Lagrangian Definition and 1000 Threads

In Sydney Coleman Lectures on Quantum field Theory (p48), he finds : $$D\mathcal{L} = e^{\mu} \partial _{\mu} \mathcal{L}$$ My calulation, with ##\phi## my field and the variation of the field under space time tranlation ##D\phi = e^{\mu} \frac{\partial \phi}{\partial x^{\mu}}## ...

$$\partial^\beta F_{\beta\alpha} +\partial^\beta A_\mu A^\mu \delta^\alpha_\sigma \delta^\rho_\beta+\mu^2 A_\alpha = 2A_\mu (\partial_\rho A^\rho) +\frac {4\pi}{c}J_\alpha$$

The euler lagrange equation I am using is: $$\frac {\partial^\beta \partial L}{\partial(\partial^\beta A^\alpha) }= \frac {\partial L} {\partial A^\alpha}$$ Now the proca lagrangian i am using is $$L= -\frac {1}{16\pi} F_{\alpha\beta} F^{\alpha\beta} + \frac {\mu^2} {8\pi} A_\alpha A^\alpha -...

In Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws (starting on page 291 here), the authors introduce the concept of a generator function G, where the transformation generated by G is given by (equation 12.29 on page 292 in the text)...

Hello, It might sound silly, but when I try to calculate the kinetic energy of a rotating rod to form the Langrangian (and in general), why it has both translational and rotational kinetic energy? Is it because when I consider the moment of Inertia about the centre I need to include the...

I am doing a project where the final scope is to find an extra operator to include in the proca lagrangian. When finding the new version of this lagrangian i'll be able to use the Euler-Lagrange equation to find the laws of motion for a photon accounting for that particular extra operator. I...

This is just something unexpected that I noticed recently, and I hadn't heard anyone mention it before. The relativistic Lagrangian for a particle moving under a scalar potential ##\Phi## is this: ##L = \frac{1}{2} m g_{\mu \nu} \dfrac{dx^\mu}{d\tau} \dfrac{dx^\nu}{d\tau} - \Phi## This leads...

I have the following lagrangian density: $$L = \bar{\psi}i \gamma^\mu \partial_\mu \psi - g\bar{\psi}(\sigma + i\gamma^5\pi)\psi + \frac{1}{2}(\partial_\mu \sigma)^2+ \frac{1}{2}(\partial_\mu \pi)^2 -V(\sigma^2 + \pi^2)$$ where $\pi$ and $\sigma$ are scalar fields. I have show that this...

Hey guy, I'm having problems to understand the final part of this section. The book says we have the lagrangian from one particle (106.16), then we have some explanation and then the total lagrangian is given(106.17). For me is everything fine until the 106.16, then i couldn't get what is going...

Problem: Solution: My question: My reasoning was that if x is max at the point then the gradient vector of g at the point has only x component; that is ##g_y=0,\, g_z=0##. This way I got: $$\begin{cases} 4y^3+x+z=0\\ \\ 4z^3+x+y=0\\ \\ \underbrace{x^4+y^4+z^4+xy+yz+zx=6}_\text{constraint...

I have heard many times that it does not matter where you put the zero to calculate the potential energy and then ##L=T-V##. But mostly what we are doing is taking potential energy negative like in an atom for electron or a mass in gravitational field and then effectively adding it to kinetic...

I think it is quite simple as an exercise, following the two relevant equations, but at the beginning I find myself stuck in going to identify the lagrangian for a relativistic system of non-interacting particles. For a free relativistic particle I know that lagrangian is...

What I first did was setting the reference system on the left corner. Then, I said that the position of the mass ##m_2## is ##x_2##. I also supposed that the pendulum makes an angle ##\theta## with respect to the vertical axis ##y##. So the generalized coordinates of the system would be ##x_2##...

https://dst-public.s3-us-west-2.amazonaws.com/lagrangian.png

I tried 1. using the Lagrangian method: From ##y=-kx^2## I got ##\dot y = -2kx \dot x## and ##\ddot y = -2k \dot x^2 - 2 kx \dot x##. (Can I use ##\dot y = g## here due to gravity?) This gives for kinetic energy: $$T = \frac{1}{2} mv^2 = \frac{1}{2} m (\dot x^2 + \dot y^2) = \frac{1}{2} m (\dot...

Summary:: Can anyone introduce an informative resource with solved examples for learning variation principle? For example I cannot do the variation for the electromagnetic lagrangian when ##A_\mu J^\mu## added to the free lagrangian and also some other terms which are possible: $$ L =...

