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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199 views

Equation of motion for cyclic model of the universe

I recently started to study about cyclic universe. I came across this article [1]. My question is about the action that used for describing the cyclic model: $$S = \int d^{4}x\sqrt{-g}(\frac{1}{16\pi ...
6
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312 views

Mechanical similarity in Landau

I've read this very short paragraph from Landau & Lifshitz's Mechanics (Chap.2, Par.10) (that you can find here) about Mechanical similarity. I was looking for some more detailed explanations of ...
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57 views

Hamiltonian Operator for nonrenormalizable Effective Field Theories?

Assuming we have a Effective Field Theory, for example a Real Scalar Field Theory, defined through a Lagrangian density of the form $\mathcal{L}_{eff} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi - ...
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425 views

Is Feynman talking about the Zeroth Law of Thermodynamics?

In Volume 1 Chapter 39 of the Feynman Lectures on Physics, Feynman derives the ideal gas law from Newton's laws of motion. But then on page 41-1, he puts a caveat to the derivation he has just ...
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456 views

General equation of motion for elementary particles

Elementary particles can be grouped into spin-classes and described by specific equations, see below: Is there a general Lagrangian density from which all these equations can be derived? A ...
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92 views

Gauge Invariance of Yang Mills Lagrangian

I am trying to show the invariance of the following Yang Mills Lagrangian: $$L= -\frac{1}{4} F^a_{\mu \nu} F_a^{\mu\nu} + J_a^\mu A_\mu^a$$ under the following gauge transformation ($\theta$ being a ...
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102 views

Gauge Invariance of the Non-abelian Chern-Simons Term

I'm trying to prove that, under the gauge transformation $$A_{\mu} \rightarrow A_{\mu}^{\prime} = g^{-1} A_{\mu} g + g^{-1} \partial_{\mu} g,$$ the non-abelian Chern-Simons Lagrangian density: ...
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133 views

How do you determine the Lagrangian?

I have always been puzzled by how do you arrive at Lagrangians? That is, how do you know that the functional you need to get Newton's equations is $L$ = $T-V(x)$ Do you derive the Lagrangian first ...
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30 views

Any good resources for Lagrangian and Hamiltonian Dynamics?

I'm taking a course on Lagrangian and Hamiltonian Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics. So far at my university ...
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60 views

Understanding the effective low-energy Lagrangian for hadrons

My course in Higgs Physics is discussing a two-nucleon low-energy effective theory of hadron interaction. With $\psi=(p,n)$, the pion is defined as $\vec{\pi}= i \bar{\psi}\vec{\tau} \gamma_5 ...
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64 views

Parity invariance of Einstein-Hilbert Lagrangian

How can we show that the Einstein-Hilbert action is Parity invariant? $$ S_{EH}=\int \sqrt{-g}R d^4x $$
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68 views

Is the “Force” of Gravity Simply Hamilton's Principle on a Curved Spacetime?

It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it ...
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73 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial ...
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57 views

Does a local symmetry transformation cause a change in kinetic energy?

Consider a local transformation $$ \varphi_i^{\prime}= \varphi_i + \alpha(x) \delta\varphi $$ If this is a symmetry of the action, the Lagrangian is given by $$ \mathcal{L}^{\prime}=\mathcal{L}+ ...
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251 views

Derivation of Euler-Lagrange equations for Lagrangian with dependence on second order derivatives

Suppose we have a Lagrangian that depends on second-order derivatives: $$L = L(q, \dot{q}, \ddot{q})$$ If we're working on the variational problem for this Lagrangian, then I know that we'll wind up ...
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84 views

Isn't the Jacobi constant just the Lagrangian times 2?

