Lagrangian Definition and 1000 Threads

the question is that there is a particle in 3 spatial Euclidean dimensions in cylindrical coordinates. I want to find a symmetry for the lagrangian if the potential energy is function of r and k.theta+z V=V(r,k.theta+z) any help please ?

This is a problem from the Goldstein text. It gives two point masses ##m_1## and ##m_2## connected by a string (negligible mass), where ##m_2## is suspended by the string through a hole in a smooth table; ##m_1## rests on the table. It is important to note that ##m_2## only travels in a vertical...

One way to solve the simple LC circuit with 1 inductor and 1 capacitor is to use the Lagrangian formulation of mechanics and consider charge q as the generalized coordinate. When writing down your Lagrangian, the energy of the inductor \frac{1}{2}L(\frac{dq}{dt})^2 is treated as the kinetic...

I'm reading Leonard Susskind's The Theoretical Minimum Vol. 1. 1. The problem: I'm on the section in which he asks the readers to derive the Lagrangian for a particle on a rotating carousel in polar coordinates. 2. Relevant ideas: The same Lagrangian in Cartesian coordinates is given as...

Homework Statement So I just learned how to derive the equation of motion under the Lagrangian formulation which involves finding the euler-lagrange equation when setting the change in action to zero, chain rule, integration by parts etc.. Then I learned how to find the equations of motion...

Hi all, I have a (hopefully) quick conceptual question that I'd like to clear up with your help. Is the following argument for why we treat position and velocity as independent variables in the Lagrangian correct? (referring to classical mechanics) : The Lagrangian, \mathcal{L} of a given...

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

Definition/Summary The Lagrangian is a function that summarizes equations of motion. It appears in the action, a quantity whose extremum (minimum or maximum) yields the classical equations of motion by use of the Euler-Lagrange equation. In quantum mechanics, the action, and thus the...

This isn't really a homework question, but may be similar to a typical example problem so I posted it here. Homework Statement I want to find the max and min dot product of a 3d vector and all points in a sphere constrained by angles in spherical coordinates. Homework Equations A point...

Hi guys. I hope this isn't a bad place to post my question, which is: I'm reading some lecture notes on Lagrangian mechanics, and we've just derived the Euler-Lagrange equations of motion for a particle in an electromagnetic field. It reads: m \ddot{\vec{r}} = -\frac{e}{c} \frac{\partial...

Greetings, I have two semi-related questions. 1. When making the Lagrangian formalism of electrodynamics, why is it that we use the Lagrangian density \mathcal{L}, rather than the plain old regular Lagrangian L? Is this something that is necessary, or is it more that it is just very...

I have just started studying Lagrangian Mechanics, and I can find decent material on the internet that describes the theory behind it, several proofs on equivalence and even some good solved examples. However, I would really appreciate if someone could recommend a book that has some of the...

I got this Lagrangian on the exam and it just seems weird to me: L = \frac{m}{2}(ẋ²+ẏ²) – eBẋy What I mean by "asymmetric" is that it doesn't seem to behave in a same way on x as on y, because there is ẋ in the last term and y but there is no ẏ and x. Deriving Euler-Lagrange equations I get: ẍ...

http://imgur.com/QhYG54l The image seems to be not showing here is the link : http://imgur.com/QhYG54lWhat does Landau mean here by the Lagrangian remaining ''unchanged''. Is it the value of the lagrangian as a function that may not change or is it the form that may not change? Also how does...

Hello everyone, I've been having trouble with the following reasoning for a while. The book I use for learning is Landau and Lifschitz vol1. When the concept of the Lagrangian is introduced in textbooks it is some abstract function of the position vector, velocity vector and time. Then they try...

Hello everyone! I was reading the following review: http://relativity.livingreviews.org/open?pubNo=lrr-2009-4&page=articlesu23.html And I got stuck at the first equation; (10.1) So how I understand this is that there are two variations, \tilde{q}(t)=q(t)+\delta q(t)...

