Lagrangian Definition and 1000 Threads

I recently posted another thread on the General Physics sub forum, but didn't get as much feedback as I was hoping for, regarding this issue. Let's say I have two Lagrangians: $$ \mathcal{L}_1 = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\mu A^\mu)^2 $$ $$...

Hey everyone, I wasn't really sure where to post this, since it's kind of classical, kind of relativistic and kind of quantum field theoretical, but essentially mathematical. I'm reading and watching the lectures on Quantum Field Theory by Cambridge's David Tong (which you can find here), and...

Question 1 When I take the derivatives of the Lagrangian, specifically of the form: \frac{\partial L}{ \partial q} I often find myself saying this: \frac{\partial \dot{q}}{ \partial q}=0 But why is it true? And is it always true?

Hello, I understand the classical Lagrangian which follows the Principle of Least Action(A) A=∫L dt But what is Lagrangian density? Is it a new concept? A=∫Lagrangian density dx^4 Here 4 is the four vector? One time-like and 3 space-like co-ordinates? QFT uses Lagrangian to...

Homework Statement A Particle of mass m is threaded on a frictionless rod that rotates at a fixed angular frequency Ω about a vertical axis. A spring with rest length Xo and spring constant k has one of it's ends attached to the mass and the other to the axis of rotation. Let x be the length...

Homework Statement While doing a problem I have found the Lagrangian to be L=\frac{1}{2}m \dot{r}^2 \left( 1 + 4a^2r^2 \right) + \frac{1}{2}mr^2 \dot{\theta}^2 -mgar^2. I have also shown that the angular momentum l is constant and is equal to l=mr^2 \dot{\theta}. I want to calculate the energy...

Homework Statement Consider a bead of mass m moving on a spoke of a rotating bicycle wheel. If there are no forces other than the constraint forces, then find the Lagrangian and the equation of motion in generalised coordinates. What is the possible solution of this motion?Homework Equations...

Homework Statement [/b] The attempt at a solution[/b] I have done the first bit but don't know how to show that phi(r,t) is a solution to the equation of motion.

Hello fellow PF members I was wondering how one would go about finding the lagrangian of a problem like the following: A particle is constrained to move along the a path defined by y = sin(x). Would you simply do this: x = x y = sin(x) x'^2 = x'^2 y'^2 = x'^2 (cos(x))^2...

Hello, I have a very basic question: Degrees of freedom for a particle describes the formal state of a physical system. Like a particle in 3 dimension space has 3 co-ordinates and if it moves in 3 velocity components, then it has 6 degrees of freedom. Lagrangian also measures this, right?

Homework Statement I'm asked to solve the typical intro level box on an inclined plane problem but I need to do it using the lagrangian. My difficulty with it is that the axis I am required to use are not the typical axes used when solving this using Newtonian mechanics. Instead of the...

This is probably a minor point, but I have seen in some QFT texts the Euler-Lagrange equation for a scalar field, \partial_{\mu} \left(\frac{\delta \cal{L}}{\delta (\partial_{\mu}\phi)}\right) - \frac{\delta \cal L}{\delta \phi }=0 i.e. \cal L is treated like a functional (seen from the...

Homework Statement I don't know why I'm having trouble here, but I want to show that, if we let t = t(\theta) and q(t(\theta)) = q(\theta) so that both are now dependent coordinates on the parameter \theta , then L_{\theta}(q,q',t,t',\theta) = t'L(q,q'/t',t) where t' =...

Homework Statement Consider the following Lagrangian in Cartesian coordinates: L(x, y, x', y') = 12 (x^ 2 + y^2) -sqrt(x^2 + y^2) (a) Write the Lagrange equations of motion, and show that x = cos(t); y =sin(t) is a solution. (b) Changing from Cartesian to polar coordinates, x = r...

In Lagrangian mechanics, the Euler-Lagrange equations take the form $$\frac{\partial L}{\partial x} = \frac{\mathbb{d}}{\mathbb{d}t}\frac{\partial L}{\partial \dot{x}}$$ From this, we can define the left side of the equation as force, and by carrying out the actual derivative, we get $$F =...

a particle of mass m is attracted to a center force with the force of magnitude k/r^2. use plan polar coordinates and find the Lagranian equation of motion. so i thought for the kinetic energy it would be.. K=\frac{1}{2}m(r2\dot{θ}2) since v2 = r2\dot{θ}2 but no.. the kinetic energy...

hi, silly question but would someone please show me how \frac{∂L}{∂q}=\dot{p}? L being the lagrangian, p being the momentum, and q being the general coordinate.

Hi Can somebody please explain fundamentally what is the difference between these two methods of modelling contact interfaces? I would prefer a more qualitative explanation (physics concept based ) rather than a more mathematical description.

