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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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Transformation of a Lagrangian

$$L(\lambda,\mu,\dot{\lambda},\dot{\mu})=\frac{m}{2}(\lambda^2+\mu^2)(\dot{\lambda}^2+\dot{\mu}^2)-\alpha \lambda^2\mu^2,$$ I'm supposed to express this Lagrangian through $x=\lambda^2-\mu^2$ $y=2\...
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How come $\frac{d}{dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}$ in Lagrangian mechanics? [duplicate]

It is written in the Goldstein's Classical Mechanics text that $$\frac{\mathrm d}{\mathrm dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}=\sum_k \...
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Deriving effective potential energy from the Lagrangian of a two-body system [duplicate]

I'm having some issues understanding how the effective potential energy of a two-body system is derived from the Lagrangian of the system. Specifically my issue is with one step... Suppose we are ...
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Dirac Brackets in General Relativity

I want to calculate Dirac brackets of different phase space variables in gravity. In case of electrodynamics, one does the same using the following steps: Looking at the momenta to find that $\Pi^...
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How do I calculate the tension in the string of a yo-yo with Lagrange? [on hold]

I can not figure out, how to solve the last equation for λ. I know the moment of inertia is $1/2·m·R^2$ Can somebody please help?
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Is it possible to create Poincaré sections using a Lagrangian?

I have a triple pendulum Lagrangian and equation of motion, and now I'm thinking of creating Poincaré sections. Do I have to find the Hamiltonian first?
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What is the physical interpretation of the action integral, without the stationary action principle?

I'm still wondering about the physical interpretation of the action integral of some mechanical system (classical theory here, to simplify things): \begin{equation}\tag{1} A = \int_{t_1}^{t_2} L(q, \, ...
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Noether, Conservation and Transformations

I have the Lagrangian $$L(\lambda,\mu,\dot{\lambda},\dot{\mu})=\frac{m}{2}(\lambda^2+\mu^2)(\dot{\lambda}^2+\dot{\mu}^2)-\alpha \lambda^2\mu^2,$$ and the transformations $$\lambda'=\lambda+\frac{s\...
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Lagrange function and a differential equation with a cylinder

I need some help with the following task. So I have a cylinder which you get by rotating the curve $z=-\frac{\alpha}{2r^2}$ around the $z$-axis. There is a mass moving on this cylinder. It can move ...
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1answer
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The physical meaning of the derivative $\frac{\partial{L}}{\partial \dot q_i}$ of a Lagrangian

The lagrangian is defined as $$L = T - V$$ where $T$ is kinetic energy and $V$ potential energy. Then the euler-lagrange-equation is $$ \frac{d}{dt} \frac{\partial{L}}{\partial \dot q_i} = \frac{\...
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Proof of the existence of the energy-momentum tensor [duplicate]

I have a problem providing or finding a general proof for this statement i found in Mussardo's statistical field theory book, section $10.3.2$: Due to the locality of the theory there exists a local ...
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Gyroscopic precession on a friction-less surface

I am having trouble understanding the total energy for a heavy spinning symmetric top (Gyroscope) on a friction-less surface. I am trying to understand it via the Lagrangian of the gyroscope. My ...
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Equivalence between Dirac and Majorana action in CFT

In Mussardo's Statistical field theory Chapter 12, section 12.3 about the conformal field theory of a free fermion field he talks about the complex fermion field (Dirac field) $$ \Psi(z,\bar{z}) = \...
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Is it possible to define an energy momentum tensor for classical point particles from a QFT?

I have a question about the semi-classical limit of a QFT that so far I have never been able to solve. Let's start with a second quantized Klein-Gordon field with Lagrangian $$\mathcal{L}(\phi)=\frac{...
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Is $F^{\mu\nu}F_{\mu\nu}$ equivalent to $A^{\mu}\nabla^{\alpha}\nabla_{\alpha}A_{\mu}$ for $U(1)$ gauge field lagrangian?

