Lagrangian Definition and 1000 Threads

So I have to find the min and max values of f(x,y,z) = x^4 + y^4 + z^ 4 given the constraint x^2 + y^2 + z^2 = 1. I've found the points (+-1/sqrt(3),1/sqrt(3) ,1/sqrt(3)), (+-1/sqrt(3),-1/sqrt(3) ,1/sqrt(3)) ... etc all of which have the f-value of 1/3 when x =/= 0 & y =/= 0 & z =/= 0 (this...

I've always found that the lagrangian for the Gravitational field is that from the Einstein-Hilbert action: L=R (R is the Ricci scalar; I'm not including the factor of \sqrt{-g}) but when variational principles are applied, we get the vacuum field equations (obviously). I'd like someone...

How to show the Proca equation by using the given Proca Lagrangian? Surely, I know the Euler-Lagrange equation, but I can't solve this differentiation!(TT) The given Proca lagrangian is, \mathcal{L}=...

Hello everyone I have difficulties in understanding some stuff in Lagrangian and Hamiltonian mechanics. This concerns the equations : \dot p = - \frac{\partial H}{\partial q} \frac{d}{dt} \frac{\partial L}{\partial \dot q} = \frac{\partial L}{\partial q} First I have to say that I'm a math...

Does anyone know the Lagrangian for the propagation of light in curved spacetime? I'm disappointed to discover that I don't actually know how to compute the action for a given null curve.

I don't know how to do the 4 questions. And I only have some ideas on questions 1. The details are written on the photos. Thanks for help. Eqtn 2.28 2.36 2.37 are given on the photo.

In most of the physical systems, if we have a Lagrangian L(q,\dot{q}), we can define conjugate momentum p=\frac{\partial L}{\partial{\dot{q}}}, then we can obtain the Hamiltonian via Legendre transform H(p,q)=p\dot{q}-L. A important point is to write \dot{q} as a function of p. However, for the...

Hi All, is there anybody to give me some help on how I can calculate the Euler Lagrangian equation associated with variation of a given functional? I am new with these concepts and have no clue about the procedure. thanks a lot

Hi, I was wondering if anyone had any examples of ALE codes in any dimension and using any method (FEM, FVM, FDM). I just need to know how the mesh velocity and the fluid velocity are dealt with. Many Thanks, H

Is there a systematically way of finding all space-time symmetries of a given Lagrangian? E.g. given a electromagnetic Lagrangian, can I somehow derive that the symmetries in question are conformal ones? Thanks.

So I was lucky enough to visit CERN earlier this year, and they were selling t-shirts with a long equation on them. And I, like an idiot tourist, somehow jumped to the conclusion this was the standard model Lagrangian and got the t-shirt. Well, it's still a pretty cool t-shirt, but then I went...

As is well known, a Dirac Lagrangian can be written in a symmetric form: L = i/2 (\bar\psi \gamma \partial (\psi) - \partial (\bar\psi) \gamma \psi ) - m \bar\psi \psi Let \psi and \psi^\dagger be independent fields. The corresponding canonical momenta are p = i/2 \psi^\dagger...

I'm going through John Taylor's book on Classical Mechanics and am having some difficulty understanding his derivation of momentum conservation in the Lagrangian section. Firstly he starts the section off with assuming that all N particles of a system are moved in space by a distance \epsilon...

We all know that the free Lagrangian for a spin-1/2 Dirac field is \mathcal{L}=\bar\psi(i\gamma_\mu\partial^\mu-m)\psi. But, if I were to invent a Lagrangian, I would have tried \mathcal{L}=\partial_\mu\bar\psi\partial^\mu\psi-m^2\bar\psi\psi. What's wrong with this second Lagrangian? Why...

I am currently reading and trying to solve most of the problems in Carroll's Geometry and Spacetime. I am generally okay at the math (I've done some mathy Riemannian Geometry type stuff), but am not overly good at some of the higher-level physics. Homework Statement (Chapter 1, Question 13)...

Hey everyone, So I'm just looking around to get a hold of some lagrangian mechanics for the GRE's coming up. Is the lagrangian always dealing with energy? Basically there was a problem I encountered with trying to find the lagrangian of a rolling ball in some setup, and once I knew that it was...

Hello, In the lagrangian of QCD, there is q which is the quark field and it is the fundamental representation of SU(3). This q is multiplied by a gamma matrix and a q bar. So, how can we have a 4x4 matrix multiplying 1x3 matrix? Thanks

Homework Statement See image (I think I forgot to rotate it, careful with your necks!): http://img600.imageshack.us/img600/7888/p1000993t.jpg The system consists of 2 point masses joined by a rigid massless bar of length 2l, which can rotate freely only in the z-x plane. The center of...

