Lagrangian Definition and 1000 Threads

Homework Statement Given the Maxwell Lagrangian ##\mathcal{L} = -\frac{1}{2} (\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu}) + \frac{1}{2} (\partial_{\mu}A^{\mu})^{2}##, show that (a) ##\frac{\partial \mathcal{L}}{\partial (\partial_{\mu}A_{\nu})} = -...

Homework Statement A string of length ##a##, mass per unit length ##\sigma## and under tension ##T## is fixed at each end. The Lagrangian governing the time evolution of the transverse displacement ##y(x, t)## is ##L = \int_{0}^{a} dx \bigg[ \frac{\sigma}{2} \Big( \frac{\partial y}{\partial...

This is a microeconomics problem that I am trying to solve. I am uncertain whether my FOCs are correct. Thank you. The objective function: ui(x1i, x2i….xLi) = Σllog[xli]; The constraint: ΣLl=1p1xl ≤ w L: Σllog[xli] + λ (w - ΣLl=1p1xl) FOCs are: L1 = 1/x1 – λ(w-p1) =0 L2 = 1/x2 – λ(w-p2)...

Homework Statement A simple pendulum of length ξ and mass m is suspended from a point on the circumference of a thin massless disc of radius α that rotates with a constant angular velocity ω about its central axis as shown in Figure. Find the equation of motion of the mass m. Homework...

Homework Statement Calculate the Lagrangian of this set up: Imagine having two ropes: They are both attached to the ceiling and have different lengths. One has length b and the other has length 4b. Say they are hooked to the ceiling a distance 4b apart. Now, the ropes are both hooked to a...

Homework Statement [/B] Show that in order for the free Lagrangian to be invariant when ## A^\mu ## is transformed by a transformation U, it has to transform as below: ## A'^{\mu}=\frac i g (\partial^\mu U) U^{-1}+U A^\mu U^{-1} ## Homework Equations [/B] The wording of the problem is a bit...

Homework Statement This isn't a homework problem, per se, in that it's not part of a specific class. That being said, the question I would like help with is finding a Lagrangian density from the vertex factor, $$-ig_a\gamma^{\mu}\gamma^5.$$ This vertex would be identical to the QED vertex...

Hi, I've been working through Cornelius Lanczos book "The Variational Principles of Mechanics" and there's something I'm having difficulty understanding on page 168 of the Dover edition (which is attached). After introducing the Legendre transformation and transforming the Lagrangian equations...

Homework Statement This is Exercise 1.19 in Goldstein's Classical Mechanics 2nd edition. Self-study, not for a class. Two mass points of mass ##m_1## and ##m_2## are connected by a string passing through a hole in a smooth table so that ##m_1## rests on the table and ##m_2## hangs suspended...

Homework Statement A wheel consists of a circular uniform disk with a circular hole in it. The disc is of radius R and mass per unit area ρ. The hole is of radius ro and an axle of radius ro passes through it. The centre of the hole is offset radially from the centre of the disk by ro. The...

My question is why there is -1/2 for ω meson?

Hello everyone! I have a (supposedly) calculus problem that I just can't seem to figure out. Basically, I'm trying to understand why alternative kinetic energy formulation does not yield the same equations of motion in problem 11 of Goldstein's Classic Mechanics 3 edition. The text of problem...

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I have a free particle moving on the surface of a sphere of fixed radius R. Gravity is ignored and m/2 is left out since its constant. The lagrangian is L = R^2 \dot{\theta^2} + R^2 sin^2{\theta} \dot{\phi^2} Using the Euler Lagrange equations I obtain sin^2{\theta} \dot{\phi} = A = const \...

For each of the four fundamental forces (or fields), must one always specify the Lagrangian and Hamiltonian? What else must one specify for other fields (like the Higgs Fields)?

Homework Statement The scenario is a pendulum of length l and mass m2 attached to a mass of m1 which is allowed to slide along the horizontal with no friction. The support mass moves along in the X direction and the pendulum swings through the x-y plane with an angle θ with the vertical. After...

Why does the following Lagrangian not have the correct non-relativistic limit? It is correct except for the derivative of proper time with respect time. But that factor goes to 1 so why is the expression wrong? ## L = -(\frac{1}{2}mu^{\mu}u_{\mu} + qu^{\mu}A_{\mu})\frac{d\tau}{dt} ##

I have started reading about the Lagrangian in General Relativity, in relation to the Einstein-Hilbert action, and there is something that does not make sense to me. The Lagrangian is split into two pieces, one derived from the Ricci curvature and the other labeled L_matter, so far so good...

