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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Is there a Lagrangian $L$ (equivalently an action functional $S$) which yields the Navier-Stokes equation?

The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws. However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\...
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What exactly did Lagrange do, historically? [migrated]

I'm tying to understand, historically, what lead to Lagrangian mechanics (LM). What did Lagrange actually do? In the time (year 1788), when Lagrange published his work (that we nowadays call "...
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Lagrangian Dynamics of an inverted Spherical Cart Pendulum

Introduction I have to come up with a PD-controller for an inverted Spherical Cart Pendulum, therefore I tried to compute the Dynamics of such a Pendulum. The Spherical Cart Pendulum is a hybrid ...
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Parity in Effective Lagrangians

Given the following Lagrangian $$\mathscr{L} = c\frac{g}{m}\bar{\psi}_A\Gamma_5\gamma^\mu\psi_B (i\partial_\mu)\phi$$ where $\Gamma_5 \in \{\gamma_5, 1\}$, for two spin one-half particles $A$ and $B$ ...
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Relativistic Euler-Lagrange equations for a four-vector (or one-form) field

I think the best way to ask my question is by considering the maxwell-Lagrangian, $$\mathcal{L}=-\frac{1}{4}F^{\mu \nu}F_{\mu \nu}=-\frac{1}{2}(\partial^{\mu}A^{\nu}\partial_{\mu}A_{\nu}-\partial^{\mu}...
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2 answers
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Proof of principle of stationary action when the Lagrangian is not $L=T-V$

The principle of stationary action claims that the action $S$ takes a stationary value in a real system, where: $$S = \int_{t_1}^{t_2} L dt\tag{1}$$ and $L$ is the Lagrangian of the system. It can be ...
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Derivation of Hamiltonian $H=T+V$ from Lagrangian $L=T-V$

I understand that the Hamiltonian is the Legendre transform of the Lagrangian: $$ \begin{split}H(q,p,t) &= \frac{\partial L}{\partial \dot{q}}\dot{q} - L(q,\dot{q},t) \\ \implies H&=p\dot{q} -...
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How to simplify the process of calculating spacetime geodesics?

I want to study the movement of a particle along geodesics in an expanding universe with metric (FRW metric) $$ ds^2 = -dt^2 + a^2(t) \left( \dfrac{1}{1-kr}dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\...
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Why does the trajectory of a relativistic particle "minimises its Minkowski distance"?

The action of a relativistic free particle is $$\mathcal{S}=\int^{t_{1}}_{t_{0}} L dt\tag{1},$$ for $$L=-\frac{mc^{2}}{\gamma}.\tag{2}$$ I understand that a particle will follow the trajectory of ...
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Would it be more insightful to teach students Lagrangian/Hamiltonian mechanics before Newtonian mechanics? [closed]

What benefits would it bring to teach analytical mechanics before Newtonian (vector) mechanics? I began thinking in this after I saw a 2019 article in the magazine Physics Today that advocates ...
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Particle moving around the inside of a semicone - how to model its position up incline?

An inverted hollow cone (cut in half) is set up with its vertex at the origin $O$ and an angle $\alpha$ between the horizontal ($x$ and $y$-axes) and the cone (so $\alpha$ close to $0$ would be a flat ...
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Chern-Simons Realization of Dijkgraaf-Witten Theory

There is a realization of $Z_N$ Chern-Simons theory (Dijkgraaf-Witten theory) using an instance of $U(1) \times U(1)$ Chern-Simons theory. As explained on page 38 of https://arxiv.org/abs/2007.05915 , ...
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How to find the fine structure constant? [closed]

Is given in the scalar field below the lagrangin. How to find the finite structure constant?
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For virtual displacement in the Lagrangian, why is $\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0$?

I am having trouble understanding why $$\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0.\tag{7.132}$$ you can see my explanation leading up to it below. I would greatly ...
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Euler-Lagrange equations in GR [closed]

This is the action: $$S = \int d\tau \left[ -e^{2a} \left(\frac{dt}{d\tau}\right)^2 + e^{-2a} \left(\frac{dr}{d\tau}\right)^2 + r^{2} \left(\frac{d\theta}{d\tau}\right)^2 + r^2 \sin(\theta)^2 \left(\...
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Kinetic energy of system consisting of rod and rolling cylinder

Suppose we have a cylinder with radius $r$ and mass $m_1$ rolling (without slipping and forward in the image below) on a table with a rod hanging on a point that's fixed to the periphery of our ...
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Help with understanding virtual displacement in Lagrangian

I know that these screen shots are not nice but I have a simple question buried in a lot of information My question Why can't we just repeat what they did with equation (7.132) to equation (7.140) ...
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Non-Abelian vertex 3-gauge-boson

I am trying to understand how the vertices depicted in page 507 of Peskin and Schroeder come about. I understand that vertex where we have 1 gauge boson and two fermions but I'm confused on the ...
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1 answer
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Derive interaction lagrangian for KG equation in QED

The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$ By ...
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12 votes
3 answers
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How are anomalies possible?