Homework Statement Prove that under an infinitesimal Lorentz transformation: x^\mu \to x^\mu+\omega^\mu_\nu x^\nu so: \phi\to\phi-\omega^\mu_\nu x^\nu\partial_\mu\phi the Lagrangian varies as: \delta \mathcal{L}=-\partial_\mu(\omega^\mu_\nu x^\nu \mathcal{L}) The Attempt at a...

Hello, I need help understanding how to apply the formula for converting a Lagrangian to an equation of motion in this following specific application. On page 4 of Zee's QTF in a Nutshell, he gives a Lagrangian (equation 1). In the following sentence he gives the corresponding equation...

Why is it the case that dual field tensors, e.g. \widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho \sigma}, aren't being included in the Lagrangian? For example, one doesn't encounter terms like -\frac{1}{4}\widetilde{F}^{\mu\nu}\widetilde{F}_{\mu\nu} in QED or...

Mega quick question: must the Lagrangian (density) be hermitian? In other words, must \mathcal{L}=\mathcal{L}^\dagger always be valid?

How do i derive the Dirac equation from L_{dirac} = \overline{ψ}_α [i(γ^μ)_{αβ} - m]ψ_β ?. I can get it for the \overline{ψ} , but I'm having trouble deriving it for ψ .

I've been given the following lagrangian: \mathcal{L}_{eff} = \bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m)\psi - \frac{G}{4}(\bar{\psi}\psi)(\bar{\psi}\psi) where I have been told that the coefficient G is real and has mass dimension -2. I will eventually need to derive the feynman rules...

Homework Statement I try to calculate the energy tensor, but i can't do it like the article, and i don't know, i have a photo but it don't look very good, sorry for my english, i have a problem with a sign in the result Homework Equations The Attempt at a Solution In the photos...

Hey, It's a simple question (hope so). How do you know (analitically) wether angular momentum is conserved based solely on the Lagrangian? Let me elaborate, for example to prove that the linear momentum is conserved you simply look for cyclic coordinates, i.e \frac{\partial L}{\partial...

Hi, This is a worked example in the text I'm independently studying. I hope this isn't too much to ask, but I am stupidly having trouble understanding how one step leads to the other, so was hoping someone could give me a little more of an in-depth idea of the derivation. Thanks. Homework...

Why are y and y' treated as independent variables, while they are not? Another slightly related question: if ' = d/dt then df'/dg' = df/dg because f' = df/dg g', but if we differentiate f' to g' we implicitly assume that df/dg is independent of g', is it?

Hello I was reading something the other day and wondered what a two-particle lagrangian would look like in SR. I'm not exactly sure what lorentz scalar we can write down for the two particles.

Homework Statement I've thought of a problem to help me with Lagrange multipliers but have got stuck. Consider a particle of mass m moving on a surface described by the curve y = x2, the particle is released from rest at t = 0 and a position x = l. I'm trying to work out the EOM's but have...

Using differential forms and their picture interpretations, I wonder if it's possible to give a nice geometric & physical motivation for the form of the Electromagnetic Lagrangian density? The Lagrangian for the electromagnetic field without current sources in terms of differential forms is F...

Now this is a bit of a mix of a math and a physics question, but I think it is best asked here. Assume we are given a Lorentzian manifold ##(Q, g)## together with a metric connection ##\nabla##. Naturally we define geodesics ##\gamma## via $$\nabla_{\dot \gamma} \dot \gamma = 0 \quad ,$$...

Homework Statement A uniform flexible chain of mass M and length L is hung under gravity over a frictionless pulley of radius a and moment of inertia I whose axle is at a fixed height above the ground. Write down the Lagrangian of this system in terms of a generalized coordinate l denoting the...

Homework Statement A rigid rod of length ##\ell## and mass ##m## has its lower end in contact with a frictionless horizontal floor. Initially, the rod is at an angle ##\alpha_o## to the upward vertical when it is released from rest. The subsequent motion takes place in a vertical plane...

Homework Statement I'm working (self-study) through Goldstein et al, Classical Mechanics, 3rd Edition, and I'm currently stuck on Problem 8.11: A particle is confined to a one-dimensional box. The ends of the box (let these be at \pm l(t)) move slowly towards the middle. By slowly we mean...

