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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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Equations of motion for a Weyl spinor in the context of SUSY

I'm learning supergravity from the textbook of Antoine Van Proeyen (this is from page 114). Suppose I'm given a Lagrangian $$ \mathcal{L} = - \partial^{\mu} \bar{Z} \partial_{\mu} Z - \bar{\chi} \...
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1answer
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Deriving the geodesic equation using a Lagrange multiplier to fix affine parametrisation

The geodesic equation can be derived using the action $$S_0 ~=~ \int d\tau \sqrt{-\dot{x}_\mu\cdot \dot{x}^\mu}\tag{1}$$ (I am using the (-+++) convention and $\dot{x} = \frac{dx}{d\tau}$). To ...
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1answer
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Lagrange Equation - Basics

The basic equation of Lagrange is given by, $$\frac{\mathrm d}{\mathrm dt} \frac{\partial L}{\partial \dot{q_j}} - \frac{\partial L}{\partial q_j} = Q_j \tag{1}$$ where $T$ is the kinetic energy, $V$ ...
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Cauchy problem for Hamilton-Jacobi equation

In Arnol'd V.I, "Mathematical methods of classical mechanics" p.257, I was asked to find a solution for the Cauchy problem $$H=\frac{p^2}{2},\ \ \ S_0=\frac{q^2}{2}$$ of the Hamilton-Jacobi equation ...
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Gimbal camera stabilizer

I want to design a camera stabilizer that stabilizes a GoPro in 3 dimension (pitch, yaw and roll). See the picture for a visualisation. I want to write a simulink model with in it the equations of ...
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Do Newton's laws imply independence of generalised coordinates and generalised velocity? [duplicate]

When deriving lagrangian formulation from Newton's laws, We use the fact that generalized velocity and generalized coordinates are independent of each other. Is there a mathematically rigorous proof ...
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1answer
38 views

Polyakov Lagrangian and Lagrange multipliers

I'm reading Polchinski's Introduction to String Theory (volume I) and something got me quite puzzled in the beginning (At the top of page 19 to be precise). This part is about the open string and the ...
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Solve equations of motion to find $\Phi(t)$ for a pendulum with an infinite period

Consider a pendulum of mass $m$ and length $l$ that can rotate in the vertical plane subject to the gravitational field $g$. Write down the Lagrangian and solve the resulting equation of ...
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1answer
109 views

Is this theory equivalent to QED?

I've found the following Lagrangian $$\mathcal{L}=i\bar{\psi}\gamma^\mu\left(\partial_\mu-ieA_\mu -ieA'_\mu\right)\psi-m\bar\psi\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}F'_{\mu\nu}F'^{\mu\nu}.$...
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Definition of generalized 4-momentum

In nonrelativistic mechanics, given a lagrangian $L$, we define the action as $$S[q]=\int L(q(t),\dot q(t))dt\tag{1}$$ and we can prove (see, for example, this answer for eq. (2)) $$\begin{align} \...
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Equation of motion for double pendulum [closed]

I have derived the Lagrangian of a double pendulum considering the pivot point of the first pendulum is the origin. Am I correctly derived the Lagrangian?
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1answer
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Lagrangian of point mass on rod inside ring with holonomic constraint

I am given a massless ring of radius $R$ that is rolling along a flat plane without slipping. There is friction. A massless rod of length $\frac {R} {2} $ is attached to the inside edge of the ring ...
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1answer
63 views

Doubt in Functional Derivative of Lagrangian

Lecture XXXIII: Lagrangian formulation of GR by Christopher M. Hirata NON-INTERACTING DUST Consider a system with a suite of particles {A} each of mass $\mu_{A}$ following some set of trajectories $...
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Classical Nambu-Goto action of a rotating string

I have just started to read the book "A First Course in String Theory" by Zwiebach and i'm trying to get some feeling about the motion of classical relativistic strings. I am looking for example at ...
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1answer
66 views

How to integrate by parts ghost fields in electrodynamics?

When applying Faddeev-Popov method to electrodynamics in the Lorenz gauge we obtain the ghost action $$S=\int d^4xd^4y\bar\eta(x)\left(\partial^2\delta(x-y)\right)\eta(y),\tag{0}$$ where $\partial^2$ ...
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Goldstein central force problem [closed]

I was reading Goldstein chapter 3 central force motion. I tried to do the exercise 21 on p. 130 of 3rd ed. but cant think of any way how to do it. any clue would be very helpful. Show that the ...
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Scattering amplitude ($s$-, $t$-, $u$-channels)

Given the Lagrangian $$\mathscr{L}=\bar{\psi}\left(i\partial\!\!\!/-m\right)\psi+\frac{1}{2}\left(\partial\phi\right)^2-\frac{1}{2}M^2\phi^2-g\bar\psi\psi\phi^2-g'\bar\psi\gamma_\mu\psi\partial^\mu\...
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30 views

Partial derivative in generalized coordinates [duplicate]

Sorry for my broken English. I'm a physics undergrad and quite poor at math. I just started to learn analytical mechanics and it really confuses me. my analytical textbook uses the equations below ...
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2answers
76 views

$U(1)$ Scalar Field Theory: Why no $| \phi |$ term?