The two seem to yield the same equation of motion is why I asked. Where of course the standard notation for exterior forms applies $dA=F$. We all know how the field strength tensor plays into the ...
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Derivation of EL equations for real scalar field

I am looking to derive Eq. (1.11) from these notes on QFT from Tong’s notes: http://www.damtp.cam.ac.uk/user/tong/qft/one.pdf But Im the second equation, is there not suppose to be a factor of 1/2 ...
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Period of coupled cycloidal pendula

If you were to couple two pendula following cycloidal paths would they still show similar properties to a single cycloidal pendulum? For example, could you in some sense still say that the period is ...
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1answer
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2 body problem question

I am taking a classical mechanics course and i am struggling to determine the equilinrium points for this potential $$Q= -(k/r) + ar + (L^2/mr^2).$$ The problem is that i have to solve a cubic ...
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Movement on a cylindersurface by using Lagrange formalism [duplicate]

A particle with mass $m$ moves under the influence of gravity smooth on the surface of a cylinder with radius $R$. The cylinderaxis is horizontal , so orthogonal to the gravity. task: Find the ...
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Shortest Time Path for a rocket with simple friction

The problem There is a rocket in space at point $a$ whose final destination is point $b$. Find the path of minimum time that goes from $a$ to $b$. To make the model realistic and avoid infinite ...
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Deriving equations of motion using Lagrangian formalism for inverted pendulum on a cart with friction between pendulum and pin

I know this is a problem with plenty of examples and solutions but I can't find one where the friction between pendulum and pin (I know it's negligible) is taken into account. My main question is if ...
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Physical Constraints

In physics, what does one mathematically mean by constraint in classical mechanics? What are the the different types/cases and how do people deal with them?
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$U(1)_V$ invariance

I'm working with an interaction Lagrangian of the form: $${\cal L}_{int} = \bar{\psi}\Theta\chi \tag1$$ Where $\Theta$ contains other operators, coupling constants, etc. I'm trying to unveil if ...
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Effect of Co-ordinate Change on Euler-Lagrange Equations for Scalar Fields

Consider a single scalar field $\phi$ on a manifold $\mathcal{M}$. Suppose in $\{x^\mu\}$ co-ordinates, the Lagrangian density is $\mathcal{L}(\phi, \frac{\partial \phi}{\partial x^\mu})$. This means ...
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Lagrangian method for finding Lorentz transformation [closed]

Consider a very general case: one reference system (x', y', z') is moving away from another fixed system (x, y, z) with constant speed $v$ (like in the fig. a little bit below). Normally, using the ...
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1answer
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Feynman Lecture Principle of Least Action: Glossed over Taylor expansion?

His initial one dimensional derivation of Newton's Second Law using the Principle of Least Action, I believe is fairly concise and easy to read. However, I did get hung up on his use of the Taylor ...
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Question about the concepts of Noether charge and Noether current

I read that a noether current occurs when the lagrangian assume vector values. Well, what are noether current and noether charge in comparison to elementary classical mechanics notions of Noether's ...
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2answers
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When the constraints are not holonomic, why is it not possible to find such $q_i$s that $\delta q_i$s are independent of each other?

In the book of Classical Mechanics by Goldstein, at page 20, it is given that However, I cannoot understand from what has been presented so far that when the constraints are not holonomics, why is it ...
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1answer
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D'Alembert's principle when the mass of the particles are changing

In the book of Classical Mechanics by Goldstein, at page 19, while deriving D'Alembert's principle, the author assumes that $$\dot p = m \ddot r.$$ However, when the mass of the bodies also changes, ...
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1answer
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Majorana fermions

If you write the Majorana spinors as $$\chi = \begin{pmatrix}\psi_L\\ i\sigma_2\psi_L^* \end{pmatrix} \tag1$$ It satisfies the Dirac equation that leads you to the Majorana equation $$i\bar{\sigma}^\...
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1answer
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Lagrangian of free particle - classical case

I have a question, more related to a mathematical aspect of physics, which seems I am not understanding very well. So, by applying Galilean transformation between two reference frames, which move at ...
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What does it mean if the Lagrangian density has explicit spatial dependence?

First off, I have seen this post here which asks seems to ask my question, but it is not properly answered. If the Lagrangian has explicit time dependence, then the total energy, and Hamiltonian, is ...
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Why the Lagrangian of a free particle cannot depend on the position or time, explicitly?