Homework Statement A bead of mass m slides under gravity on a smooth rod of length l which is inclined at a constant angle ##\alpha## to the downward vertical and made to rotate at angular velocity ##\omega## about a vertical axis. The displacement of the bead along the rod is r(t)...

Hi all, I've been playing around with spin 1/2 Lagrangians, and found the very interesting Fierz identities. In particular for the S x S product, (\bar{\chi}\psi)(\bar{\psi}\chi)=\frac{1}{4}(\bar{\chi} \chi)(\bar{\psi} \psi)+\frac{1}{4}(\bar{\chi}\gamma^{\mu}\chi)(\bar{\psi}\gamma_{\mu}...

Dear All, To give a background about myself in Classical Mechanics, I know to solve problems using Newton's laws, momentum principle, etc. I din't have a exposure to Lagrangian and Hamiltonian until recently. So I tried to read about it and I found that I was pretty weak in coordinate...

Homework Statement This is the problem: http://i.imgur.com/OJyzfhz.png?1 Homework Equations $$\frac{dL}{dq_i}-\frac{d}{dt}\frac{dL}{d\dot{q_i}}=0$$ $$ L=\frac{1}{2}mv^2-U(potential energy)$$ The Attempt at a Solution This is my attempt at question A): http://i.imgur.com/IeJVGm3.jpgDoes...

Homework Statement I need some help understanding a derivation in a textbook. It involves the Lagrangian in generalized coordinates. Homework Equations The text states that generalized coordinates {q_1, ..., q_3N} are related to original Cartesian coordinates q_\alpha = f_\alpha(\mathbf r_1...

Homework Statement Consider the Lagrange Polynomial approximation p(x) =\sum_{k=0}^n f(x_k)L_k(x) where L_k(x)=\prod_{i=0,i\neq k}^n \frac{x-x_i}{x_k-x_i} Let \psi(x)=\prod_{i=0}^n x-x_i. Show that p(x)=\psi(x) \sum_{k=0}^n\frac{f(x_k)}{(x-x_k)\psi^\prime(x)} Homework Equations None...

I tried to verify that the SYM lagrangian is invariant under SUSY transformation, but it turned out there is a term that doesn't vanish. The SYM lagrangian is: \mathscr{L}_{SYM}=-\frac{1}{4}F^{a\mu\nu}F^a_{\mu\nu}+i\lambda^{\dagger a}\bar{\sigma}^\mu D_\mu \lambda^a+\frac{1}{2}D^a D^a the...

Hello, Does anyone know what "induced gravity" is and where can one read (something not too technical) about it? I am trying to understand how a system with zero Lagrangian has something to do with gravity. Could someone explain perhaps? Thanks

I am learning Hamiltonian and Lagrangian mechanics and looking for a book that starts with Newtonian mechanics and then onto Lagrangian & Hamiltonian mechanics. It should have some historical context explaining the need to change the approaches for solving equation of motions. Also it should...

Hello! I've read thousand of explanations about how q and q-dot are considered independent in the Lagrangian treatment of mechanics but I just can't get it. I would really appreciate if someone could explain how is this so and (I've seen something about an a-priori independence but I couldn't...

In a two-body solution ( Kepler Problem) how do you find the Lagrangian, L, if the position vector is: x = (v_x * t + x_0, v_y * t + y_0, v_z * t + z_0) Action from Wikipedia is: S = \int_{t_1}^{t_2} L * dt

How are the Hamiltonian and Lagrangian different as far as preserving symmetries of a theory? Peskin and Schroeder write that the path integral formalism is nice because since it's based on the action and Lagrangian it explicitly preserves all the symmetries, but I'm wondering how/why the...

A few doubts regarding lagrnagian method to deal with motion of particles: 1) It seems like a heuristic method of solving for motion of a particle. In Newtonian mechanics, you carefully consider all the forces and find out the particle's motion. In this, based on intuition you guess the...

Homework Statement I am studying inflation theory for a scalar field \phi in curved spacetime. I want to obtain Euler-Lagrange equations for the action: I\left[\phi\right] = \int \left[\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi + V\left(\phi\right) \right]\sqrt{-g} d^4x Homework...

Homework Statement Consider the following Lagrangian of a relativistic particle moving in a D-dim space and interacting with a central potential field. $$L=-mc^2 \sqrt{1-\frac{v^2}{c^2}} - \frac{\alpha}{r}\exp^{-\beta r}$$ ... Find the velocity v of the particle as a function of p...

Another Exercise... Ohanian Exercise 6 The Euler-Lagrange Equation given: ##\frac{\partial}{\partial x^\mu} \frac{\partial \mathcal{L}}{\partial (\partial_\mu h^{\alpha\beta})} - \frac{\partial \mathcal{L}}{\partial h^{\alpha\beta}} = 0## ##\frac{\partial \mathcal{L}}{\partial...