When we write down the lagrangian of a general $U(1)$ scalar field theory we generally write $$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi^* - \frac{m^2}{2}\phi \phi^* - V(|\phi|^2)$$...
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Deriving gyroscope basic equation from Lagrangian formalism [duplicate]

The following represents the equation of motion of a gyroscope: $$ \frac{d\vec{L}}{dt} = \vec{r} \times m\vec{g} $$ where $\vec{L}$ is the gyroscope's angular momentum, $\vec{r}$ is the vector ...
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Variation with respect to a traceless symmetric tensor

Suppose we have an action variation like $$\delta S[G]=\int \mathfrak{H}^{\mu\nu}\delta G_{\mu\nu} \,\, d^Nx,$$ where $\mathfrak{H}^{\mu\nu}$ is a tensor density. If the variation with respect to $...
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1answer
73 views

Maxwell equations of motion from $S = \frac{-1}{2} \int F \wedge \ast F$

I'm trying to understand the following equation, used in the derivation of the equations of motion. Let $S = \frac{-1}{2} \int F \wedge \ast F$ and $F = dA$. Let $\delta$ denote variation. Then $$...
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1answer
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Correction to the fermion propagator

Given the Lagrangian $$\mathscr{L}=\bar{\psi}\left(i\partial\!\!\!/-m\right)\psi +\frac{1}{2}\left(\partial\phi\right)^2- \frac{1}{2}M^2\phi^2 - g\bar{\psi}\psi\phi^2,$$ calculate the propagator ...
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2answers
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Must the varied paths in the action be physically possible?

For simplicity without loss of generalization, consider a free particle. When using the Principle of Least Action, I imagine all variations of the true path between $t_1, t_2$ regardless of whether ...
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1answer
34 views

Relativistic action is a constant?

Say that you want to find the equations of motion of a free relativistic massive point particle by minimizing the action $$S=-m\int\mathrm{d}\tau\,\sqrt{\eta_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\...
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1answer
48 views

Application of Darboux's theorem to magnetic field line flows

In this famous paper by Cary and Littlejohn on noncanonical Hamiltonian mechanics and its application to magnetic field line flow, they claim that as a result of Darboux's theorem, it is always ...
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Calculus of Variations. Finding the extremals of a perturbed Lagrangian [closed]

Im trying to solve the following problem: Approximate with an error of $O(\epsilon ^3)$ the extremals of the Lagrangian $$L(y,y',x) = y^2 + (y')^2 - 2y \sin(x) + \epsilon y^3$$ with $y(0)=1$ ...
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1answer
27 views

Angular momentum conservation reduces degree of freedom

In 2 dimention dynamics, if angular momentum is conserved: mr^2(theta dot)=constant, does that mean degree of freedom is reduced from 2 to 1? I think it should since r and theta(although theta dot ...
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Harmonic Oscillator and Shifts in Derivative Operators

What symmetries/symmetry breaking arises from shifts in the derivative operators? To explain what I mean let's study an example. The classical one particle one dimensional harmonic oscillator has the ...
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Is it possible to derive the correct QED Lagrangian without demanding local gauge invariance?

Usually, the correct interaction term $A_\mu \bar{\Psi} \gamma_\mu \Psi$ in the Lagrangian is derived by demanding local gauge invariance. Is there any other argument that fixes the form of the ...
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1answer
47 views

What is the relationship between velocity-dependent potentials and non-Abelian gauge fields?

My (limited) understanding of non-Abelian gauge fields is that they arise from the construction of a theory using a non-Abelian Lie group (as a generalization of the Abelian group underlying E&M) ...
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1answer
66 views

Schwinger's variation of the action of point particle with *both* time and position as independent variables

In Chapter 8, pages 86-87, equations (8.5)-(8.11) of Julian Schwinger et al., Classical Electrodynamics, the equations of motion for the following action principle of a point particle in an external ...
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Lagrangian perturbation theory

I have a system of coupled non-linear differential equations which stem from a Lagrangian and the Euler-Lagrange Equations. I want to solve them with perturbation theory. I know that in Quantum ...
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1answer
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Variation of Antisymmetric tensor's trace?