On p. 5 in $\S$3 pf the book of Mechanics by Landau & Lifshitz, it is claimed that [...] for a free particle, the homogeneity of space and time implies that Lagrangian cannot depend on ...
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Effective low energy actions in string cosmology

I wonder why can SUGRA actions be used in string cosmology, since in some cases they are used to model big bang scenarios (for example the pre big bang scenario in chapter 4 of Gasperini's "Elements ...
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Newtonian mechanics: pendulum spring system [closed]

I am thinking about the following pendulum-spring system. But something is off with my equation of motion. Problem We have a uniform rod of length L with mass m pivoted at one end. We ...
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Obtaining Spinor Lagrangian from Copying 4-Vector Lagrangian

I am reading Schwartz's book on QFT and I am currently on writing a lagrangian for spinors. We note that $\psi^\dagger_R\psi_R$ and $\psi^\dagger_R\vec{\sigma}\psi_R$ can be written as a 4-vector (as ...
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Lagrangian of EM field: Why the $B$-field term has a minus sign in front of it in the Lagrangian?

I know that $L = T - U$ and that, in the non-relativistic case $$L= \frac{1}2mv^2 - q\phi(r,t) + q\vec{v}\cdot\vec{A}(r,t).\tag{1} $$ My lecturer used the following form of the Lagrangian density ...
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Integrate inertial forces in static stability analysis

I'm calculating the maximum containerload of a reachstacker. This is a vehicle that stacks containers in harbors. Therefore, I started by doing a static stability analysis. This means calculating the ...
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1answer
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Discrepancy in Lagrangian to Hamiltonian transformation results? [closed]

I know, $$ L=T-V \;\;\; \; \;\;\; [1]\;\;\; \; \;\;\; ( Lagrangian) $$ $$ H=T+V \;\;\; \; \;\;\;[2] \;\;\; \; \;\;\; (Hamiltonian)$$ and logically, this leads ...
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Using the Lagrangian Method to Solve Bead on Rotating Rod [closed]

The following problem appeared on the Princeton University Physics Competition in 2017: A bead of mass $m$ is free to slide along a thin rod of length $L$ tilted at angle $\phi$ to the vertical. ...
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1answer
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Equilibria and conserved quantities in a superposition of central potentials

I have a Lagrangian made up of a superposition of $k$ central repulsive potentials centered at $(a_k,b_k)$. Each potential is somewhat strange - they take the form $V(r)=-\ln\left(r\right)$. The whole ...
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2answers
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Finding the value of the holonomic constraint forces

So let's say I have a Lagrangian augmented with some holonomic constraints. $$L' = L + \sum_i \lambda_i(t) f_i(q,t).\tag{i}$$ The solutions is the system of differential equations: $$\frac{\partial ...
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Lagrange's equation of second kind - how to find constant for a pendulum motion solution?

This could be a more general question about pendulums but I'll show it on an example. We have a small body (mass $m$) hanging from a pendulum of length $l$. The point where pendulum is hanged moves ...
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Maxwell Lagrangian where $F$ and its derivatives are the variables (i.e., without replacing $F={\rm d}A$) [duplicate]

The way the electromagnetic Lagrangian is usually constructed is by noticing that the EM fields are always constrained to satisfy ${\rm d}F=0$ (half of Maxwell's equations). We can immediately solve ...
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Is Higgs mechanism generate mass parameter? Or physical mass?

Is Higgs mechanism generate mass parameter in the Lagrangian? Or physical mass that we can observe in experiment? It is obscure for me how Higgs mechanism produce mass for elementary particle. ...
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Definition of integral functional [duplicate]

I'm reading the section of Marion and Thornton devoted to basics on the Calculus of Variations, and came across this definition for the functional: $$J = \int f(y(x), y'(x);x) dx$$ implying that $f$ ...
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3answers
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Confusion over use of contravariant notation in Noether's theorem and Lagrangian filed theory

The variational principle clearly gives $$\frac{\partial \rho}{\partial t} + \overrightarrow{\nabla}\cdot \mathbf{J} = 0.$$ So the sign is positive. However in my lecture notes it is claimed that ...
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Generalized coordinates for a floating planar linkage

I am trying to derive the Euler-Lagrange equation of motion for a planar bipedal robot that consists of the body (a link) and 2 two-link legs for a total of five links. $$D(q)\ddot{q}+C(q,\dot{q})\...
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Is it possible for the Action $S$ to *not* have a stationary point?

So the path of an object in configuration space is given by Hamilton's principle, which states that the path which the particle travels on is the one on which the action is stationary: $$\delta S = \...