Lagrangian Definition and 1000 Threads

I understand the definitions of both the classical and relativistic (SR) Lagrangians. But I cannot find a precise mathematical definition of Lagrangian Density. Please assist. Thanks in advance.

When going from the Lagrangian to the Hamtiltonian, we define p_i=\frac{\partial L}{\partial \dot{q}_i} as the independent variables in place of \dot{q}. This change of variables is possible if and only if the Hessian matrix \frac{\partial^2L}{\partial \dot{q}_i\partial\dot{q}_j} is...

Homework Statement Given an interaction lagrangian L = i \, g \, \bar \psi(x)_i (\lambda^a)_{ij} \gamma_5 \, \psi(x)_j \phi(x)_a where \psi_i are three Dirac fermions with mass M and \phi_a are eight real scalar fields of mass m and \lambda_a are the generators of SU(3). I have to find...

Homework Statement A ball is sitting on a frictionless seesaw with no inclination at the beginning, and a constant angular velocity \phi. Find the position of the ball as a function of timeHomework Equations L=T-V, T=(m\dot{}x2+m\dot{}y2)/2, V=mgyThe Attempt at a Solution The first problem I...

Homework Statement Find the potential energy density of a hanging string of mass density m/L that has been displaced from equilibrium at a point a distance d up from the bottom of the string. This point is displaced a distance X in the x direction, and a distance Y in the y direction. The...

Homework Statement Show that the complex Klein-Gordon Lagrangian density: L=N\left(\partial_\alpha\phi^{\dagger}(x)\partial^\alpha\phi(x)-\mu^2\phi^{\dagger}(x)\phi(x)\right) is invariant under charge conjugation: \phi(x)\rightarrow C\phi(x)C^{-1}=\eta_c \phi^\dagger (x) Where C...

In Mechanics by Landau-Lifgarbagez there is a step during the derivation of the Lagrangian where.. \int_{t_1}^{t_2} L(q+\delta q, \dot q + \delta \dot q, t ) \, \mathrm{d}t - \int_{t_1}^{t_2} L(q, \dot q, t ) \, dt then they write "when this difference is expanded in powers of...

If we take the the Dirac Lagrangian and decompose into Weyl spinors we find \mathcal{L} = \bar{\psi} ( i \gamma^\mu \partial_\mu - m ) \psi = i U^\dagger_- \sigma^\mu \partial_\mu u_- + i u^\dagger_+ \bar{\sigma}^\mu \partial_\mu u_+ - m(u^\dagger_+ u_- + u^\dagger_- u_+ ) =0 So far I have...

Hello! I`m looking for Lagrangian Systems with Lagrangian function containing higher derivatives in t. I would be really happy if someone can tell some higher order Lagrangians with physical relevance. Thanks, Viktor

Homework Statement A meter stick stands on a frictionless surface and leans against a frictionless wall as shown. It is released to fall when it makes an angle of 1 degree from the vertical. Use Lagrange and Euler to find how long it takes the stick to fall to the ground. The Attempt...

The Hamiltonian operator in quantum field theory (of Klein Gordon Lagrangian) is H=\frac{1}{2} \int \frac{d^3p}{(2 \pi)^3} \omega_{\vec{p}} a_{\vec{p}}^\dagger a_{\vec{p}} after normal ordering Now we construct energy eigenstates by acting on the vacuum |0 \rangle with a_{\vec{p}}^\dagger...

Homework Statement In a shallow layer of water, the velocity of water in the z direction may be ignored and is therefore (\dot{x},\dot{y}). We can define the Lagrangian coordinates such that the depth of water h is satisfied by the relations Given that h = \frac{1}{\alpha} and \alpha =...

Homework Statement A cylinder of mass m and radius R rolls without slipping down a wedge of mass M. The wedge slides on a frictionless horizontal surface. The angle between the wedge's hypotenuse & longest leg (which lies on the frictionless ground) is beta. The wedge's hypotenuse DOES have...

Hello, is it possible to write a Lagrangian (L) for a simple RC (or RL) circuit? Normally L = kinetic - potential energy, but how would you write this for an RC circuit? thanks!

Please teach me this problem: It seem that following Haag's theorem there not exist quantized equation of motion for interacting fields.So I don't understand how to know the form of interacting Lagrangian has form of product of fields(example Lagrangian of Fermi field interacting with...

1. A rigid straight uniform bar of mass m and length l is attached by a frictionless hinge at one end to a fixed wall so that it can move in a vertical plane. At a distance a from the hinge it is supported by a spring of stiffness constant k, as shown in the figure Ignoring gravitational...