From Matthew D. Shwartz Quantum Field Theory textbook, he writes: "Most of the time, a symmetry of a classical theory is also a symmetry of the quantum theory based on the same Lagrangian. When ...
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Problem 6 of Sheet 1 - Quantum field theory David Tong - Variation of Lagrangian density

The Problem reads: Consider the infinitesimal form of the Lorentz transformation derived in the previous question: $x^\mu \rightarrow x^\mu +\omega^{\mu}_\nu x^\nu$. Show that the scalar field ...
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1 answer
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How to derive the Lagrangian for a system of first order equations of motion?

Please be lenient if this question has been asked before in a similar manner. My background is in CS, but I am working on the physics-based modeling of complex systems and I am studying physics as an ...
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Cart Pole kinetic energy

As explained in [1], the kinetic energy of a Cart Pole is: $$ \frac{1}{2} (M+m)\dot x^2 + \frac{1}{2} m L^2 \dot \theta^2 - m L cos(\theta) \dot \theta \dot x $$ Where $m$ is the mass at the tip of ...
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Relative signs between interaction terms

What is the interpretation / meaning of relative signs between interaction terms in a Lagrangian density? (If there is none, are they even physically reasonable?) Example: Let $\phi$ be a scalar field,...
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2 votes
1 answer
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Is my expression of the Noether current $J^\mu$ for a local $\rm U(1)$ symmetry correct? If not, what's wrong?

The Lagrangian of electrodynamics reads $$\mathcal{L}=i\bar\psi\gamma^\mu D_\mu\psi-m\bar\psi\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ where $D_\mu=\partial_\mu+iqA_\mu$. It is unchanged under the set of ...
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Lagrangian of a bead free to move on the spoke of a bike wheel

I am trying out this system for a final project in my Physics class. So the wheel is vertical as in a bike wheel and it is being driven so that the wheel is not a part of the system or the Lagrangian....
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Deriving isospin factors in phenomenological pion nucleon nucleon interactions

Preface One commonly finds this interaction Lagrangian in phenomenology (ignoring constants): $$\mathscr{L} \propto \bar\psi \gamma^5\gamma^\mu \vec\tau\psi \cdot \partial_\mu \vec\pi \tag{1}$$ ...
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Inequality constraint in Lagrangian

An example for an equality constraint: \begin{equation} x \geq x_a \end{equation} which can be used in the lagrangian: \begin{equation} \mathcal{L} = E(x) + \lambda(x-x_a) \end{equation} but ...
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Chiral symmetry of the Dirac Lagrangian

I need to show that in the mass to zero limit the lagrangian density: $$\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi$$ is invariant under the transformations: $$\psi'=e^{i\alpha\gamma^5} \psi$...
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Lagrange equation of motions for a particle moving in a surface in the presence of gravity

I have to model the dynamic behaviour of a particle solid in a gravitational field.It is for a control theory course. And my background in dynamics is not the greatest. The particle can move left and ...
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2 answers
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Why is derivative of Lagrangian with respect to generalized position and velocity equal to this?

I'm currently studying Lagrangian mechanics, and in the process, I've met the following equations in a couple of proofs. $$ \frac{\partial \mathcal{L}}{\partial q_i} = \dot p_i $$ $$ \frac{\partial \...
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1 answer
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Variational operator confusion

Let $L=L(X, \dot X)$ such that the first variation of $L$ is given by $$\delta L=\frac{\partial L}{\partial X}\delta X+\frac{\partial L}{\partial \dot X}\delta \dot X.\tag{1}$$ This is pretty standard ...
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Can a primary constraint contain spatial derivative of the field?

