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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293 views

Lorentz force from velocity-dependent potential and Lagrangian

There is something i'm missing. I am at page 22-23 of Goldstein Classical Mechanics 3rd ed. Lorentz force can be derived from a potential $$U=q\phi-q\mathbf{A}\cdot\mathbf{v}$$ Where $\phi(t,x,y,z)$ ...
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1answer
77 views

Lagrange multiplier dependent on time

At the moment I am following a course on variational methods for mathematicians. Last week we derived the Euler-Lagrange equations for a functional under a constraint. In this derivation we found that ...
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1answer
165 views

Working with a Routhian for a specific system

I asked a more general question earlier about the Routhian, but I'm still having trouble working with it. Here's my specific case. Given the following Lagrangian: ...
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1answer
256 views

How do you derive Lagrange's equation of motion from a Routhian?

Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\frac{d}{dt}\frac{∂R}{∂\dot{\phi}}-\frac{∂R}{∂\phi}=0$$ And ...
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1answer
227 views

Virtual displacement and generalized coordinates

I have a doubt regarding the expression of a virtual displacement using generalized coordinates. I will state the definitions I'm taking and the problem. The system is composed by $n$ points with ...
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1answer
94 views

Landau Lifshitz energy for uniform rotation

Landau Lifshitz claim in their Mechanics book (39.11) that for a uniform rotation we have $ E = \frac{mv^2}{2} - \frac{m}{2} (\omega \times r)^2 + U,$ where the rotation is given by $v' = v + \omega ...
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1answer
75 views

Finding the EOM for a charged relativistic particle

For an exercise sheet of a course in general relativity I'm asked to derive the equations of motion for a charged particle in an EM-field given by a potential $A^\mu$. I am give the action: $$S = ...
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2answers
993 views

Expression of kinetic energy in polar coordinates

Expression for kinetic energy in Cartesian coordinate: Expression for kinetic energy in polar coordinate (applying the transformation of coordinates): Why can't we express it in the following ...
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1answer
307 views

Why does Quantum Field Theory use Lagrangians rather than Hamiltonains? [duplicate]

Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains? I heard many reasons, but I'm not sure which is true. Some say it's just a matter of beauty, so Lagrangians are more ...
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4answers
588 views

Is there a proof from the first principle that the Lagrangian L = T - V?

Is there a proof from the first principle that for the Lagrangian $L$, $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$ in classical mechanics? Assume that Cartesian coordinates are ...
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2answers
482 views

Small oscillations of the double pendulum

From the Lagrangian I've got the following equations of motion for the double pendulum in 2D. (The masses are different but the lengths of the two pendula are equal.) Let $m_2$ be the lowest-hanging ...
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1answer
74 views

Are there two types of D-term and two types of F-term in SUSY?

I've noticed that one can obtain D-terms either by integrating a vector superfield (the vector multiplet) over superspace or by integrating a Kahler potential over superspace. In both cases we get ...
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1answer
137 views

Hamiltonian from Euclidean lagrangian?

Can somebody help me in deriving the Hamiltonian of system starting from Euclidean Lagrangian? Say we are given the Minkowski Lagrangian $$L_m = \frac{\dot{\phi}^2}{2} - V(\phi).$$ The Hamiltonian ...
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0answers
52 views

A posteriori solution to the Hamilton Jacobi equation

I was wondering about the following: For many simple systems it is far too cumbersome to solve the Hamilton Jacobi equation compared with the Hamilton or Lagrange formalism. Now I was wondering, ...
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2answers
314 views

Global phase symmetry for complex scalar field theory

I have started to study QFT. And I have some difficulties in such classical situation. Suppose i want to calculate $\frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\phi$ for lagrangian ...
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1answer
231 views

Non-relativistic limit of complex scalar field

In page 42 of David Tong's lectures on Quantum Field Theory, he says that one can also derive the Schrödinger Lagrangian by taking the non-relativistic limit of the (complex?) scalar field Lagrangian. ...
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2answers
316 views

Derivation of the Polyakov Action

As is usually done when first presenting string theory, the Nambu-Goto Action, $$ S_{\text{NG}}:=-T\int d\tau d\sigma \sqrt{-g} $$ ($g:=\det (g_{\alpha \beta})$ is the induced metric on the ...
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5answers
441 views

What is Quantization?

In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & ...
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1answer
111 views

Can the Solar System be assumed a single body concentrated in the Sun?

This question springs from a comment against my question posted on the Space SE My questions may seem inane, or obvious to most of you real physics people too ... Any number of sources put the peg ...
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32 views

How could I show that ${\mathcal{L}\left(e\right)}$ is ${SU\left(2\right)_{L}\times U\left(1\right)_{Y}}$ invariant?

How can I show that ${\mathcal{L}\left(e\right)}$ is ${SU\left(2\right)_{L}\times U\left(1\right)_{Y}}$ invariant by checking explicitly that the ${\chi_{L}}$, ${e_{R}}$, ${\vec{W}_{\mu}}$ and ...
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1answer
117 views

Is it possible to project a problem of mechanics in a lower dimensionality?

