Lagrangian Definition and 1000 Threads

Consider two arbitrary scalar multiplets ##\Phi## and ##\Psi## invariant under ##SU(2)\times U(1)##. When writing the potential for this model, in addition to the usual terms like ##\Phi^\dagger \Phi + (\Phi^\dagger \Phi)^2##, I often see in the literature, less usual terms like: $$\Phi^\dagger...

Total Lagrangian is very complex,but in concrete theory we use a part of Lagrangian.My question is:Why the results of a theory are the same when we use only some terms of the total Lagrangian?

Hi everyone, I have a question that can't solve. Does exist a lagrangian for the relativistic angular momentum (AM)? I can't even understand the question because it has no sense for me... I mean, the lagrangian is a scalar function of the system(particle,field,...), it isn't a function FOR the...

The EM Lagrangian is $$\mathcal{L} = -\frac{1}{2}[(\partial_\mu A_\nu)(\partial^\mu A^\nu) - (\partial_\mu A_\nu)(\partial^\nu A^\mu)]$$ In the QFT notes from Tong the EM Lagrangian is written in the form $$\mathcal{L} = -\frac{1}{2}[(\partial_\mu A_\nu)(\partial^\mu A^\nu) - (\partial_\mu...

Homework Statement I'm stuck at my particle physics exercise about 4-component chiral fields. The following problem is given: "Derive the expression for the QED Lagrangian in terms of the four component right-handed and left-handed Dirac fields ##\Psi_R(x)## and ##\Psi_L(x)##, respectively."...

90 years have gone by since P.A.M. Dirac published his equation in 1928. Some of its most basic consequences however are only discovered just now. (At least I have never encountered this before). We present the Covariant QED representation of the Electromagnetic field. 1 - Definition of the...

Assuming generlized variables, q, we have a Lagrangian in mechanics as the kinetic energy, K, minus potential energy, U, with a dependency form such that L(q,dq/dt) = K(q, dq/dt) - U(q) Can someone provide examples of Lagrangians in other disciplines?

This is an example from "Noether's Theorem" by Neuenschwander. Chapter 5, example 4, page 74-75. He gives the Lagrangian for a charged particle in an electromagnetic field: ##L=\frac12 m \dot {\vec{r}}^2+e \dot{\vec{r}} \cdot \vec{A} -eV## And claims invariance invariance under the...

I know, $$ L=T-V \;\;\; \; \;\;\; [1]\;\;\; \; \;\;\; ( Lagrangian) $$ $$ H=T+V \;\;\; \; \;\;\;[2] \;\;\; \; \;\;\; (Hamiltonian)$$ and logically, this leads to the equation, $$ H - L= 2V \;\;\; \; \;\;\...

Hi, I am reading Landau-Lifshitz course in theoretical physics 1. volume, mechanics. The mechanics is derived using variatonal principle from the start. At first they start with point particles, that do not interact with each other. Thus the equations of motions must be independent for the...

Homework Statement The problem is attached. I'm working on the second system with the masses on a linear spring (not the first system). I think I solved part (a), but I'm not sure if I did what it was asking for. I'm not sure exactly what the question means by the... L=.5Tnn-.5Vnn. Namely, I'm...

Hi all, I'm not certain if this is the correct section of the forum for this thread but I'm trying to understand ghosts and BRST symmetry and my starting point is chapter 16 of Peskin and Schroeder where I've found a nagging issue. My issue is regarding the derivation of equation (16.6) on...

Hi here is the situation; There's a spherical particle contained with a MEMS sensor (3D accelerometer and gyroscope) moving down a bed. What we want is to estimate the total kinetic energy of the particle. The total kinetic energy has two parts, translational part and rotational part. for the...

Homework Statement I'm supposed to find the normal force acting on the box by the slab as a function of time. The problem is I don't know what the constraint is. I can't find the relation between r and theta that adds the two up to zero. Homework Equations Lagrangian equation. The Attempt...

Homework Statement A yoyo falls straight down unwinding as it goes, assume has inner radius a, outer radius b and Inertia I. What is the generalised coordinates and the lagrangian equation of motion? Homework Equations L=T-U where T is kinetic energy and U is potential The Attempt at a...

Hello, I'm trying to follow Goldstein textbook to show that the Lagrangian is invariant under coordinate transformation. I got confused by the step below So ## L = L(q_{i}(s_{j},\dot s_{j},t),\dot q_{i}(s_{j},\dot s_{j},t),t)## The book shows that ##\dot q_{i} = \frac {\partial...

Homework Statement Show that $$L=\phi\Box^2\phi$$ generates negative energy density. Homework EquationsThe Attempt at a Solution The energy density is $$E=\frac{\partial L}{\partial \dot{\phi}}\dot{\phi}-L$$ Also the Lagrangian can be rewritten (using divergence theorem) as...