Homework Statement I am studying a question in Marion's classical mechanics: I am successful in obtain the equation of motion, which is where theta is the theta shown in . However, in the second part of the solution, , it puts derivative of theta to be zero and I can't understand this. Also...

Hello, I'm sure most of you are already familiar with the book "Mechanics" by Landau and Lifshitz. There's a section that I do not understand. In section 4 towards the end they mentions that "It is easy to see that the mass of the particle cannot be negative." They then give the argument that...

Homework Statement [/B] Consider a mass m moving in a frictionless plane that slopes at an angle \alpha with the horizontal. Write down the Lagrangian \mathcal{L} in terms of coordinates x measured horizontally across the slope, and y, measured down the slope. (Treat the system as...

This is my second term in my master's and one of the courses I've taken is QFT1 which is basically only QED. In the last class, the professor said the Klein-Gordon Lagrangian has a global symmetry under elements of U(1). Then he assumed the transformation parameter is infinitesimal and , under...

Possibly very silly question in QFT. Consider the Lagrangian for a scalar field theory. A term like g/φ^2 should be renormalizable on power counting arguments. The mass dimension of g should be 2 (D-1) where D is the number of space-time dimensions.Does this make sense?

Hi. I have thought that the variables q and q' in L = L(q, q') are independent. (q' = dq/dt) Of course q and q' are functions of time t , but they are only dependent in terms of t . However, in the sight of general(or abstract? I mean, not specific) functional L(q, q'), q and q' are just...

Looking for a Book on the LaGrangian Points solution in an Eigen System calculation for all the Planets.

Homework Statement A rigid “T” consists of a long rod glued perpendicular to another rod of length l that is pivoted at the origin. The T rotates around in a horizontal plane with constant frequency ω. A mass m is free to slide along the long rod and is connected to the intersection of the...

I'm looking for a book that has problems and explanations (not necessarily background theory) about mechanical systems. Consider the picture below for reference. The idea is I want to get a hold of energy and impulse methods for solving problems like this, eventually the book would have a...

Homework Statement Given the Lagrangian ##\mathcal{L} = \frac{1}{2}(\partial_{\mu}\Phi)^{2}+\frac{1}{2}\Phi^{2}-\frac{1}{2}\Phi^{3}+\frac{\alpha}{8}\Phi^{4}##, where ##\Phi=\Phi(x)##, find the equation of motion of the system. Assume that the field ##\Phi## is spherically symmetric, i.e...

Hi. I've only ever seen Noether's theorem formulated ond proven in the framework of Lagrangian mechanics. Is it possible to do the same in Newtonian mechanics, essentially only using F=dp/dt ? The "symmetries" in the usual formulation of the theorem are symmetries of the action with respect to...

Dear all, If we consider the lagrangian to have both geometric parts (Ricci scalar) and also a field, the action would take the form below: \begin{equation} S=\frac{1}{2\kappa}\int{\sqrt{-g} (\ R + \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi -V(\phi)\ )} \end{equation} which are...

Hi there, Over the last couple of weeks, I have been learning about the relativistic description of electromagnetism through Leonard Susskind's Theoretical Minimum lectures, and although I have managed to follow it, there are some parts which I am becoming increasingly confused by, not helped...

Homework Statement Two masses are connected by a weightless spring in a friction-less semicircular well (Picture included). Derive the equations of motion with help of lagrange Homework Equations L = T - U = kinetic energy - potential energy The Attempt at a Solution ##L =...

Homework Statement So, a particle is moving in a plane under the action of a force F that is oriented at all times to the direction of the center of the force.may r be the distance from the particle to the center of the force generator. Find the potential generator expression that occurs and...

Homework Statement Ok so I need to find the Lagrangian ## L ## for this system below, I have drawn some poor sketch in paint but I think its pretty easy to see what i mean Its a wheel with mass ##m## and radius ##r## that rolls inside a big cylinder with radius ##R## and at the center of...

Does the Lagrangian L in classical mechanics have any physical meaning? In classical mechanics, the Lagrangian is defined as L=T-V, the difference between the kinetic and potential energy of the system. Does this quantity have any meaning apart from that it can be plugged into Euler-Lagrange's...