I was working on an exercise in Ohanian's book. [Appendix A3, page 484, Exercise 5] I guess he means charge conservation, but wrote ##j^\nu = 0##. The Lagrangian was given by ##\mathcal{L}_{em} = -\frac{1}{16\pi} \left( A_{\mu ,\nu} - A_{\nu ,\mu} \right) \left( A^{\mu ,\nu} - A^{\nu...

I am watching Susskind's derivation of Newton's F=ma from the Euler-Lagrange equations (53 minutes in) here for which he uses the Lagrangian of kinetic energy minus potential energy. I have seen this done elsewhere as well. As far as I can tell, and please correct me if I'm wrong, the only...

I've been reading a lot about path integrals lately, and I've found it fascinating to see at the quantum level how the extremal values of the lagrangian are basically the only ones that contribute when the action is large and therefore we get the classical path. Something that continues to...

Hello, I have the functional J = ∫ L(ψ, r, r') dψ, where r'=dr/dψ. L is written in polar coordinates (r,ψ). Now I want to constrain the motion to take place on the polar curve r = r(ψ). Can I write the constrained lagrangian as Lc=L(ψ, r, r') - λ(r - r(ψ)) and then solve the...

Homework Statement (context: I'm studying for a test, and this is a question from a past exam paper.) "A wedge of mass M with angle \phi is free to slide on a frictionless horizontal table. A solid ball of radius a and mass m is placed on the slope of the wedge. The contact between the ball...

Hi! I tried to compute an ideal barbell-shaped object's dynamics, but my results were wrong. My Langrangian is: ## L = \frac{m}{2} ( \dot{x_1}^2 + \dot{x_2}^2 + \dot{y_1}^2 + \dot{y_2}^2 ) - U( x_1 , y_1 ) - U ( x_2 , y_2 ) ## And the constraint is: ## f = ( x_1 - x_2 )^2 + ( y_1 -...

I'm trying to figure out when is \frac{\delta L}{\delta p}=\dot{q} . From L=p\frac{\delta H}{\delta p}-H I get that \frac{\delta L}{\delta p}=p\frac{\delta^2 H}{\delta p^2}=p \frac{d}{dp}\dot{q} . For this to equal just \dot{q} , it must obey the dfe \frac{d}{d ln p}\dot{q}=\dot{q}...

Let say I want to study electron-proton scattering (without considering proton's quarks, i.e. no QCD), which is the Lagrangian? I've seen two different answers to this question :confused: First one: L=\bar{ψ}e(i∂-me)ψe+\bar{ψ}p(i∂-mp)ψp-\frac{1}{4}Fμ\nuFμ\nu-e\bar{ψ}eγμψeAμ+e\bar{ψ}pγμψpAμ...

Is the lagrangian used in QFt because its the only information of motion we can obtain about a system at relativistic speeds? Does the lagrangian reflect the conservation of energy ? Is this why the lagrangian must be invariant... Meaning that it must be constant... Meaning that energy is...

Homework Statement Thank you for answering my question about setting the Euler-Langrangian expression to zero separately for each coordinate (ehild ans.=yes). Now my question is: Can the Lagrangian be derived, or is it the expression, when inserted into the Euler-Lagrange equation(s), that...

Homework Statement Consider the setup shown in the gure below. The cart of mass M moves along the (horizontal) x axis. A second mass m is suspended at the end of a rigid, massless rod of length L. The rod is attached to the cart at point A, and is free to pivot about A in the x-y plane...

So I was going through an ODEs textbook and in a section discussing physical problems, decided that it would be interesting to come up with the equations of motion using Lagrangian mechanics for the examples they posted. For the first example, a falling rock, this easily worked. The second...

No this is not homework. http://imgur.com/zAZxmuC http://imgur.com/zAZxmuC Ok i am struggling to even start this question. I see it has a constraint so i would be tempted to use Lagrangian but from there i don't see how px and qy fit into it? Some assistance on the tools needed to...

Homework Statement Write down the Lagrangian for a one-dimensional particle moving along the x-axis and subject to a force: F=-kx (with k positive). Find the Lagrange equation of motion and solve it. Homework Equations Lagrange: L=T-U (kinetic energy - potential energy) The Attempt...

In reading Ryder's book on quantum field theory he advocates reading off the Feynman rules directly from the Lagrangian in the path integral quantization method. I can sort of do this in phi-four theory, but it is not obvious in for example Yang-Mills theory, so I wondered if someone could...

Hey, I'm having trouble with part (d) of the question displayed below: I reckon I'm doing the θ Euler-Lagrange equation wrong, I get : \frac{\mathrm{d} }{\mathrm{d} t}(\frac{\partial L}{\partial \dot{\theta}})-\frac{\partial L}{\partial \theta}=\frac{\mathrm{d} }{\mathrm{d}...