily construct the action of a free particle $$S=\alpha \int d\tau \underbrace{\...
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I have problem understanding something from the variational principle for free particle motion (James Hartes' book, chapter 5)

I am currently studying general relativity from James Hartle's book and I have trouble undestanding how he goes to equation (5.60) from equation (5.58). It's about the variational principle for free ...
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Lagrangian for scalar field in terms of klein Gordon equation

I am Studying Peskin and Schroeder, at page 287 , Lagrangian for scalar field is $$L={1\over 2}(\partial _\mu \phi )^2-{1\over 2}m^2 \phi^2.$$ It can be rewritten as $$L={1\over 2} \phi (-\...
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What are the necessary conditions on the energy of the Euler-Lagrange equation to have an oscillating solution? [closed]

Which condition(s) should the energy satisfy, such that the solution of the corresponding Euler-Lagrange equation is oscillating?
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Boundary terms in AdS space

Given the metric in AdS space $$ ds^2=\frac{r^2}{L^2}(-dt^2+d\vec{x}^2)+\frac{L^2}{r^2}dr^2 $$ I am trying to calculate the variation of the action of the KG equation in this metric. What would be ...
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Gauge transformation of the gauge-fixing term in the QED action

In the classroom my teacher stated that the Gauge-fixing term in the action $$\frac{1}{2\alpha}\int d^4x (\partial_\mu A^\mu(x))^2$$ transforms under $A_\mu(x) \rightarrow A_\mu(x)+\partial_\mu \...
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A particle without gauge interactions

Please, Can anyone helps me to understand What does it mean a particle with no gauge interactions?
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Linear system in polar coordinates [closed]

Unlike the Cartesian coordinates, I find navigating through polar coordinates difficult. Is the system defined by the following Lagrangian $L$ defined in polar coordinates linear? $$L = \frac{1}{2} m \...
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1answer
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Hamiltonian of a quantum field that is minimally coupled to gravity

The action for the gravitational field is known as the Einstein-Hilbert action: $$\begin{equation} S_{G}=\int d^4 x \sqrt{|g|} R \end{equation}$$ where $R$ is the Ricci scalar. The ...
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How are body deformations modeled in Lagrangian mechanics?

With rigid-body systems, we choose a finite number of generalized coordinates to model a system, i.e. a pendulum. However, I've read that deformable bodies like elastomers have "infinite" degrees of ...
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Physical interpretation of transport, mutual and relative kinetic energy?

Let $u_{ik}$ denote the $i^{th}$ component of the position vector of the $k^{th}$ particle. Then kinetic energy of the system of $N$ particles is given by: $$T=\frac{1}{2}\sum^N_{k=1}\sum^3_{i=1}m_k\...
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How to deal with coupled equations of motions in equilibrium analysis?

Consider a system including two generalized coordinates $q_1$ and $q_2$ whose dynamics is supposed to be obtained using first-kind Euler-Lagrange (E-L) formalism $$\frac{d}{dt}\frac{\partial L}{\...
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Lagrange formalism in field theory

I recently had a discussion with a friend of mine who is like me studying physics. And we might got used to a misconception about the Lagrange-Formalism in field theory. In common field theory books ...
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How to see that kinetic energy depends on generalised coor., gen. vel. and time?

Let $u_{ik}$ denote the $i^{th}$ component of the position vector of the $k^{th}$ particle. Then kinetic energy of the system of $N$ particles is given by: $$T=\frac{1}{2}\sum^N_{k=1}\sum^3_{i=1}m_k\...
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1answer
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Notation of derivatives in field theory

Some textbooks write $$ \frac{\delta F_{\mu\nu}}{\delta(\partial_\sigma A_\kappa)} $$ which sort of implies the derivative of a functional. Some other textbooks write $$ \frac{\partial F_{\mu\nu}}{\...
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On-shell SUSY-transformations for interacting Wess-Zumino model

I'm learning SUSY with Quevedo, Cambridge Lectures on Supersymmetry and Extra Dimensions. Setup: The SUSY transformations of the component fields of a chiral field $\Phi$ are given by (p.41) \begin{...

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