A couple questions, please: I know that the Lagrangian points 1, 2 and 3 are unstable and special Lissajous orbits plus some station-keeping are required to place a spacecraft around them. But I was wondering if they are so totally unstable that they can't temporarily "capture" a passing...

i have a mathematical question which is quite similar to one asked before, still a bit different https://www.physicsforums.com/threads/derivative-of-first-term-in-lagrangian-density-for-real-k-g-theory.781472/the first term of KG-Lagrangian is: \frac{1}{2}(\partial^{\mu} \phi)(\partial_{\mu}...

Hi. I am working through a QFT book and it gives the relativistic Lagrangian for a free particle as L = -mc2/γ. This doesn't seem consistent with the classical equation L = T - V as it gives a negative kinetic energy ? If L = T - V doesn't apply relativistically then why does the Hamiltonian H =...

Hey I'm trying to follow the derivation given here: http://lampx.tugraz.at/~hadley/ss1/studentpresentations/Bloch08.pdf Homework Statement As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. L=-1/4...

I know this has been asked before: "Why is there a negative in the Lagrangian: L = T - V" I have read the answers and am not happy with them so I tried to formulate my own justification and now ask if anyone could comment on it? First, I am not happy with those who say "Because it works and...

1.) The Problem Statement: a.) Find the Lagrangian of a pendulum where the height of the hinge is oscillating in the y direction and is is defined as a function ##y_0=f(t)## b.) Add a function (a gauge transformation) of the form ##\frac{d F(\theta,t)}{dt}## to the original lagrangian...

Homework Statement Find velocity, acceleration, stream function and vorticity. Prove that velocity is equal to the acceleration. Functions given: X_1(t,e_1,e_2)= (e^\lambda)^t[e_1cos\omega t+e_2sin(\omega t)] X_2(t,e_1,e_2)= (e^-\lambda)^t[-e_1sin\omega t+e_2cos(\omega t)] Homework Equations...

I am disappointed by my graduate-level classical mechanics course, and especially the treatment of Lagrangian/Hamiltonian mechanics. Now, I scanned my notes and some crazy idea popped into my head, further fueling my discontent towards this course, because all the problems covered in class were...

Homework Statement I'm given a driven, dampened harmonic oscillator (can it be thought of as a spring-mass system with linear friction?) Is it possible to solve the equation of motion using Lagrangian mechanics? I could solve it with the usual differential equation x''+βx'+ωₒ²x=fₒcos(ωt) but as...

I am now reading Lagrange's equations part in Taylor's Classical Mechanics text. It says: When a system of interest involves constraint forces, F_cstr, and all the nonconstraint forces are derivable from a potential energy(U), then the Lagrangian for the system L is L = T - U, where U is the...

Homework Statement What is the lagrangian of a free reletavistic particle in a electro-magnetic field? And what are the v(t) equations that come from the Euler-Lagrange equations (given A(x) = B0/2 crosProduct x) (B/2 is at z direction) Homework EquationsThe Attempt at a Solution I've got to: L...

Homework Statement I am asked to find the Hamiltonian of a system with the following Lagrangian: ##L=\frac{m}{2}[l^2\dot\theta^2+\dot{\tilde{y}}^2+2l\dot{\tilde{y}}\dot{\theta}\sin{\theta}]-mg[\tilde{y}-l\cos{\theta}]## Homework Equations ##H = \dot{q_i}\frac{\partial L}{\partial...

Homework Statement The lagrangian is given by: L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi + \frac{\alpha}{2} \phi \partial_{\mu} \phi \partial^{\mu} \phi And the question is to find the feynman rules. Homework EquationsThe Attempt at a Solution I started by using the...

Homework Statement Show that if a transformation ##\Phi \rightarrow \Phi + \alpha \partial \Phi/ \partial \alpha## is not a symmetry of the Lagrangian, then the Noether current is no longer conserved, but rather ##\partial_{\mu}J^{\mu} = \partial L/ \partial \alpha##. Use this result to show...

I was solving the double pendulum problem via Lagrangian methods but something bothered me quite a lot. (Consider the two bobs are of equal mass and the pendula are of equal length). Then the potential energies are conveniently written as ##V_1=-mgl\cos\theta## and...