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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254 views

Euler-Lagrange equation (equation of motion) solution with hairy Lagrangian [closed]

I'm going through Zwiebach Chapter 6 on relativistic strings to try to solve a similar problem. I got all the way to my equation of motion \begin{eqnarray*} \delta S & = & [ p' \delta \...
3
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1answer
393 views

How do I treat the Lagrangian in the case of a rigid body?

Here's Exercise 1.11 from Goldstein's Classical Mechanics 3rd edition (the first one after having derived the Lagrangian basically): Exercise 1.11: Consider a uniform thin disk that rolls without ...
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0answers
101 views

What's the conserved stress energy tensor? [closed]

I've worked on this problem for forever and still don't really see the solution. Any help appreciated. Say we have the Lagrangian for a scalar field that's $U(1)$ charged,$$\mathcal{L} ={1\over4}(F_{\...
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0answers
86 views

Classical toy models of particles with intrinsic spin

Related to my question here (spacetime torsion, the spin tensor, and intrinsic spin in einstein cartan theory), I'd like to be able to put test particles on a manifold with non-zero torsion and see ...
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0answers
284 views

How is the method of Lagrange multipliers used for multiple constraints of multiple variables? [closed]

Let's say for example that I have two constraints $f(x,\dot{x},y,\dot{y})$ and $g(x,\dot{x},y,\dot{y})$ and a Lagrangian $L(x,\dot{x},y,\dot{y})$. What are the Euler-Lagrange equations of the first ...
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148 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 \bar{\...
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95 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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91 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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299 views

Question about an integration by parts in Feynman's Quantum Mechanics [closed]

I have begun reading Feynman & Hibbs Quantum Mechanics and Path Integrals. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I ...
3
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1answer
236 views

Deriving field equation in Yang Mills theory

Trying to show that $$D_\mu\vec{F^{\mu \nu}} = \partial_{\mu}\vec{F^{\mu \nu}} + g \vec{A_\mu} \times \vec{F^{\mu \nu}} = 4 \pi \vec{J^\nu},$$ or (correct me if I'm wrong) $$ \partial_{\mu} F^{\mu \...
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175 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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64 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}+ j^{...
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0answers
2k 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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0answers
251 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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136 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: $$\pi_n=\frac{\partial\...
3
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0answers
88 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 ...
3
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2answers
208 views

Is there something more to Noether's theorem?

From the definition of Lagrangian mechanics, Noether's theorem shows that conservation of momentum and energy comes from invariance vs time and space. Is the reverse true? Are Lagrangian mechanics ...
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6answers
5k views

How are constraint forces represented in Lagrangian mechanics?

Suppose we try to obtain the movement equation for a particle sliding on a sphere (no friction, ideal bodies...). The only forces acting on the particle are its weight and - here's my problem - a ...
2
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2answers
107 views

Why don't we take this term $D_{\mu}D_{\nu}F^{\mu\nu}$ in Lagrangians?

Why don't we take $$D_{\mu}D_{\nu}F^{\mu\nu}$$ in Lagrangians?
2
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2answers
227 views

How to find the rank of the matrix $\frac{\partial ^2\mathcal{L}}{\partial \dot{X^\mu} \partial \dot{X^\nu} }$ for the Nambu-Goto string Lagrangian?

In this case $$\mathcal{L}~=~-T\sqrt{-\dot{X^2}X'^2+(\dot{X}\cdot X')^2}.$$ I was reading some books and papers about the constraints in the Nambu-Goto action, and all of them say something like ...
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2answers
401 views

Why is ${\partial^i}{\partial_i\phi}$ = ${\partial^i {\phi}}{\partial_i{\phi}}$?

This notation can be found on page 254 of Victor Stenger's Comprehensible Cosmos and in David Tong's Lectures on QFT (Equation 2.4 http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf), and in EDIT: on ...
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1answer
1k views

What gauge is used in the Lagrangian for a non-relativistic point particle in an electromagnetic potential

For the Lagrangian $$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$ of a non-relativistic point particle in an electromagnetic potential, what gauge is used for the electromagnetic ...
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1answer
5k views

Derivation of Dirac equation using the Lagrangian density for Dirac field

How can I derive the Dirac equation from the Lagrangian density for the Dirac field?
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1answer
739 views

Proof that total derivative is the only function that can be added to Lagrangian without changing the eom

So I was reading this: Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time and while the answers for the first question are good, nobody gave much ...
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1answer
678 views

What is meant by a local Lagrangian density?

What is meant by a local Lagrangian density? How will a non-local Lagrangian look like? What is the problem that we do not consider such Lagrangian densities?
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2answers
725 views

First integral of relativistic Euler-Lagrange equations

Connsider a pseudo-Riemannian ($4$-dimensional) manifold $M$ with a pseudometric $g_{ab}$. The Lagrangian of a free particle in $M$ (in analogy to the flat case) is $$\mathcal L=\frac{1}{2}g_{ab}\frac{...
2
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2answers
753 views

Lorentz invariance of the action for free relativistic particle

I tried to check the Lorentz invariance of the standard special relativity action for free particle directly: ($c=1$) $$ S=\int L dt=-m\int\sqrt{1-v^{2}}dt $$ Lorentz boost: $$ dt=\frac{dt^{'}+udx^...
2
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1answer
78 views

Poincare non-invariance in real world and field theory

This may be a very blunt question but I wonder why we always use Poincare invariant Lagrangians in field theory. After all, the entire world around us is by no means homogeneous, isotropic and so on. ...
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2answers
1k views

Adding a total time derivative term to the Lagrangian

This is proof that $L'$ represents same equation of motion with $L$ through Lagrange eq. I understand $L'$ satisfies Lagrange eq, but how does this proof mean $L'$ and $L$ describe same motion of ...
2
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1answer
94 views

Why is $\mu_0$ missing in EM formulas in Peskin and Schroeder?