Hello! Can anybody suggest me some articles and books on Lagrangian and Hamiltonian systems with Grasmann variables? Thank you for your help! O

Hi all I've found a way to include dissipation in the kinetic energy of the lagrangian for simple systems and I want to know if its ok to do this. My understanding is that dissipation is typically included using the Rayleigh dissipation function which is separate from the Lagrangian. The...

Homework Statement http://img708.imageshack.us/img708/4375/81747300.jpg http://img837.imageshack.us/img837/5850/42434333.jpg [PLAIN][PLAIN]http://img696.imageshack.us/img696/3518/50793668.jpg Homework Equations...

Ive been doing some research on the title concepts... And would love it if someone could answer some questions because I can't seem to find the answer anywhere. 1) How was the lagrangian found? I know its kind of defined, and there are other lagrangians- but is there an idea behind it or was...

Hey, can somebody show me how to derive that the lagrangian in classical phyics is L=T-V i have seen this formula so many times, but i have no idea where it really comes from?

Homework Statement http://www.facebook.com/photo.php?fbid=1612917330439&set=a.1250823718325.2039017.1461460506&ref=fbx_album" The Attempt at a Solution I don't want to ask about the full solution for this thing, but only one thing: in that expression, which is the kinetic energy and...

Homework Statement A combination of masses along the z-axis is separated by a distance 'a' with middle mass at origin. The potential is V = \frac{1}{2}kx^2 . What is the force of constraint using Lagrange multiplier? Homework Equations L = T - V + \lambda f The Attempt at a...

Verify that the Lagrangian density L= \frac{1}{2} \partial_\mu \phi_a \partial^\mu \phi_a - \frac{1}{2} m^2 \phi_a{}^2 for a triplet of real fields \phi_a (a=1,2,3) is invariant under the infinitesimal SO(3) rotation by \theta \phi_a \rightarrow \phi_a + \theta \epsilon_{abc} n_b \phi_c...

Homework Statement I am trying to get an equation of motion for the following (seemingly simple) setup. You place on a rod on a pivot. The rod's centre of mass is precisely over the pivot. Think of balancing a ruler horizontally on your finger. Gravity, of course acts downward. The...

I have to create a simulation of the pendulum shown in the .pdf at the bottom of the page. The 3 rods are free to rotate around their pivots in a plane. The two edge rods are connected as close to their edges as possible. There is no friction. Unfortunately my equations of motion are spitting...

I'm probably making a mistake, but looking at the free field lagrangian for QED \mathcal{L} \propto (-F^{\mu\nu}F_{\mu\nu}) \propto (\mathbf{E}^2 - \mathbf{B}^2) it appears to me that the action is not bounded from above, nor from below. Does that mean the equations of motion we obtain by...

Homework Statement Eulerian velocity: V_{1}=-z_{1}^{2} V_{1}=\frac{dz_{1}}{dt} z_{1}(t=0)=x_{1} This is supposed to become the Lagrangian velocity of: z_{1}=\frac{x_{1}}{1+tx_{1}} I don't understand how to take the Eulerian velocity and transform it to Lagrangian. Homework EquationsThe...

Homework Statement http://img85.imageshack.us/gal.php?g=hw1y.jpg Its an imageshack gallery Homework Equations Book gives completely irrelevant equations. The Attempt at a Solution I couldn't even solve A. I have no clue how to start this. The instructor isn't providing any...

Homework Statement [PLAIN]http://mityaka.com/users/kolodny/img/lagprob.png Homework Equations L = T - V T = \frac{1}{2}*m*U2 Vs = \frac{1}{2}*k*x2The Attempt at a Solution I worked out the equations of motion as: FL = m*\ddot{y}+k*y-k*b*\theta FL*e =...

Find the Euler – Lagrange Equation when L = -1/2 (D_p a_u)(D^p a^u) \sqrt{-g} dx^4 Use g_u_v to raise/lower indices D_p is the covariant derivative I am very new at this notation and am having a lot of trouble getting anywhere with this. I know I have to take the action: S = \int Ldt...

Homework Statement When writing down the Lagrangian and the writing down Euler-Lagrange equation I'm having some difficulties with reasoning something. Homework Equations Lagrangian is: \mathcal{L}=\frac{1}{2}mv^2-q\phi+\frac{q}{c}\vec{v}\cdot\vec{A}. Euler-Lagrange eq...

Homework Statement A spring of rest length L_0 (no tension) is connected to a support at one end and has a mass M attached at the other. Neglect the mass of the spring, the dimension of the mass M, and assume that the motion is confined to a vertical plane. Also, assume that the spring only...