I'm going to run through a derivation I've seen and ask a few questions about some parts that I'm unsure about. Firstly the theorem: For every symmetry of the Lagrangian there is a conserved quantity. Assume we have a Lagrangian L invariant under the coordinate transformation qi→qi+εKi(q)...

Hi everyone. I'm studying Heavy Quark Effective Theory and I have some problems in proving an equality. I'm am basically following Wise's book "Heavy Quark Physics" where, in section 4.1, he claims the following identity: $$ \bar Q_v\sigma^{\mu\nu}v_\mu Q_v=0 $$ Does any of you have an...

Homework Statement A uniform disk of mass 2M, radius R, is mounted on a frictionless horizontal pivot through its principal axis. The disk has an additional point-mass, M, fixed to a point on its circumference. (a) Give the Lagrangian for this system. (b) Find the frequency of small...

Last semester I had intermediate mechanics, and we spent a good amount of the class studying the LaGrangian. One thing that I never got an explanation for was why ##L = T-V##, as opposed to ##T+V##. The only reason I can think of is the "give and take" relationship that Kinetic and Potential...

Homework Statement F(x, y) = 96xy - 4x subject to constraint of 11 = x + y Form the lagrangian. Homework Equations F(x, y) = 96xy - 4x subject to constraint of 11 = x + y The Attempt at a Solution My only question is solving 11 = x + y My book says the answer is: L...

Homework Statement A ladder of length 2l and mass m leans against a smooth wall and rests on a smooth floor. The ladder initially makes an angle θ0 to the vertical. It slides downwards maintaining contact with both the wall and the floor. Calcula the the Lagrangian and the conjugate momentum...

Lagrangian is a function of ... Since Lagrangian is a function of q, q dot & time, then why in describing the Hamiltonian (H), L does not involve time explicitly? as H = (p*q dot) - L (q, q dot).

I'm just learning this theory and the maths is really trivial but the theory is slightly confusing me. I understand that if we have some function z=f(x,y) and we graph this on a three dimensional set of axis we will have some surface, we can then extend this by creating level curves in the...

Homework Statement I am providing a solution up to the point when I'm having a little issue with defining the generalized force. An eccentrically hollow cylinder of radius r rolls down a plane of inclination angle \alpha. Inside the cylinder, there is a cylinder-shaped hole of radius...

Given a system's Lagrangian, How can we calculate the equilibrium points corresponding to the system? also how can we determine if that's a stable equilibrium?

Homework Statement Suppose we have a skateboard half-pipe. The half-pipe has a radius R. We take a bicycle wheel of radius ρ and let it roll into the half-pipe. The wheel rolls without slipping ( and doesn't fall over, of course). Let Θ be the angle from the vertical to the line connecting the...

Homework Statement Two masses m_1 and m_2 (m_1 ≠ m_2) are connected by a rigid rod of length d and of negligible mass. An extensionless string of length l_1 is attached to m_1 and connected to a fixed point P . Similarly, a string of length l_2 (l_1 ≠ l_2) connects m_2 and P...

Homework Statement Two equal masses are constrained by the spring-and-pulley system shown in the accompanying sketch. Assume a massless pulley and a frictionless surface. Let x be the extension of the spring from its relaxed length. Derive the equations of motion by Lagrangian methods...

From the attached image problem: When deriving the third term in the Lagrangian: \lambda_{2}(w^{T}∑w - \sigma^{2}_{\rho}) with respect to w, are w^{T} and w used like a w^{2} to arrive at the gradient or am I oversimplifying and it just happens to work out on certain problems like this? (∑...

i have L of particle m in 1D motion, but how i can find the coordinate of particle x at time t?

Consider the configuration below shown in the attached picture! The wedge can slide on the inclined plane and the cube on the wedge.Their motion is described by x_1 and x_2 respectively. There is no friction and the inclined plane doesn't move. Here's the Lagrangian of the system...

Hi, This is probably a trivial question, but I just wanted to check my understanding. Is the following expression for the Dirac Lagrangian correct...