According to my CM text, Lagrangian Mechanics can be used to derive Newton's laws. We define the Lagrangian as L=T-V. Now, how do we know what T is? Is it defined to be 1/2mv^2? The only way I know how to derive that is using the work energy theorem which feels like 'cheating' since I am...

Please teach me this: Do the Lagrangian(before broken in symmetry) and corresponding hidden symmetry broken Lagrangian describe the same physics or not?Because the field in the Lagrangian before broken is shifted by a constant in comparision with the field in the broken Lagrangian,but at...

Homework Statement I am trying to prove that Lagrangian L is not uniquely defined, but only up to a time derivative of a function: \frac{d\Lambda}{dt}, \Lambda(\vec{q}, t) So L > L+\frac{d\Lambda}{dt} = L+\frac{\partial \Lambda}{\partial q}~\dot{q}+\frac{\partial \Lambda}{\partial t} But...

Homework Statement I want to check that the QED lagrangian \mathcal{L}=-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta} + \bar\Psi(i\displaystyle{\not} D - m)\Psi where F^{\alpha\beta} = \partial^\alpha A^\beta - \partial^\beta A^\alpha, \ D^\mu = \partial^\mu - ieA^\mu is invariant under charge...

So I type a differential equation into Wolfram Alpha, like so: And one of the things that W-A outputs is the "Lagrangian" of that equation, which is so: My question is, what is this Lagrangian, what does it describe, and how do I find it?

Hi all, I'm doing some analysis of a bicycle mechanics problem and at one point the approximations I'm making mean that the problem reduces to the classic inverted pendulum. I'm very confused, as the equations I've worked out appear to have an unphysical singularity in them, and I can't see...

Homework Statement In my notes i have the following two equations written with no explanation where thehy came from... can someone help please!? L=(u, x )= -mc\sqrt(u^{\beta} u_\beta)-\frac{q}{c}A^{\alpha}u_\alpha, L(v,r, t) = -mc^2(1-\frac{v^2}{c^2})-\phi +\frac{q}{c}vA L is...

Please teach me this: Why the minima of potential of classical Lagrangian is called the ''vacuum expectation value of Phi(field function)''.Is it really a vacuum expectation value of field operator at the vacuum states(at this state,the potential part of classical Lagrangian equals zero)...

I started to answer this question, and I have quite a bit an answer, but still not complete, let's say that we write a Lagrangian in QFT, which an unknown function of the scalar field \phi and its derivative \partial \phi. We can always Taylor-expand it and get: L(\phi,\partial\phi) = a + b \phi...

Hi, Would someone know where I can find a derivation of the lorentz-invariant lagrangian density? This lagrangian often pops-up in books and papers and they take it for granted, but I was actually wondering if there's a "simple" derivation somewhere... Or does it take a whole theory and...

Homework Statement Trying to solve a Lagrangian, got down this integral. Unfortunately the zeroth-solution isn't good enough since the constant k is close to 1 for our experimental set-up. \int_{0}^{x}dx(\frac{xsin(x)}{1+kcos^2(x)}})^\frac{1}{2} Any hints? I'm not sure where to get...

We were working on some Lagrangian and trying to solve it using an ansatz. Due to some problems in the results we got, we started to doubt the correctness of the method in use. Here is a very simple Lagrangian which shows the problem very easily: L=\frac{1}{2}\dot{x}^{2}-gx We took m=1 for the...

Hi, Hope some one can help me with a problem I am working on: It involves working out: \frac{\delta L}{\delta A_\nu} of the following Lagrangian: L=\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{2} (D_{\mu} \Psi)^{*} D^{\mu}\Psi The solutions show that this is equal to: \frac{\delta...

I'm looking for a summary of what invariance or symmetry of the Action in Feynman's path integral has on the equations of motion and on measurement. Do different symmetry groups of the Action integral result in different equations of motion for different particles? Is the least action principle...

I actually have 2 questions. 1)How do you decompose the Lagrangian into kinetic and potential energy? 2)Knowing the Lagrangian, how do we find out if energy of the system is conserved. Example: L=q'^2*sin(q)+q'*exp(q)+q q' is the time derivative of q. Thanks in advance

According to this site http://quantummechanics.ucsd.edu/ph130a/130_notes/node453.html a good choice of Lagrangian for the electromagnetic field is L = - \frac {1}{4} F_{\mu\nu}F_{\mu\nu} + \frac {1}{c} j_\mu A_\mu where F_{\mu \nu} = \frac {\partial A_\nu}{\partial...