I had the intuition that, in classical mechanics, when the trajectory of a body is known, then analysis of its motion can be done in the linear space of that trajectory, if all forces are projected on ...
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0answers
108 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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1answer
166 views

Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0 = ...
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0answers
58 views

Setting Lagrangian [closed]

Can you help me to set Lagrangian? I found that $$\vec r_A=b\sin\theta\vec i+b\cos\theta\vec j$$ $$\dot{\vec r_A}=b\dot\theta\cos\theta\vec i-b\dot\theta\sin\theta\vec j$$ For point $G$ I've got ...
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256 views

Scalar field lagrangian in curved spacetime

I am studying inflation theory for a scalar field $\phi$ in curved spacetime. I want to obtain Euler-Lagrange equations for the action: $$ I\left[\phi\right] = \int ...
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1answer
666 views

Equations of motion for a spherical pendulum in a non-inertial reference frame

Take a spherical pendulum with bob mass $m$, rod length $\ell$ and physical coordinates $\theta$, $\phi$ (spherical angles) and $h$ (the hinge height with respect to the coordinate origin). The rod is ...
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1answer
266 views

Euler's equations of rigid body motion from least action principle

I would like to derive Euler's equations of rigid body motion from least action principle. Suppose we are in free space so we have no gravity so Lagrangian is equal to kinetic energy. $$ L = T = ...
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1answer
272 views

What is Maupertuis' principle good for?

The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \delta \int p dq =0$$ and I am not sure ...
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66 views

Determining the Action of the Electromagnetic Field - Examples

In Landau Volume 2 (page 71) an expression for determining the entire electromagnetic Lagrangian is given. What would be an explicit numerical examples of working this idea out along the lines Landau ...
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1answer
166 views

Beads sliding on a hoop

Two particles, $P_1$ and $P_2$, of equal masses $m$ are linked by a spring of stiffness $k$ and natural length $a$. They are sliding freely without friction along a horizontal fixed hoop of radius ...
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881 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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1answer
240 views

D'Alembert's principle

Actually I have some troubles to understand what this principle is all about, so I want to use the simple pendulum in order to get the idea. Since I have read a few passages that dealt with this ...
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1answer
258 views

How to introduce the electromagnetic field in Quantum Field Theory?

There are many ways to introduce the electromagnrtic field in Quantum Field Theory(QFT), such as canonical quantization method which introduces the creation and annihilation oprators by treating the ...
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2answers
392 views

Symmetry of Euler-Lagrange equations and conservation laws

Continuous symmetry of the action implies a conservation law, but what if equations of motion have a continuous symmetry? Does it imply a conservation law? Also is symmetry of equations of motion ...
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2answers
308 views

How do I read the simple, but contradictory, Lagrangian ($\mathcal{L} = x + v$)?

I understand the lagrangian formulation of classical mechanics, to a degree. I can derive the Euler-Lagrange equations from the "least" action principle, and equivalently can determine the equations ...
3
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2answers
325 views

Lagrangian and conservation of energy

If Lagrangian of the motion is $$\mathcal{L}=\frac{1}{2}m\left(a^2\dot\phi^2+a^2\dot\theta^2\sin^2\phi\right)+mga\cos\phi,$$ how can I show that total mechanical energy is conserved? I've read ...
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1answer
126 views

Applying Lagrange's equations ignoring normal reaction

A small bead is sliding on a smooth vertical circular hoop of radius $a$, which is constrained to rotate with constant angular velocity $\omega$ about its vertical diameter. $\theta$ is an angle ...
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1answer
155 views

Lagrange Multipliers Versus Generalized Coordinates

When forced to explain to someone why one could either set up a general Lagrangian & then incorporate constraints using Lagrange multipliers, as opposed to just setting up a Lagrangian with ...
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3answers
1k views

What is canonical momentum?

What does the canonical momentum $\textbf{p}=m\textbf{v}+e\textbf{A}$ mean? Is it just momentum accounting for electromagnetic effects?
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3answers
694 views

Maxwells Equation from Electromagnetic Lagrangian

In Heaviside-Lorentz units the Maxwell's equations are: $$\nabla \cdot \vec{E} = \rho $$ $$ \nabla \times \vec{B} - \frac{\partial \vec{E}}{\partial t} = \vec{J}$$ $$ \nabla \times \vec{E} + ...
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2answers
226 views

The “stationary potential energy” condition for static equilibrium in mechanical systems

I've often read that, for a mechanical system which can be described by $n$ generalized coordinates $q_1,...,q_n$, a point $\mathbf{Q}=(Q_1,...,Q_n)$ is a point of equilibrium if and only if the ...
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1answer
274 views

Are Lagrangians and Hamiltonians used by Engineers?

Analytical Mechanics (Lagrangian and Hamiltonian) are useful in Physics (e.g. in Quantum Mechanics) but are they also used in application, by engineers? For example, are they used in designing bridges ...
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1answer
169 views

Definition of Kinetic energy

In class we had that $ T= \frac{1}{2}T_{ij}v_iv_j$ where we used the Einstein summation convention. Hitherto we only discussed examples where the kinetic energy was dependent of the square of one ...
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2answers
160 views

Dimensions in lagrangian potential

According to Mankowski flat space dimensions We can write, $$L= \int \text{dt} \text d^d{x} \left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
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1answer
89 views

Cyclic integral?

Can anyone explain me what is cyclic integral and give me some instructions how to show that there exist cyclic integral for Lagrangian $$L~=~\frac{1}{2}ma^2{\dot {\theta}}^2+\frac{1}{2}ma^2{\dot ...
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2answers
155 views

What's an “Action” and what does the Lagrangian equation mean exactly?

How and why would a particle take the shortest path? $L=KE-PE$? What's the $KE-PE$ mean in English? I understand the 'mechanics' but not the idea itself. Please explain simply, I do know Calculus ...
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0answers
266 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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1answer
126 views

Finding out Energy value

A Lagrangian is given by, $$L= \left(\frac{\pi}{2}\right)^2 R^d \left[\frac{1}{2}\dot A^2 - V(A_{max})\right]$$ $$E=\left(\frac{\pi}{2}\right)^2R^d V(A_{max}) $$ where V (A) now includes nonlinear ...
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0answers
153 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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2answers
137 views

A kind of Noether's theorem for the Hamiltonian

How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law using ...