Why, in lagrangian mechanics, do we calculate: ##\frac{d}{dt}\frac{\partial T}{\partial \dot{q}}## to get the (generalised) momentum change in time instead of ##\frac{d T}{dq}##? (T - kinetic energy; q - generalised coordinate; p - generalised momentum; for simplicity I assumed that no external...

It is the first time that I am faced with a complex field, I would not want to be wrong about how to solve this type of problem. Usually to solve the equations of motion I apply the Euler Lagrange equations. $$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...

In the book "The Variational Principles of Mechanics" by Cornelius Lanczos, the following statement is made about a lagrangian ##L_1## where time is given as an dependent parameter, and a new parameter ##\tau## is introduced as the independent variable, see (610.3) and (610.4) pg. 186,187 Dover...

I was going over the Einstein-Hilbert action derivation of the Einstein field equations and came across a term that does not seem to be explicitly defined. That term is the Langragian for the matter fields. What exactly is matter in General relativity in the context of the Lagrangian? Here is...

this figure form ( https://en.wikipedia.org/wiki/Effective_mass_(spring%E2%80%93mass_system) ) massive spring ; m K.E. of total spring equal to ## K.E. = \frac{1}{2} \sum dm_i v_i^2 = \frac{1}{2} \sum \rho dy (Vy/L)^2## V is the speed at the end of the spring and V are same speed of mass M...

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With free part L=-½(∂Φ)^2 -½m^2 Φ^2 and interaction term L=½gΦ^2Any help would be appreciated, thank you.

I must find the Lagrangian for an undamped pendulum using the diagram showed below, I've no idea what to do with the second angle φ2 because is measured from the line that joins the two pivot points. The ecuations I must obtain are as follows I get so many different things but I can't reach...

Hello! I need some help with this problem. I've solved most of it, but I need some help with the Hamiltonian. I will run through the problem as I've solved it, but it's the Hamiltonian at the end that gives me trouble. To find the Lagrangian, start by finding the x- and y-positions of the...

Is there a relationship between the Lagrangian ‘hill diagram’ and the spacetime curvature embedment graphs? The Lagrangian map shows effective potential, which deals with centrifugal force. As centrifugal force is a fictitious force (and gravity is as well), I would assume the underlying...

I’m reading Lancaster & Blundell, Quantum field theory for the gifted amateur (even tho I”m only an amateur...) and have a problem with their explanation of symmetry breaking from page 242. They start with this Lagrangian: ## \mathcal{L} = (\partial_{\mu} \psi^{\dagger} - iq...

Pendulum plane, which suspension executes a horizontal harmonic motion $$x = acos(\gamma t)$$ Position P, orientation x to right and y points below, phi is the pendulum's angle wrt y. $$P = (acos(\gamma t) + lsin(\phi(t)), lcos(\phi(t)) )$$ So executing all that is necessary, i found it...

Helo, The Lagrangian in general relativity is written in the following form: \begin {aligned} \mathcal {L} & = \frac {1} {2} g ^ {\mu \nu} \nabla \mu \phi \nabla \nu \phi-V (\phi) \\ & = R + \dfrac {16 \pi G} {c ^ {4}} \mathcal {L} _ {\mathcal {M}} \end {aligned} with ## g ^ {\mu \nu}: ## the...

I am trying to get a foothold on QFT using several books (Lancaster & Blundell, Klauber, Schwichtenberg, Jeevanjee), but sometimes have trouble seeing the forest for all the trees. My problem concerns the equation of QED in the form $$ \mathcal{L}_{Dirac+Proca+int} = \bar{\Psi} ( i \gamma_{\mu}...

A stick is pivoted at the origin and is arranged to swing around in a horizontal plane at constant angular speed ω. A bead of mass m slides frictionlessly along the stick. Let r be the radial position of the bead. Find the conserved quantity E given in Eq. (6.52). Explain why this quantity is...

Hi.I am looking for a book to learn about discrete mechanics (i.e. working in a 3D lattice instead of ##n## generalized coordinates). I am particularly interested in how to derive the discrete E-L equations by extremizing the action. I have checked Gregory and Goldstein but they do not deal...

a) I think I got this one right. Please let me know otherwise We have (let's leave the ##x## dependence of the fields implicit :wink:) $$\mathscr{L} = N \Big(\partial_{\alpha} \phi \partial^{\alpha} \phi^{\dagger} - \mu^2 \phi \phi^{\dagger} \Big) = \partial_{\alpha} \phi^{\dagger}...