At this wikipedia page the Jacobi constant is expressed as: $$C_J=2\left(\frac{v^2}{2}-U\right)$$ where $U$ is the potential energy and $v$ is velocity. If kinetic energy $T$ is defined (as it ...
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85 views

Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: ...
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163 views

Angular momentum of particle in dipole magnetic field

Basically I'm just trying to find the expression for the angular momentum of a particle of mass $m$ and charge $q$ in a dipole magnetic field. In cylindrical coordinates, ...
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42 views

Boundary term in Einstein-Hilbert action

Why is the boundary term in the Einstein-Hilbert action, the Gibbons-Hawking-York term, generally "missing" in General Relativity courses, IMPORTANT from the variational viewpoint, geometrical setting ...
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89 views

Hoop rolling inside a circular hole

A hoop of radius $b$ and mass $m$ rolls without slipping within a stationary circular hole of radius $a > b$ and is subject to gravity. Use the generalized coordinates the rotation angle $\phi$ of ...
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0answers
51 views

Equations of motion for controlled/driven classical systems? Does D'Alembert's principle apply?

I'm puzzled about how to derive the equations of motion for certain classical systems where some entity is controlling some of the DOFs. For example, consider a double-pendulum, with lengths $l_1$ ...
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52 views

Local symmetry and General Relativity

First I want to consider an example of 1D motion. Lagrange equation: $$ \frac{d}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0 $$ If we transform $ L \rightarrow L+a $ ...
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54 views

Building free energy directly from Lagrangian

Energy can be built from Lagrangian when considering the symmetry of time $(\frac{\partial L}{\partial t}=0)$. Free energy is a generalization of energy when the system exchanges heat with the ...
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71 views

Feynman rules of a theory in non-standard form

I am currently studying lecture notes by Akhmedov on interacting scalar field theory in de Sitter space. In these notes, he considers a scalar field theory of the form ...
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38 views

Translation symmetry and the non-conserved momentum in Viscous fluids

Even though a viscous fluid has a translation symmetry (invariance) for its Lagrangian , it still 'waste' Linear momentum. How come ?, isn't the rule that every symmetry yields a conservation law ?
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160 views

Lagrange's Equations for a Tetherball

I'm trying to write down the equations of motion for a tetherball moving around a pole while the string is getting shorter. --- MAJOR EDIT --- I started with Lagrange: $$ x(t)=l(t) \sin (\theta) ...
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113 views

Parity violating Dirac particle

We normally write down the Dirac Lagrangian as \begin{equation} {\cal L} _D = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi \end{equation} but are the Lagrangian's, \begin{equation} ...
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237 views

Trouble with calculating Christoffel symbols of FLRW metric using Lagrangian method

The FLRW metric which I am using is $$ds^2 = dt^2 - \frac{a(t)^2}{c^2} \left( dx^2 + dy^2 + dz^2 \right)$$ where $a(t)$ is the so-called 'scale factor'. I did not want to calculate the Christoffel ...
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51 views

What's the value of the coupling constant in interacting field theories?

Consider this Lagrangian : $L = \frac{1}{2}(\partial_\mu \Phi)^2 - \frac{M^2}{2}\Phi^2 +\frac{1}{2}(\partial_\mu \phi)^2 -\frac{m^2}{2} \phi^2 -\mu\Phi\phi^2$ Its interaction term is given by : ...
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33 views

Total Vs Partial in Lagrange density?

I have a question regarding the red term below. This is the integration by parts during the derivation of the Euler-Lagrange equation for continuous systems. Why is this not the time derivative ...
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30 views

Equations of motion with replacing the Lagrangian by irrep diagrams generating functional

I have read that equations of motion of ghosts is equal to $$ \tag 1 \frac{\delta \Gamma}{\delta \bar{c}^{a}(x)} = -\partial^{\mu}_{x}\frac{\delta \Gamma}{\delta K^{\mu , a}(x)}, $$ where $\Gamma = W ...
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41 views

Intuition behind the principle of virtual work

To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset ...
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55 views

Constructing conserved current given the lagrangian

Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ ...
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62 views

Lagrangian Systems

Given a manifold $M$, Arnold's "Mathematical Methods of Classical Mechanics" defines a Lagrangian system as a pair $(M,L)$ where $L$ is some smooth function on the tangent bundle $TM$. The function ...
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54 views