Homework Statement Given the Lagrangian density: L= -\frac{1}{2} \partial_{\mu}A_\nu \partial^{\mu}A^\nu -\frac{1}{c}J_\mu A^\mu (a) find the Euler Lagrange equations of motion. Under what assumptions are they the Maxwell equations of electrodynamics? (b) Show that this Lagrangian...

Homework Statement Given the Lagrangian density: L= -\frac{1}{2} \partial_{\mu}A_\nu \partial^{\mu}A^\nu -\frac{1}{c}J_\mu A^\mu (a) find the Euler Lagrange equations of motion. Under what assumptions are they the Maxwell equations of electrodynamics? (b) Show that this Lagrangian...

Homework Statement The specific Lagrangian for a cantilever beam is given by: \overline{L}=\frac{1}{2}m[\dot{u}^2(s,t)+\dot{v}^2(s,t)]-\frac{1}{2}EI[\psi ^{\prime}(s,t)]^2 where m,EI are mass and bending stifness, respectively. \dot{u},\dot{v} are velocities in u,v directions...

Hello, Just a few questions about a couple of terms in and not in the SM Lagrangian. I'll talk in particular about these fields, and their representations in SU(3) x SU(2) x U(1) Q (3,2,1/6) (left-handed quarks, fermion) U (3,1,2/3) (right-handed up-quarks, fermion) \phi (1,2,-1/2) (higgs...

Homework Statement Find the Lagrangian and the Lagrangian equations for this pendulum (see the picture). Radius of circle is a, mass of the bob is m and l is the length of the pendulum when it hangs straight down. Homework Equations The Attempt at a Solution I obtain...

I know that, for time-independent potentials, we have E=sum (Vi*partial dL/dVi) - L What if one or more of the potentials are time-dependent? Is the relationship between energy and the lagrangian then "total dE/dt = - partial dL/dt "?

Hey, I was just playing about with some lagrangian mechanics and tried to work out the angular momentum of the earth; Starting with the Lagrangian \mathcal{L} = \left(\frac{1}{2}m (r')^2 + \frac{1}{2}m r^2 (\theta ')^2\right)+\frac{G m M}{r} Applying the Euler Lagrange eqn to prove...

Homework Statement I worked a textbook problem earlier where I had to write the Lagrangian for a pendulum (of mass m and length l) connected to a massless support moving along the x-axis. I chose the angle theta as my generalized coordinate, since the problem specified that the acceleration of...

Euler Lagrange Equation : if y(x) is a curve which minimizes/maximizes the functional : F\left[y(x)\right] = \int^{a}_{b} f(x,y(x),y'(x))dx then, the following Euler Lagrange Differential Equation is true. \frac{\partial}{\partial x} - \frac{d}{dx}(\frac{\partial f}{\partial y'})=0...

What's the most persuasive argument for using the potential phi and A as independent deegres of freedom in the electromagnetic Lagrangian instead of the more physical field E and B? Why does the cannonical approach break down for E and B?

Homework Statement Assuming that transformation q->f(q,t) is a symmetry of a lagrangian show that the quantity f\frac{\partial L}{\partial q'} is a constant of motion (q'=\frac{dq}{dt}). 2. Noether's theorem http://en.wikipedia.org/wiki/Noether's_theorem The Attempt at a Solution...

I need the general expression for the lagrangian density of a linear elastic solid. I haven't been able to find this anywhere. Thanks.

In flat space time the Lagrangian for the EM potential is (neglecting the source term) \mathcal{L}_{flat}=-\frac{1}{16\pi}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}) which is a scalar for flat spacetime. I would have expected the...

Hi, My question is. Can in principle, a Lagrangian density for some theory be a pseudo-scalar. Normally people say that the Lagrangian needs to be a scalar, but it case it is a pseudo-scalar it would also be a eigaen function of the parity operator. This topic could well be on the...

My question Does the Sun influence the stability of the Lagrangian points in the Earth-Moon system? Apparently the concept of Lagrangian points works both in the Earth-Moon system and the Earth-Sun system. However, In the Earth-Moon system the Sun, as a third body, has a big influence on...

Homework Statement A bead of mass m is free to move on a stationary frictionless hoop of radius R. The hoop is in a horizontal plane (no need to take gravity into account) and it is located a distance d from a stationary wall. The bead is attached to the wall by a spring (constant k and...