I remember when I learned some basic continuum mechanics, Lagrangian is just a integral of lagrangian density over space, which is quite easy to accept because it's just a continuous version of L=T-U. Now I'm trying to start a bit QFT and notice that Lagrangian is an integral over space-time...

Hi, hopefully someone can check my logic. I have a lever with a mass on top which is rising towards the vertical on a frictionless pivot. The length of the lever can change. (In reality this is a robot rocking onto a foot and straightening its leg). The intention is to bring the lever to a...

In Ryder's text, he defines the dual tensor as the anti-symmetric \tilde F^{\nu \mu} = \epsilon^{\nu \mu \alpha \beta} F_{\alpha \beta}. Later he plops down the complex scalar field Lagrangian as L = (D_\mu \phi)(D^\mu \phi *) - m^2 \phi * \phi - \frac{1}{4} F^{\nu \mu}F_{\nu \mu} where...

The Lagrangian finite strain tensor is defined as: E_{i,j}=\frac{1}{2}\left(\frac{\partial x_k}{\partial X_i}\frac{\partial x_k}{\partial X_j}-\delta _{i,j}\right) Is it in Einstein Notation so that there is a summation symbol missing, i.e. would it be the same thing if one wrote it as...

Hello I am using Landau's mechanics Vol I for classical mechanics. On page 4 he mentions for Lagrangian of a system composed of two systems A and B which are so far away so that their interactions can be neglected. then for the combined system we have L = LA + LB I'm trying to...

Basic exercise for finding a Lagrangian from the Landau's "Mechanics" Hello everyone! Homework Statement I've just started preparing for the classical mechanics course using only Landau & Lifgarbagez, so I'm doing everything according to their formulation. And so I solved an exercise...

Does anyone know where I can find the lagrangian for this? From memory I believe it looks something like S = \frac{1}{2} \int \frac{d\tau}{e}[\dot{X}^2 +i \dot{\psi}{\psi}-2ie\nu \dot{X} \psi] where e is the graviton and nu is the gravitino. Does anyone know of a reference that...

Starting on the topic of the Lagrangian, I have been told not to try to make intuitive sense, but just accept the nice differential equations which it goes into. Fine, but it should at least make basic sense. That L= T-V should be stationary means then d(T-V) = dT-dV = 0, i.e., dT=dV, which...

Is a symmetric Lagrangian leads to a symmetric Stress-Energy Momentum ？

Hey, I'm doing some examples in QFT and I don't want to go too far with this one: Doing gauge symmetries, we first introduce the Unitary spacetime-dependent gauge transformation that gives us a gauge potential. With the new gauge added Lagrangian, I want to take its variation to confirm the...

Is the Lagrangian of the neutral Proca field \mathcal{L}=-\frac{1}{16\pi}\left(F^{\mu\nu}F_{\mu\nu}-2m^2 A_{\mu} A^{\mu}\right) symmetric? And How to make sure whether it's symmetric.

"invariant" Lagrangian or action Hello everyone, I tried to describe my question but it seems getting too complicated and confusing to write down my thoughts in detail, so I am trying to start with the following question... Are invariance of the Lagrangian under a transformation and...

L=\frac{1}{2}m(\dot{q}_1-\dot{q}_2)^2-V(q_1,q_2) Because if we put p_1=\frac{\partial L}{\partial \dot{q}_1} p_2=\frac{\partial L}{\partial \dot{q}_2} we get p_1=-p_2=m(\dot{q}_1-\dot{q}_2) We can't invert to get \dot{q_1} in terms of the two momenta. We can still write down a...

I am self trying to understand Lagrangian mechanics and I have come across with Degree of freedom and constraints which I think I understood in bits. So please try to explain these terms to me. I use Goldstein's Classical Mechanics.

Anyone know of a Lagrangian given in terms of E and B (or equivalently the tensor F) that yields Maxwell equations? A link or reference would be appreciated. I can write down such a Lagrangian which yields the two second-order Maxwell equations, but not the usual four 1st order equations...

I heard that in classical field theory, terms in the Lagrangian cannot have more than two derivatives acting on them. Why is this? In quantum field theory, I read somewhere that having more than two derivatives on a term in the Lagrangian leads to a violation of Poincare invariance. Is this...

Homework Statement Hi all, I need to derive differantial equations of system with lagrange multiplier method, a disk is rolling and a bar is fixed onto the point of a disk http://img130.imageshack.us/img130/1669/adsziss.jpg By deniz120 at 2010-05-31 Homework Equations The...