Lagrangian Definition and 1000 Threads

I'm having trouble following a proof of what happens when the Dirac Lagrangian is put into the Euler-Lagrange equation. This is the youtube video: and you can skip to 2:56 and pause to see all the math laid out. I understand the bird's eye results of the Dirac Lagrangian having an equation of...

I've learned that in canonical quantization you take a Lagrangian, transform to a Hamiltonian and then "put the hat on" the fields (make them an operator). Then you can derive the equations of motion of the Hamiltonian. What is the reason that you cannot already put hats in the QFT Lagrangian...

I understand the process of the calculus of variations. I accept that a proper Lagrangian for Dynamics is "Kinetic minus potential" energy. I understand it is a principle, the same way F=ma is a law (something one cannot prove, but which works) Still... what do you say to students who say "I...

Hi all experts! I would like to read about the Lagrangian of a classical (non-quantum), real, scalar, relativistic field and how it is derived. What is the best book for that purpose?Sten Edebäck

Hi, I am reading Robert D Klauber's book "Student Friendly Quantum Field Theory" volume 1 "Basic...". On page 48, bottom line, there is a formula for the classical Lagrangian density for a free (no forces), real, scalar, relativistic field, see the attached file. I like to understand formulas...

In https://arxiv.org/pdf/1709.07852.pdf, it is claimed in equation (1) and (2) that when we take non-relativistic limit, the following Lagrangian (the interaction part) $$L=g \partial_{\mu} a \bar{\psi} \gamma^{\mu}\gamma^5\psi$$ will yield the following Hamiltonian $$H=-g\vec{\nabla} a \cdot...

Lagrangian principle is easier to solve any kind of problem. But we always "forget" (not really. But we don't take it into account directly.) of Tension in a system when looking at Lagrangian. But some questions say to find Tension. Since we can get the equation of motion from Newton's 2nd law...

Setting up coordinates for the problem ##L=\frac{1}{2}M_1 \dot{x}^2+\frac{1}{2}M_2(\dot y-\dot x)^2+M_1gx+M_2g(l_a-x+y)## After using Euler Lagrange for x component and y component separate and substitute one to another then I get that ##\ddot{x}=\frac{M_1-2M_2}{M_1}g## whereas on the...

What are the rules for writing a good Lagrangian? I know that it should be a function of the position and its first order derivatives, because we know that we only need 2 initial conditions (position and velocity) to uniquely determine the future of the particle. I know that the action has to be...

If the standard model Lagrangian were generalized into what might be called "core capabilities" what would those capabilities be? For example, there are a lot of varying matrices involved in the standard model Lagrangian and we can generalize all of them as the "core capability" of matrix...

I'm stuck in a problem of a spring mass system with a pendulum attached to it as showed in the figure below: My goal is to find the movement equation for the mass, using Lagrangian dynamics. If the spring moves, the wire will move the same amount. Therefore, we can write the x and y position...

Hello, I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse. So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...

How would you unify the two Lagrangians you see in electrodynamics? Namely the field Lagrangian: Lem = -1/4 Fμν Fμν - Aμ Jμ and the particle Lagrangian: Lp = -m/γ - q Aμ vμ The latter here gives you the Lorentz force equation. fμ = q Fμν vν It seems the terms - q Aμ vμ and - Aμ Jμ account for...

I need to use hermiticity and electromagnetic gauge invariance to determine the constraints on the constants. Through hermiticity, i found that the coefficients need to be real. However, I am not sure how gauge invariance would come into the picture to give further contraints. I think the...

According to @vanhees71 and his notes at https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf under certain conditions one can choose ##\tau## as the parameter to parametrize the Lagrangian in special relativity. For instance if we have, $$A[x^{\mu}]=\int d\lambda...

This is from Coleman Lectures on Relativity, p.63. I understand that he uses integration by parts, but just can't see how he gets to the second equation. (In problem 3.1 he suggest to take a particular entry in 3.1 to make that more obvious, but that does not help me.)