Homework Statement I want to show that the propagator of Proca Lagrangian: \mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}M^2A_\mu A^\mu Is given by: \widetilde{D}_{\mu \nu}(k)=\frac{i}{k^2-M^2+i\epsilon}[-g_{\mu\nu}+\frac{k_\mu k_\nu}{M^2}]Homework Equations Remember that...

Homework Statement I have the Lagrangian $$L=-\frac{1}{2}\phi\Box \phi-\frac{1}{2}m^2\phi^2$$ and I need to show that the propagator in the momentum space I obtain using this lagrangian (considering no interaction) is the same as if I consider the free Lagrangian to be...

Hi, If I have a massive particle constrained to the surface of a Riemannian manifold (the metric tensor is positive definite) with kinetic energy $$T=\dfrac 12mg_{\mu\nu} \dfrac{\text dx^{\mu}}{\text dt} \dfrac{\text dx^{\nu}}{\text dt}$$ then I believe I should be able to derive the geodesic...

Hello! If you have a Lagrangian (say of a scalar field) depending only on the field and its first derivative and you want to calculate the ground state configuration, is it necessary a constant value? I read about Spontaneous symmetry breaking having this Lagrangian $$L= \frac{1}{2}(\partial_\mu...

Homework Statement This is a continuation of this problem. I will rewrite it here too: The Lagrangian density for the ##h=h^{00}## term of the Einstein gravity tensor can be simplified to: $$L=-\frac{1}{2}h\Box h + (M_p)^ah^2\Box h - (M_p)^b h T$$ The equations of motion following from this...

<<Moderator's note: Moved from a technical forum, no template.>> Description of the system: The masses m1 and m2 lie on a smooth surface. The masses are attached with a spring of non stretched length l0 and spring constant k. A constant force F is being applied to m2. My coordinates: Left of...

Homework Statement I'd like to show, if possible, that rotational invariance about some axis implies that angular momentum about that axis is conserved without using the Lagrangian formalism or Noether's theorem. The only proofs I have been able to find use a Lagrangian approach and I'm...

Is there any proof for the Lagrangian: $$L = T - U $$ And why L = T - U ? Any help is much appreciated. Thank you.

These images have been taken from Goldstein, Classical Mechanics. Why do we need Lagrangian formulation of mechanics when we already have Newtonian formulation of mechanics? Newtonian formulation of mechanics demands us to solve the equation of motion given by equation 1. 19. for this we need...

A single particle Hamitonian ##H=\frac{m\dot{x}^{2}}{2}+\frac{m\dot{y}^{2}}{2}+\frac{x^{2}+y^{2}}{2}## can be expressed as: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{x^{2}+y^{2}}{2}## or even: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{\dot{p_{x}}^{2}+\dot{p_{x}}^{2}}{4}##...

Hello! I have a classical Lagrangian of the form $$L=A\dot{x_1}^2+B\dot{x_2}^2+C\dot{x_1}\dot{x_2}cos(x_1-x_2)- V$$ the potential is irrelevant for the question and A, B and C are constants. When doing $$\frac{d}{dt}\frac{\partial L}{\partial \dot{x_1}}$$ the solution gives this...

Homework Statement $$ L = -\frac{1}{2} (\partial_{\mu} A_v) (\partial^{\mu} A^v) + \frac{1}{2} (\partial_{\mu} A^v)^2$$ calculate $$\frac{\partial L}{\partial(\partial_{\mu} A_v)}$$ Homework Equations $$ A^{\mu} = \eta^{\mu v} A_v, \ and \ \partial^{\mu} = \eta^{\mu v} \partial_{v}$$ The...

Homework Statement This is derivation 2 from chapter 8 of Goldstein: It has been previously noted that the total time derivative of a function of ## q_i## and ## t ## can be added to the Lagrangian without changing the equations of motion. What does such an addition do to the canonical momenta...

Homework Statement A mass m slides down a frictionless plane that is inclined at angle θ. Show, by considering the force of constraint in the Lagrangian formulation, that the normal force from the plane on the mass is the familiar mg cos(θ). Hint: Consider the Normal force to be the result of...

Homework Statement I would like to solve for Y an optimisation problem Homework Equations Max Y'C + Y'Br + αr0 Subject to : k=sqrt(Y'ΣY) Y'e + α = 1 Where Y, C and B are columns vector of n lines. Σ is symetric matrix of n order e =(1,...1)' and α is a reel parameter. I did calculus with...

Hello guys, I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here...

Homework Statement I want to be able, for an arbitrary Lagrangian density of some field, to derive the energy-momentum tensor using Noether's theorem for translational symmetry. I want to apply this to a specific instance but I am unsure of the approach. Homework Equations for a field...