Lagrangian for FRW metric

For the metric $$ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ $$L= \sqrt{-g_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}}$$ How does this become $$L= \sqrt{1-a^2 (\frac{dx}{dt})^2}~? $$ I guess ...
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27 views

Source material desired for behavior of derivatives of action

I'm basically looking for concise commentary, and especially source material/ short discussion pertaining to the following, which I will (emphasizing loosely) state as follows: Suppose a given action ...
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0answers
37 views

Hamiltonian flow is volume preserving

I was reading about advantage of Hamiltonian over Lagrangian. One of the advantage is "Hamiltonian flow is volume preserving". Can anyone help me to understand this? Means what is advantage of ...
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0answers
52 views

path integral quantization of EM field derived from canonical quantization?

In Peskin's QFT book page 294, he formally addressed the quantization of EM field, $$propagotor_{EM}=\frac{-ig_{\mu\nu}}{k^2+i\epsilon}$$ Now that we have the functional integral quantization ...
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24 views

The equation of the location of L1

On http://en.wikipedia.org/wiki/Lagrangian_point#L1 it says that the location of L1 can be determined as $\frac{M_1}{(R-r)^2}=\frac{M_2}{r^2}+\left(\frac{M_1}{M_1+M_2}R-r\right)\frac{M_1+M_2}{R^3}$, ...
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0answers
78 views

Restrained double pendulum

The equations of motion of a double pendulum are well-known. Usually you'd have the them expressed in the rotations $\theta_1(t)$ and $\theta_2(t)$. There are two degrees of freedom. Now consider the ...
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404 views

Ball rolling without slipping inside a hollow cylinder

A small ball of radius $r$ performes small oscillations within a hollow cylinder of radius $R$. What would be the angular frequency of the oscillations given that the rolling is without slipping? The ...
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80 views

Derivation of equations of motion in Nordstrom's theory of scalar gravity?

Nordstrom's theory of a particle moving in the presence of a scalar field $\varphi (x)$ is given by $$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d ...
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78 views

How to analyze this constraint question

Let $\gamma$ be a smooth curve in the plane, and introduce curvilinear coordinates $q_1,q_2$ on a neighborhood of $\gamma$; $q_1$ is the direction of $\gamma$ and $q_2$ is distance from the curve. ...
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101 views

A discrete approach to the catenary

I'm trying to work out a model for the system above, that is, $N$ particles of unitary mass subject to the constraints: $$1=\varphi _i(\mathbf r _1,\mathbf {r}_2,...,\mathbf r _n)=|\mathbf ...
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37 views

Where are the L3, L4, and L5 of a hyperbolic orbit?

Do the L3, L4, L5 points exist in hyperbolic orbits? If yes, then where do they lie?
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77 views

How to introduce generating function in Hamiltonian formalism for field theories?

Let's have hamiltonian $$ H(\psi , P ,\partial_{i}\psi ) = P\partial_{0}\psi - L(\psi , \partial_{\mu}\psi ), \quad P = \frac{\partial L}{\partial (\partial_{0}\psi)}. \qquad (.0) $$ I tried to ...
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113 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
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1k views

Equations of motion for a pendulum and spring system

The question is available here: I've modeled the building as a rod on a torsional spring (with a pendulum hanging from the top). $\phi$ is the angle from the centre for the pendulum and $\theta$ ...
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135 views

Proving conservation of angular momentum in an elliptic billiard problem

This is for a course focusing on the connections between Newtonian, Lagrangian and Hamiltonian formalisms. We're given an elliptic billiard table with foci 1 and 2, where $L_1$ and $L_2$ are the ...
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100 views

Lagrangians for non-local equations of motion

Say I have a multicomponent field $X_a(x,t)$ such that I know it Fourier modes satisfy the following equation of motion, $(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(k,t) = e^t \int \frac{d^3p ...