I am currently studying the Hamiltonian formulation of GR and I have problems understanding this definition of primary constraint. In the textbooks, primary constraint occurs when a momentum conjugate ...
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Confusion with the variational operator $\delta$ and finding variations

I have recently started studying String Theory and this notion of variations has come up. Suppose that we have a Lagrangian $L$ such that the action of this Lagrangian is just $$S=\int dt L.$$ The ...
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What does "conformally coupled scalar" mean?

"Conformally coupled scalar $\phi$" - I encounter it a lot, but I can't find what it means.
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-1 votes
1 answer
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Struggling to find kinetic energy for Lagrangian

The system I am looking at is follows. To find the Lagrangian $T-V$, I need $T$, which is the sum of the kinetic energies of the three masses shown (rods are assumed to be light). I know that the ...
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Lagrangians related by field redefinition

Is there a sufficient criteria in the form of a theorem to check if two Lagrangian densities $\mathscr{L}$ are related via field redefinitions $\phi\rightarrow f(\phi)$, where $f(\phi)$ is an analytic ...
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1 vote
1 answer
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Geodesic equations with varying mass and the variational principle

Consider the action, $$ S = \int d\lambda\ \phi(x) \left( -g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{dx^\nu}{d\lambda} \right)^{1/2}. $$ Using the variation principle we obtain, $$ \delta S = \int d\...
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1 vote
2 answers
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Euler-Lagrange solution of $L=\ddot{q}^2$?

I'm new to Calculus of variations and have a very basic question. Suppose we want to solve the Lagrangian $L=\ddot{q}^2$ using the Euler-Lagrange equation. My intuition tells me that the solution ...
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-1 votes
1 answer
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Interpretations of Lagrangian vs. Hamiltonian mechanics

This might seem like a duplicate question; however, rest assured, it is not. My question is pointed and particular: Some background: Given a system we describe its Lagrangian $L$ as $T-V$, where each $...
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21 votes
3 answers
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Equations of motion only have a solution for very specific initial conditions

An exercise made me consider the following Lagrangian $$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$ If I didn't make a mistake the equations of motion should be given by: ...
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$n$-link pendulum solution methods

I want to simulate a $n$-link pendulum without using the Lagrangian method. The reason is that the formula for the Lagrangian becomes large and depends on the number of the links $n$. Therefore, I ...
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2 votes
1 answer
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Notation and Terminology Questions from Schwartz' QFT Book

I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing. First off, on page 34 he defines a translation of a field to first order as $$...
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What is the lagrangian interaction term of a charged current weak interaction?

For example, if you have an electron coupling to a $W^{-}$ and producing an electron neutrino $e\rightarrow \nu_{e}W^{-}$. Is something like $$\frac{-g_{w}}{2\sqrt{2}}\bar{\nu}_{e}\gamma^{\mu}W^{-}_{\...
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2 votes
1 answer
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Feynman path integral in an EM field

I'm studying Feynman and Hibbs, Quantum Mechanics and Path Integrals In problem 4-2, the book says for a particle of charge $e$ in an EM field the Lagrangian is $$L=\frac{m}{2}\dot{\boldsymbol{x}}^2+\...
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3 votes
1 answer
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Noether/Hilbert energy-momentum tensor

In chapter 4 of Carroll's book Spacetime and geometry he finds using the Hilbert action that the energy-momentum tensor for a scalar field is (see eq. (4.79)) $$T_{\mu\nu}^{\phi}=\nabla_\mu\phi\nabla_\...
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Express the power spectrum by displacement field in Lagrangian perturbation theory

Recently I'm reading this paper, Resumming Cosmological Perturbations via the Lagrangian Picture, to learn the application of Lagrangian perturbation theory in the modelling of large-scale structures. ...
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Deriving spherical pendulum equations with an Applied torque

I'm and electrical engineer and I'm trying to derive a state space model for a spherical pendulum with an applied torque to the system, not energy. So if I understand correctly I can't apply the ...
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1 vote
3 answers
181 views

Lagrangian first integral

I want to extremize $$\int dt \frac{\sqrt{\dot x ^2 + \dot y ^2}}{y}.$$ I have thought that, since the Lagrangian $L(y, \dot y, \dot x)$ is $t$ dependent only implicitly, that i could use the fact ...
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Step in derivation of Lagrangian mechanics

There is a step in expressing the momentum in terms of general coordinates that confuses me (Link) \begin{equation} \left(\sum_{i}^{n} m_{i} \ddot{\mathbf{r}}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\...
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