I want to share my recent results on the foundation of classical mechanics. Te abstract readWe construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, several concepts from analytical mechanics will be rediscovered...

I am stuck deriving the gauge field produced in curved spacetime for a complex scalar field. If the underlying spacetime changes, I would assume it would change the normal Lagrangian and the gauge field in the same way, so at first guess I would say the gauge field remains unchanged. If there...

This is from Taylor's classical mechanichs, 11.4, example of finding the Lagrangian of the double pendulum Relevant figure attached below Angle between the two velocities of second mass is $$\phi_2-\phi_1$$ Potential energy $$U_1=m_1gL_1$$ $$U_2=m_2g[L_1\cos(1-\phi_1)+L_2(1-\phi_2)]$$...

I know that the Action has units Energy·time or Momentum·position. A second fact is that the derivative of the action with respect to time is Energy and similar with momentum-position, consistent with a units ie. dimensions check.Is it a coincidence that both are Noether conserved quantities...

There is a problem from a Russian textbook in classical mechanics. Consider a scalar equation $$\ddot x=F(t,x,\dot x),\quad x\in\mathbb{R}.$$ Show that this equation can be multiplied by a function ##\mu(t,x,\dot x)\ne 0## such that the resulting equation $$\mu\ddot x=\mu F(t,x,\dot x)$$ has...

I am fairly certain that the answer here is to differentiate partially with respect to time rather than fully. In Landau and Lifshitz' proof of energy conservation one of the hypotheses is that the partial of L wrt time is zero. Am I on the right track?

In classical mechanics to establish the Euler-Lagrange equations of motion of a particle we "minimize" the action, that is the integral of the Lagrangian, prescribing as the integral limits the initial and final positions of the particle. Usually, for a problem in mechanics we do not know the...

Hi, I am working on the following optimization problem, and am confused how to apply the Lagrangian in the following scenario: Question: Let us look at the following problem \min_{x \in \mathbb{R}_{+} ^{n}} \sum_{j=1}^{m} x_j log(x_j) \text{subject to} A \vec{x} \leq b \rightarrow A\vec{x}...

Hi Pfs When instead of the variables x,x',t the lagrangiean depends on the trandformed variables q,q',t , time may be explicit in this lagrangian and q' (the velocity of q) may appear outside. I am looking for a toy model in which tine is not explicit in L but where the velocities appear somhere...

Lagrangian mechanics is built upon calculus of variation. This means that we want to find out function which is a stationary point of particular function (functional) which in Lagrangian mechanics is called the action. To know what this function is, action needs to be defined first. Action is...

Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?

I understand that it is a system with two degrees of freedom. And I chose as generalized coordinates the two angles shown in the pic I posted. I am having troubles in finding the kinetic energy of this system, cause the book tells me that the kinetic energy is something different then what I...

Klein Gordon Lagrangian is given by \mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^2\phi^2 I saw also this link https://www.pas.rochester.edu/assets/pdf/undergraduate/the_free_klein_gordon_field_theory.pdf Can someone explain me, what is...

I think I undeerstand Lagrangian mechanics but I have a question that will help to clarify some concepts. Imagine I throw a pencil. For that I have 5 generalised coordinates (x,y,z and 2 rotational). When I express Kinetic Energy (T) as: $$T = 1/2m\dot{x^{2}}+1/2m\dot{y^{2}}+1/2m\dot{z^{2}} +...

I am sorry for all these questions this morning. Could someone read the attached and tell me where I am going wrong?

As I have read, the effective Lagrangian are non local sometimes, does that mean they break causlaity ? Are they non local because the heavy particles ( propagators) are integrated out?

Some introduction books on Lagrangian and Hamiltonian mechanics use classical mechanics as the theoretical framework, and when it come to special relativity it goes back to the basics and force language again. I would like to ask for some recommendations on good books that introduces Lagrangian...

Hi I have been doing a question on Lagrangian mechanics. I have the solution as well but i have a problem with the way the question is asked regarding dimensions. The 1st part of the question says that a particle of mass m with Cartesian coordinates x , y , z moves under the influence of...

Hi, I was trying to solve the classical two body problem with Lagrangian Principle. I replaced the angular velocity before taking the partial derivatives (which respect to the distance to the virtual particle) and the result was completely different. I would like to ask, therefore, which...

I have sometimes seen the claim that one advantage of Lagrangian mechanics is that it works in any frame of reference, instead of like Newtonian mechanics which will hold only in the inertial frame of reference. However isn't this gain only at the sacrifice that the Lagrangian will need to take...

Hi, there. Suppose I write down a Langrangian as ##L(t, x, y, z, f)## where ##t##, ##x##, ##y## and ##z## are identified as variables, and ##f=f(t, x, y, z)##. Now the Euler-Lagrange equations$$\sum_{\mu=0} ^3 \frac d {dx^{\mu}}\left (\frac {\partial L }{\partial (\partial_\mu f)} \right )...

So Noether's Theorem states that for any invarience that there is an associated conserved quantity being: $$ \frac {\partial L}{\partial \dot{Q}} \frac {\partial Q}{\partial s}$$ Let $$ X \to sx $$ $$\frac {\partial Q}{\partial s} = \frac {\partial X}{\partial s} = \frac {\partial...

My understanding of the system from the image (which was given in book) I could see there's 3 tension in 2 body. Even I had seen 2 tension in a body. It was little bit confusing to me. I could find tension in Lagrangian from right side. But left side was confusing to me...

Hi Guys Finally after a great struggle I have managed to develop and solve my Lagrangian system. I have checked it numerous times over and over and I believe that everything is correct. I have attached a PDF which has the derivation of the system. Please ignore all the forcing functions and...

Hi :) This is a problem from David Tong's Classical Dynamics course, found here: http://www.damtp.cam.ac.uk/user/tong/dynamics.html. In particular it is problem 6ii in the first problem sheet. Firstly we can see the lagrangian is ##L = \frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2) -...

We have Rayleigh's dissipation function, defined as ## \mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right) ## Also we have transformation equations to generalized coordinates as ##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...

Hi Guys I am looking for some guidance with regards to a Lagrangian problem I am trying to solve. Please refer to the attached documents. Please neglect all the forcing functions for the time being. I am currently just trying to simulate the problem using initial conditions only I have...

> A particle is subjected to the potential V (x) = −F x, where F is a constant. The particle travels from x = 0 to x = a in a time interval t0 . Assume the motion of the particle can be expressed in the form ##x(t) = A + B t + C t^2## . Find the values of A, B, and C such that the action is a...

Hi Guys Please refer to the attached file. I have not included any of the derivatives or partial derivatives as it does get messy, I just just included the kinetic and potential energy equations and the holonomic constraint. The holonomic constraint can be considered using Lagrange...

Hello, I have posted a similar thread on this question before, but I'd like to get some help to simplify the answers I've got so far in order to match the solutions provided. If anyone could help me, I would really appreciate it. Since (c) is quite similar to (b), I'll leave here what I've done...

Hello, I'm having some trouble understanding this solution provided in Landau's book on mechanics. I'd like to understand how they arrived at the infinitesimal displacement for the particles m1. I appreciate any kind of help regarding this problem, thank you!

So I've been studying classical mechanics and have come across a small doubt with the solution provided to the problem in question from Landau's book. My question is: why are the coordinates for the particle given as they are in the solution? I imagine it has something to do with the harmonic...

I read somewhere that Morse originally applied his theory to the calculus of variations. I'm wondering, is this application useful in physics and mechanics, like maybe it sheds light on lagrangian mechanics? Does anyone know?

The given lagrangian doesn't seem to correspond to any of the basic systems (like simple/ coupled harmonic oscillators, etc). So I calculated the momentum ##p## which is the partial derivative of ##L## with respect to generalized velocity ##\dot{q}##. Doing so I obtain $$p =...