Lagrangian Definition and 1000 Threads

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...

Homework Statement To try and relate the three ways of calculating motion, let's say you have a particle of some mass, completely at rest, then is acted on by some force, where F equals a constant, C, times time. (C*t). I want to find the equations of motion using Lagrangrian, but also Newton...

Homework Statement So, basically there is a stick, mass m and length l, that is pivoted at its top end, and swings around the vertical axis with angular frequency omega. The stick always makes an angle theta with the direction of gravity. I am told there are 2 degrees of freedom (theta...

Homework Statement for particle with lagrangian L = m/2 dx/dt^2 + fx where x is constant force, what is ScL (classical action) Homework Equations d/dt (∂L/∂(dx/dt)) = ∂L/∂x ScL = ∫m/2 dx/dt^2 + fx dt from ti to tf The Attempt at a Solution d/dt (∂L/∂(dx/dt)) = ∂L/∂x implies f =...

Homework Statement Consider the following Lagrangian: \begin{equation} L = \frac{m}{2}(a\dot{x}^2 + 2b\dot{x}\dot{y} + c\dot{y}^2)- \frac{k}{2}(ax^2 + 2bxy + cy^2)\end{equation} Assume that \begin{equation} b^2 - 4ac \ne 0 \end{equation}Find the equations of motion and examine the cases...

I have been looking at the problem of 2 point masses connected by a spring in polar coordinates. The problem is solved using the center of mass coordinate R and the relative coordinate r where M=total mass and m=reduced mass. The Euler-Lagrange equations then give equations for P(a vector) and...

Hi everyone! I've been thinking about a certain problem for a while now. And that is a Lagrangian formulation of Newtonian gravity. I know there is a Lagrangian formulation for general relativity. But I was hoping to find a Lagrangian for Newtonian gravity instead (for some continuous mass...

Hello, this is probably one of those shoot yourself in the foot type questions. I am going through Landau & Lifshits CM for fun. On page 7 I do not understand this step: L' = L(v'^2) = L(v^2 + 2 \vec{v} \cdot \vec{\epsilon} + \epsilon^2) where v' = v + \epsilon . He then expands the...

Homework Statement Build the lagrangian of a set of N electric dipoles of mass m, length l and charge q. Find the equations of motion. Find the corresponding difference equations. Homework Equations Lagrange function L=T-V Lagrange's equations \frac{d}{dt}\left(\frac{\partial L}{\partial...

The full Lagrangian for electrodynamics ##\mathcal{L}## can be expressed as ##\mathcal{L}=\mathcal{L}_\textrm{field}+ \mathcal{L}_\textrm{interaction}+\mathcal{L}_\textrm{matter}##. Practically every textbook on relativity shows that...

If you don't have any charges or currents, the electromagnetic Lagrangian becomes ##\mathcal{L}=-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}##. The standard way to derive Maxwell's equations in free space is to replace ##F_{\alpha\beta}## by ##\partial_\alpha A_\beta -\partial_\beta A_\alpha## and...

The question is that a mass m of weight mg is attached to a fixed point by a light linear spring of stiffness constant k and natural length a. It is capable of oscillating in a vertical planne. Let θ be the angle of the pendulum wrt to the vertical direction, and r the distance between the mass...

The Lagrangian for a point particle is just L=-m\sqrt{1-v^2}. If instead we had a continuous distribution of matter, what would its Lagrangian density be? I feel that this should be very easy to figure out, but I can't get a scalar Lagrangian density that reduces to the particle Lagrangian in...

Hi there, I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are the same is presented in the book I am reading as a rule, commutativity, and...

I have been reading Lagrangian from Classical Mechanics by John R. Taylor. I have adoubt in a derivation which invloves differential calculus. I have attached snapshot of the equation , can someone please explain. Here y,η are functions of x but α is s acosntant. Please let me know if I...

What is the Lagrangian of interaction of photon and spin zero charge scalar?The vertex of photon and spin 1/2 charge fermion is proportional with e multiplied vertor gamma matrix,but I do not know what is the vertex of photon and charge scalar.I hear that a vertex is proportional with polynomial...

This thread is supposed to be a continuation of the discussion of this thread: (1) https://www.physicsforums.com/showthread.php?t=88570. The previous thread was closed but there was a lot of things I did not understand. This is also somewhat related to a recent thread I created: (2)...

I recently posted another thread on the General Physics sub forum, but didn't get as much feedback as I was hoping for, regarding this issue. Let's say I have two Lagrangians: $$ \mathcal{L}_1 = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\mu A^\mu)^2 $$ $$...