In Solution https://www.slader.com/textbook/9780201657029-classical-mechanics-3rd-edition/67/derivations-and-exercises/20/ In the question say the wedge can move without friction on a smooth surface. Why does the potential energy of the wedge appear in Lagrangian? (You can see the Larangian...

Suppose I had a Lagrangian $$L = q+ \dot{q}^2 + t.$$ This has explicit time dependence. Now consider another Lagrangian: $$L = q+ \dot{q}^2 .$$ Which has no explicit time dependence. But after solving for the equations of motion, I get $$\dot{q} = t/2 + C.$$ So I could now write my Lagrangian...

Hello! I have some problem getting the correct answer for (b). My FBD: For part (a) my lagrangian is $$L=T-V\iff L=\frac{1}{2}m(b\dot{\theta})^2+mg(b-b\cos\theta)-\frac{1}{2}k\boldsymbol{x}^2,\ where\ \boldsymbol{x}=\sqrt{(1.25b-b)^2+(b\sin\theta)^2}-(1.25b-0.25b)$$ Hence my equation of...

In this article [1] we can read an explanation about Wilson's approach to renormalization I have read that Kenneth G Wilson favoured the path integral/many histories interpretation of Feynman in quantum mechanics to explain it. I was wondering if he did also consider that multiple worlds...

I know that by extremizing lagrangian we get equations of motions. But what if we extremize the energy? I am just little bit of confused, any help is appreciated.

Hello, I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it. ∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1) L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2) Here is Eq (1) the...

I am new to Lagrangian mechanics and I am unable to comprehend why the Euler Lagrange equation works, and also what really is the significance of the lagrangian.

Hello, I am having trouble applying Lagrangian to this problem: A uniform thin circular rubber band of mass ##M ## and spring constant k has an original radius ##R##. Now it is tossed into the air. Assume it remains circular when stabilized in air and rotates at angular speed ##\omega## about...

Hello, I have been working on the three-dimensional topological massive gravity (I'm new to this field) and I already faced the first problem concerning the mathematics, after deriving the lagrangian from the action I had a problem in variating it Here is the Lagrangian The first variation...

I copy again the statement here: So, I think I solved parts a to c but I don't get part d. I couldn't even start it because I don't understand how to set the problem. I think it refers to some kind of motion like this one in the picture, so I'll have a maximum and a minimum r, and I can get...

This is from Griffiths particle physics, page 360. We have the full Dirac Lagrangian: $$\mathcal L = [i\hbar c \bar \psi \gamma^{\mu} \partial_{\mu} \psi - mc^2 \bar \psi \psi] - [\frac 1 {16\pi} F^{\mu \nu}F_{\mu \nu}] - (q\bar \psi \gamma^{\mu} \psi)A_{\mu}$$ This is invariant under the joint...

Hello, I've got to rationally analice the form of the solutions for the equations of motion of a simple pendulum with a varying mass hanging from its thread of length ##l## (being this length constant). I approached this with lagrangian mechanics, asumming the positive ##y## direction is...

In order to compute de lagrangian in spherical coordinates, one usually writes the following expression for the kinetic energy: $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 )\ ,$$ where ##\theta## is the colatitud or polar angle and ##\phi## is the...

Homework Statement:: ... Relevant Equations:: . What is the minimum mathematic requirement to the Lagrangian and hamiltonian mechanics? Maybe calc 3 and linear algebra?

Given the following $$L(\theta,\dot{\theta},\phi,\dot{\phi}) = \frac12ml^2((\dot{\theta})^2 + (sin(\theta)^2)\dot{\phi}^2) + k\theta^4$$ This is my attempt: I am not understanding if the conserved quantities (like angular momentum about the z-axis) impacts my formulation of the Hamiltonian or...

Would the following: $$ L = m(\vec{\dot r + \vec v})$$ (constant velocity added to above eq.) lead to equivalent euler-lagrange equations due to the fact that the ratio of T and V is unaffected by an increase in constant velocity? And would this be an example of energy conservation?

I know that from the given problem, I need to find the expression for Kinetic energy, KE = 1/2 m [r(dot)]^2 and Potential energy, PE = 1/2 k r^2 So L = 1/2 m [r(dot)]^2 - 1/2 k r^2 Hence H = 1/2 m [r(dot)]^2 + 1/2 k r^2 I assume that the fixed length r0 is provided to find the value of end...