In this post, $\hbar=c=1$ units are used throughout. It is well known that the action of classical electromagnetism is given by $$\mathcal S_{\text{Maxwell}} = \int d^4x\left\{-\frac{1}{4\mu_0}F_{\mu\...
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1answer
260 views

Total divergence term and corresponding Feynman Diagram

A total divergence term added to the Lagrangian doesn’t affect the action because the integral of a total divergence vanishes. But if one attempts to derive the Feynman rules from the Lagrangian with ...
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3answers
373 views

What is the definition of momentum when a mass distribution $\rho(r,t)$ is given?

This question is Edited after recieving comments. What is the definition of momentum when a mass distribution $\rho(r,t)$ is given? Assuming a particle as a point mass we know the definition of ...
2
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1answer
87 views

Equivalence between principle of least action and minimum potential energy

Are the principle of least action and the principle of minimum potential energy equivalent? How does one show that? Also, are Newton's laws of motion equivalent to the principle of least action? How ...
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1answer
687 views

Obtain the Lagrangian from the system of coupled equation [closed]

In this particular paper, "Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation" by C.K.Law, PhysRevA.51.2537 \begin{equation} \ddot{Q}_{k}=-\omega^{2}_{k}Q_{k}+2\...
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2answers
270 views

Hamiltonian mechanics really useful for numerical integration? Lagrangian can become 1st-order

(I'm talking about the classical mechanics.) Many texts say that Euler-Lagrange equations are difficult to treat numerically because they are second-order ODEs, ${f_i(\boldsymbol{q, \dot{q}, \ddot{q}}...
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2answers
306 views

Why does the classical electrodynamics Lagrangian density equation have a “field” term and an “interaction” term?

On Wikipedia's page on classical electrodynamics, they state the Lagrangian density equation as follows \begin{equation} \mathcal{L} = \mathcal{L}_{\text{field}} + \mathcal{L}_{\text{int}} = -\frac{...
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1answer
322 views

Difference in “momentum” names in Lagrangian mechanics

In the context of Lagrangian formulation of classical mechanics, the following names keep occurring in most textbooks, which confuse me a lot, are they different in any way? Momentum Generalized ...
2
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4answers
575 views

Poincare invariant Lagrangians

The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean $$ \partial_\mu \mathcal{L}=0~? $$ If this is the case doesn't the ...
2
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1answer
213 views

A Question on Hamilton's Principle

In some literatures, the Hamilton's principle for conservative systems is introduced by this equation: $$\delta \int_{t_1}^{t_2}(T-V) ~\mathrm{d}t~=~0$$ In some others, this principle is introduces ...
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2answers
411 views

Intuition Behind Conservation of Angular Momentum

I'm having a fairly hard time understanding the intuition behind Noether's derivation of the conservation of angular momentum from the rotational invariance of the Lagrangian, though I do understand ...
2
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1answer
309 views

Is Lagrangian a scalar?

I may be wrong: Lagrangian are scalars. They are NOT invariant under coordinate transformations. The simplest example is when you have a gravitational potential ($V=mgz$) and you translate $z$ by $a$ ...
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2answers
7k 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 ...
2
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1answer
260 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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2answers
395 views

Bertrand's theorem

I found in Goldstein's Classical Mechanics that the condition for closed orbits is given by $\frac{d^2 V_{eff}}{dr^2}>0$.(bertrand's theorem). Can somebody explain to me, how this inequality is ...
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1answer
772 views

How to tell local and non-local in QFT?

I'm taking QFT course in this term. I'm quite curious that in QFT by which part of the mathematical expression can we tell a quantity or a theory is local or non-local?
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2answers
127 views

Is angular momentum conserved for a mass fixed to a horizontal guide

I have the system shown below. Mass 1 confined to a vertical guide, and mass 2 confined to a horizontal guide joined together by a spring. My question is very simple: is the total angular momentum ...
2
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1answer
115 views

Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
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2answers
98 views

Is there any loss or gain of information if a physics law is changed from one form to another? [closed]

Is there any loss or gain of information if a physics law is changed from one form to another such that the parameter appearing in them is changed from vector to a scalar? For example, consider the ...
2
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1answer
97 views

Degrees of freedom of a point mass sliding on a rigid curved wire without friction

I am very new to the subject and am going through Structure and Interpretation of Classical Mechanics. One exercise asks to find the degrees of freedom of a number of systems, one of which is a ...
2
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1answer
130 views

Lagrangian density for the electromagnetic field

I want to know how the Lagrangian density for the electromagnetic field is written in the following form: