Lagrangian Definition and 1000 Threads

I'm basically trying to understand the 2-D case of the catenary cable problem. The 1-D case is pretty straightforward, you have a functional of the shape of a cable with a constraint for length and gravity, and you get the explicit function of the shape of a cable. But if you imagine a square...

Hi, I'm in the masters year of a theoretical physics course which begins this September. I'm reading the classical mechanics notes ahead of time, and I came across the idea of holonomic and non-holonomic constraints. I understand that in the case of a holonomic system, you can use the...

Just refer to my profile picture to see what the issue is! :biggrin: Here's the problem: a ball of mass m is connected to a vertical pole with an inextensible, massless string of length r. The angle between the string and the pole is θ. The pole rotates around the z axis with a constant angular...

The equation of motion of a pendulum with a support oscillating horizontally sinusoidally with angular frequency ##\omega## is given by (5.116). (See attached.) I get a different answer by considering the Euler-Lagrange equation in ##x## and then eliminating ##\ddot{x}## in (5.115): Referring...

Do I substitute A_\mu + \partial_\mu \lambda everywhere A_\mu appears, then expand out? Do I substitute a contravariant form of the substitution for A^\mu as well? (If so, do I use a metric to convert it first?) I’m new to Ricci calculus; an explanation as to the meaning of raised and lowered...

The form of the Lagrangian is: L = K - U When cast in terms of generalized coordinates, the kinetic energy (K) can be a function of the rates of generalized coordinates AND the coordinates themselves (velocity and position); a case would be a double pendulum. However, the potential energy (U)...

How does one write a Lagrangian of a coherent state of vector fields (of differing energy levels) in terms of the the individual Lagrangians? I desperately need to know how to know to do this, for a theory of mine to make any progress. Please stick with me, if I didn't make sense just ask...

hi, I would like to put into words that I really wonder how these lagrangian or lagrangian densities are created. For instance in the link at 59.35 suskind says $$\int A^u dx^u$$ is invariant or action integral. How is this possible ?Could you provide me with the proof?

From the Eulerian form of the continuity equation, where x is the Eulerian coordinate: \frac {\partial \rho}{ \partial t } + u \frac {\partial \rho}{\partial x} + \rho \frac { \partial u}{\partial x} = 0 The incremental change in mass is, where m is the Lagrangian coordinate: dm = \rho dx...

Only thing I know about them is that they are alternate mechanical systems to bypass the Newtonian concept of a "force". How do they achieve this? Why haven't they replaced Newtonian mechanics, if they somehow "invalidate" it or make it less accurate, by the Occam's razor principle? Thanks in...

I'm trying to derive the conservaton of energy for electromagnetic fields with currents from the action principle, but I have some trouble understanding how the interaction term in the Lagrangian fits into this. The approach I have seen so far has been to express the Lagrangian density as...

Hi, in gravitational theory the action integral is: I = ∫( − g ( x))^1/2 L ( x) d 4 x, but I do not know why there is a square root -g . Could you give me the proof of this integral? I mean How is this integral constructed? What is the logic of this? Thanks in advance...

Homework Statement Attached. Homework Equations I am assuming the coordinate transformation is \vec{x}' = \vec{x} + \alpha\vec{\gamma} ? Then you have \vec{v}' = \vec{v} + \alpha\frac{d\vec{\gamma}}{dt} And r is the magnitude of the x vector. The Attempt at a Solution Part A. So to get the...

I'm trying to understand how we set up the lagrangian for a charged particle in an electromagnetic field. I know that the lagrangian is given by $$L = \frac{m}{2}\mathbf{\dot{r}}\cdot \mathbf{\dot{r}} -q\phi +q\mathbf{\dot{r}}\cdot \mathbf{A} $$ I can use this to derive the Lorentz force law...

Homework Statement We have a particle of mass m moving in a plane described by the following Lagrangian: \frac{1}{2}m((\dot{x}^2)+(\dot{y}^2)+2(\alpha)(\dot{x})(\dot{y}))-\frac{1}{2}k(x^2+y^2+(\beta)xy) for k>0 is a spring constant and \alpha and \beta are time-independent. Find the normal...

Homework Statement Hi, I'm working on understanding how a time independent point transformation . effects the Lagrangian and to see how what values are co and contra variant.Homework Equations Would these be correct formulations, or have I overlooked something? The Attempt at a Solution and...

Homework Statement We have a mas m attached to a vertical spring of length (l+x) where l is the natural length. Homework Equations Find the Lagrangian and the hamiltonian of the system if it moves like a pendulum The Attempt at a Solution we know that the lagrangian of a system is defined as...

Homework Statement Hi everybody! I would like to discuss with you a problem that I am wondering if I understand it correctly: Find expressions for the cartesian components and for the magnitude of the angular momentum of a particle in cylindrical coordinates ##(r,\varphi,z)##. Homework...

Homework Statement Mass 1 can slide on a vertical rod under the influence of a constant gravitational force and and is connected to the rod via a spring with the spring konstant k and rest length 0. A mass 2 is connected to mass 1 via a rod of length L (forms a 90 degree angel with the first...

Homework Statement In this exercise, we are given a discrete Lagrangian which looks like this: http://imgur.com/TL0P61r. We have to minimize the discrete S with fixed point r_i and r_f and find the the discrete equations of motions. In the second part we should derive a discrete trajectory for...

Homework Statement Hi there! So I have a problem regarding a particle of mass m moving down an inverted cone under the force of gravity. The cone is linear with equation z(r) = r, in cylindrical coordinates (r, theta, z) A. Write down the Lagrangian, include the constraint that the particle...

Ok, I'm reading up on Lagrangian mechanics, and there is a problem that I don't really understand: the double pendulum (in this case, without a gravitational field). So, I want to take it step by step to make sure I understand all of it. We've got a pendulum (1) with a weight mass m=1kg...

Homework Statement Please see attached image :) Homework Equations Euler-Lagrange Equation \frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}}} = 0 L = T - V The Attempt at a Solution a. The potential energy V is the potential energy from the spring and the...

Could someone explain how can one go from $$ \int dx\ \frac{-1}{4}F^{\mu \nu}F_{\mu \nu}$$ where $$F_{\mu \nu} = \partial_{\mu} \phi_{\nu}-\partial_{\nu} \phi_{\mu}$$ to $$\int dx\ \frac{-1}{2}(\partial_{\mu} \phi^{\nu})^2 + \frac{1}{2}(\partial_{\mu} \phi^{\mu})^2 $$ I assume it has...

Homework Statement Hi everybody! I have a crazy problem about Lagrangian on which I've been working on for two days without being able to figure it out. I have a solution, but I think there is a flaw in it. First here is the problem: A line rotates with constant angle velocity ##\omega##...

A student wishes to minimize the time required to gain a given expected average grade, 𝑚, in her end-of-semester examinations. Let $${t}_{i}$$ be the time spent studying subject i$$\in$${1,2}. Suppose that the expected grade functions are $${g}_{1}$$($${t}_{1}$$) = 40+8$$\sqrt{{t}_{i}}$$ and...

Homework Statement Hi everybody! I'm back with another lagrangian problem :) Although I think (or hope) I have made progress on the topic, I always learn a lot by posting here! A pendulum with point-shaped mass ##m_1## hangs on a massless string of length ##l##. The suspension point (also a...

Hello, Consider the the following Lagrangian of the $\phi ^4$ theory: $$\begin{align*} \mathcal{L} = \frac{1}{2} [\partial ^{\mu} \phi \partial _{\mu} \phi - m^2 \phi ^2] - \frac{\lambda}{4!} \phi ^4 \end{align*}$$ Now I'm interested in Feynman diagrams. 1. The second term gives the...

Homework Statement Homework Equations L = T-V The Attempt at a Solution I got a forumla for the lagrangian as

I am stuck at this part page 1, $$\frac{\partial{L}}{\partial{\dot{φ}}}=\mu{r^2}\dot{φ}=const=l------->\dot{φ}=\frac{l}{\mu{r^2}}......(8)$$ Why is this a constant? Isn't r and dφ/dt variables of time? Source: http://www.pha.jhu.edu/~kknizhni/Mechanics/Derivation_of_Planetary_Orbit_Equation.pdf

Hi everyone, I am trying to calculate the equation of motion of a charged particle in the field of a monopole. The magnetic field of a monopole of strength g is given by: \textbf{B} = g \frac{\textbf{r}}{r^3} And the Lagrangian by: \mathcal{L} = \frac{m\dot{\textbf{r}}^2}{2} +...

Homework Statement This was supposed to be an easy question. I have a question here that wants you to describe a yoyo's acceleration (in one dimension) using Lagrangian mechanics. I did and got the right answer. Now I want to use Hamilton's equations of motion but I get a wrong number. Here is...

Hey all. I've been experimenting with Lagrangian mechanics (and numerical simulation of physical systems), and I've come across a problem. By finding the Lagrangian, then using the Euler-Lagrange formula, I can find equations of motion (in generalized angular coordinates with respect to the...

I came across a simple equation in classical mechanics, $$\frac{\partial L}{\partial \dot{q}}=p$$ how to derive that? On one hand, $$L=\frac{1}{2}m\dot{q}^2-V$$ so, $$\frac{\partial L}{\partial \dot{q}}=m\dot{q}=p$$ On the other hand...

The following is taken from page 40 of Matthew Schwartz's "Introduction to Quantum Field Theory." The Lagrangian for the graviton is heuristically ##\mathcal{L}=-\frac{1}{2}h\Box h + \frac{1}{3}\lambda h^{3}+Jh,## where ##h## represents the gravitational potential. We are ignoring spin and...

Homework Statement Verify that the Lagrangian density ##\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi_{a}\partial^{\mu}\phi_{a}-\frac{1}{2}m^{2}\phi_{a}\phi_{a}## for a triplet of real fields ##\phi_{a} (a=1,2,3)## is invariant under the infinitesimal ##SO(3)## rotation by ##\theta##, i.e...

Homework Statement The motion of a complex field ##\psi(x)## is governed by the Lagrangian ##\mathcal{L} = \partial_{\mu}\psi^{*}\partial^{\mu}\psi-m^{2}\psi^{*}\psi-\frac{\lambda}{2}(\psi^{*}\psi)^{2}##. Write down the Euler-Lagrange field equations for this system. Verify that the...

I know how to implement Lagrangian mechanics at a mathematical level and also know that it follows the approach of calculus of variations (i.e. optimisation of functionals, finding their stationary values etc.), however, I'm unsure whether I've grasped the physical intuition behind the...

Homework Statement The energy-momentum tensor ##T^{\mu\nu}## of the Klein-Gordon Lagrangian ##\mathcal{L}_{KG} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}## is given by $$T^{\mu\nu}~=~\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}.$$ Show...

What are Hamiltonian/Lagrangian Mechanics and how are they different from Newtonian? What are the benefits to studying them and at what year do they generally teach you this at a university? What are the maths required for learning them?

Homework Statement 1. Show directly that if ##\varphi(x)## satisfies the Klein-Gordon equation, then ##\varphi(\Lambda^{-1}x)## also satisfies this equation for any Lorentz transformation ##\Lambda##. 2. Show that ##\mathcal{L}_{Maxwell}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## is invariant under...

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...

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...

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?