Homework Statement Given a system with a Lagrangian ##L(q,\dot{q})## and Hamiltonian ##H=H(q,p)## and that the Lagrangian is invariant under the transformation ##q \rightarrow q+ K(q) ## find the generating function, G. Homework EquationsThe Attempt at a Solution ##\delta q = \{ q,G \} =...

Bathe (reference below) outlines the updated Lagrangian (UL) and total Lagrangian (TL) approaches using the second Piola Kirchhoff (PK2) stress. Others (i.e., Ji, et al. and Abaqus) define the UL and TL formulations in terms of the Kirchhoff or the Cauchy stress in rate form. This form requires...

Homework Statement [/B] In a homogeneous gravity field with uniform gravitational acceleration g, a pointmass m1 can slide without friction along a horizontal wire. The mass m1 is the pivot point of a pendulum formed by a massless bar of constant length L, at the end of which a second...

Consider a Lagrangian: \begin{equation} \mathcal{L} = \mathcal{L}(q_1\, \dots\, q_n, \dot{q}_1\, \dots\, \dot{q}_n,t) \end{equation} From this Lagrangian, we get a set of ##n## equations: \begin{equation} \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q}_i} - \frac{\partial...

Homework Statement Not sure if the link is showing. But it's imgur.com/a/LEvd0 Homework Equations The steps I've taken so far as written in the attempt section below is correct. The solution provided then proceeds with letting ##z = x + iy## and setting ##\ddot z+i \omega \dot z = 0##. Then...

In Lagrangian mechanics, both q(t) and dq/dt are treated as independent parameters. Similarly, in Hamiltonian mechanics q and p are treated as independent. How is this justified, considering you can derive the generalized velocity from the q(t) by just taking a time derivative. Does it have...

Homework Statement Homework Equations The Attempt at a Solution

The Lagrangian in classical mechanics is known to be a difference of the kinetic and potential energy. My first question is - why? I.e. are there any reasons (except for "because it works this way") to have it as this difference of energies? The second question is why is it this very value...

(note: I'm going to represent the lagrangian as simply L because I don't know how to do script L in latex.) Homework Statement Two particles of equal masses m are confined to move along the x-axis and are connected by a spring with potential energy ##U = \frac{1}/{2}kx^2## (here x is the...

Homework Statement A bead of mass ##m## slides (without friction) on a wire in the shape, ##y=b\cosh{\frac{x}{b}}.## Write the Lagrangian for the bead. Use the Lagrangian method to generate an equation of motion. For small oscillations, approximate the differential equation neglecting terms...

Homework Statement A particle of mass ##m## moves without slipping inside a bowl generated by the paraboloid of revolution ##z=b\rho^2,## where ##b## is a positive constant. Write the Lagrangian and Euler-Lagrange equation for this system. Homework Equations...

I'm trying to show that the theta term in the QCD Lagrangian, ##\alpha G^a_{\mu\nu} \widetilde{G^a_{\mu\nu}}##, can be written as a total derivative, where ##\begin{equation} G^a_{\mu\nu} = \partial_{\mu} G^a_{\nu} - \partial_{\nu}G^a_{\mu}-gf_{bca}G^b_{\mu}G^c_{\nu} \end{equation} ##...

The QCD Lagrangian is ##\mathcal{L}=-\frac{1}{4}G^{a}_{\mu\nu}G^{a\mu\nu}+\sum\limits_{j=1}^n \left[\bar{q}_j\gamma^{\mu}iD_{\mu}q_j - (m_jq^{\dagger}_{Lj}q_{Rj}+h.c.)\right]+\frac{\theta g^2}{32\pi^2}G^{a}_{\mu\nu}\widetilde{G}^{a\mu\nu}## Why is it so often quoted as just...

I am working through the first chapter of Lessons on Particle Physics by Luis Anchordoqui and Francis Halzen. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf I am on page 22. Equation 1.5.61: ##L_{Dirac}=\psi \bar ( i\gamma^\mu \partial_\mu-m)\psi## where ##\psi bar =...

So the first term of the Lagrangian is proportional to ##{F_{\mu \nu}}{F^{\mu \nu}}##. I wrote out the matrices for ##{F_{\mu \nu}}## and ##{F^{\mu \nu}}## and multiplied at the terms together and added them up, but some of the terms didn't cancel like they should have. Should I have used minus...

Homework Statement I am given the Hamiltonian of the relativistic free particle. H(q,p)=sqrt(p^2c^2+m^2c^4) Assume c=1 1: Find Ham-1 and Ham-2 for m=0 2: Show L(q,q(dot))=-msqrt(1-(q(dot))^2/c^2) 3: Consider m=0, what does it mean? Homework Equations Ham-1: q(dot)=dH/dp Ham-2: p(dot)=-dH/dq...