Does an action of the form $$S=\int f(\text{tensors})g^{ij}\nabla_kA^k_{ij}\ d^4x$$ where $A$ is antisymmetric in the lower indices produce, upon variation, any nontrivial equation? Given that the ...
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1answer
75 views

Why does the Hamiltonian represent something different after plugging in the solution?

so I am beginning to learn Hamiltonian mechanics. We have learned that the Hamiltonian is a function of q, p, and t. Once we have a Hamiltonian, we can use the Hamiltonian equations to derive the ...
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2answers
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Generalized force arising from a velocity-dependent potential

On slide 16 of this presentation it is stated without proof that given a velocity-dependent potential $U(q,\dot q, t)$, the associated generalized force is $$Q_j = - \frac{\partial U}{\partial q^j} + \...
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1answer
44 views

Intuition for free-particle action principle?

Consider a particle constrained to a manifold $\mathbb Q$ embedded within standard Euclidean 3-dimensional space that experiences no forces other than constraint forces keeping it within $\mathbb Q$. ...
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Finding a symmetry in a Lagrangian that is not an ignorable coordinate?

I have particle in a fixed electromagnetic field $$ E = E_0 \hat y\\ B = B_0 \hat z $$ The Lagrangian for such a system is: $$ L = \frac{1}{2} m \dot x^2 + \frac{1}{2} m \dot y^2 + \frac{1}{2} m \...
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Independence of generalized coordinates and generalized velocities [duplicate]

I think this might be a very basic doubt, but in the Lagrangian method of classical mechanics, we assume the generalized coordinate $q_{i}$ and the corresponding velocity $\dot q_{i}$ are independent. ...
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1answer
74 views

Show that a theory is scale invariant

I'm a bit new to this invariant transformations for fields so I've been having trouble manipulating them and I would appreciate any guidance. I saw in this wikipedia article that, for example, a $\...
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1answer
48 views

Conserved Charges

If we are studying a physical system which has some symmetries, why do we calculate conserved charges? What do the conserved charges tell about the system that the symmetries do not?
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Derivation of the energy-momentum tensor from matter Lagrangian

We know that the energy momentum tensor for a perfect fluid is given by $$ T_{\mu\nu} = \left(\rho+{p\over c^2}\right) v_\mu v_\nu + p g_{\mu\nu}. $$ How can we derive it from a Lagrangian? Which ...
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Self-modifying Hamiltonians (Lagrangians) and emerging intelligence? [closed]

Are there dynamical physical systems that are governed by self-modifying Hamiltonians (Lagrangians), i.e. Hamiltonians (Lagrangians) determine not only the next point in phase space, but also the form ...
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When and why is $\frac{d}{dt}\delta q^{i}=\delta \frac{dq^{i}}{dt}?$ true? [duplicate]

Following Susskind, a variation of the form $$\delta q^{i}=f^{i}\left[\left\{q\right\}\right]\varepsilon,$$ such that the consequent variation of the Lagrangian is $\delta{L}=0,$ is said to be a ...
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Inverted double pendulum - difference between hand calculation and Simulink/Simscape

I have following problem - I'm using Lagrange's equations to describe inverted double pendulum with torsion spring at each of two joints. I solved obtained ODE in Matlab, receiving functions $\Phi_1(t)...
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1answer
49 views

Functional Poincaré's lemma and the inverse Lagrangian problem

I have only encountered the inverse Lagrangian problem in mathematics books that treat Lagrangian field theory using jet bundles and homological algebra, and while I am studying this approach, I still ...
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84 views

Noether's Theorem, Boundary-Bulk, and Open Thermodynamic Systems

Before going any further, I should emphasize that I know we cannot use the action principle for locally dissipative systems or even Noether's theorem for that matter. There are plenty of stackexchange ...
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43 views

Physical meaning of the WZW action and Lagrangian

What is the (super)-WZW term physical meaning? I mean, what is the physical importance of the Wess-Zumino-Witten action/Lagrangian in superstrings/M-theory and or field theory (not stringy if ...
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1answer
52 views

Hamiltonian for a variable length pendulum

This question is taken from the book "Classical Dynamics of Particles and Systems" - Marion, problem 7.24. The problem is about a pendulum that is set into motion, it's length varies at a constant ...
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Transformation of the Stress-Energy tensor [closed]

My question is related to this one. However in my case, the Lagrangian can depend on higher order derivatives (so the second point made doesn't hold). Can someone help me with it?