Lagrangian Definition and 1000 Threads

I'm trying to derive the Geodesic equation, \ddot{x}^{α} + {Γ}^{α}_{βγ} \dot{x}^{β} \dot{x}^{γ} = 0. However, when I take the Lagrangian to be {L} = {g}_{γβ} \dot{x}^{γ} \dot{x}^{β}, and I'm taking \frac{\partial {L}}{\partial \dot{x}^{α}}, I don't understand why the partial derivative of...

Hi guys, So textbooks have it that: "Two Lagrangians differing by a total time-derivative of a function of the coordinates are equivalent". I have no idea what that means or how to use it; so I don't know which terms I can drop from Lagrangians, which is a bit of a problem. For example...

Homework Statement Consider a particle of mass m and electric charge e subject to a uniform electromagnetic field (E(x,t),B(x,t)). We must remember that the force they exert is given by: F = cE(x,t) + ex' \times B(x,t) A principle of action that represents such particle subject to the...

Homework Statement A mouse of mass m runs around the inner circumference of a vertical wheel which is free to rotate about the centre. The wheel has mass M and moment of inertia I. Let θ be the angle that the radius vector makes to the mouse from the downward vertical at time t. Write down the...

Homework Statement Consider the Atwood’s pulley shown below. The masses are 4m, 3m, and m. Let x and y be the directed distances from the centers of the fixed (i.e. inertial) top pulleys for the left and right masses as indicated. http://imgur.com/VXEygxt a) Write down the Lagrangian...

Hi, I hope I put this in the right place! I'm having trouble with some of the calculus in moving from the Klein-Gordin Lagrangian density to the equations of motion. The density is: L = \frac{1}{2}\left[ (\partial_μ\phi)(\partial^\mu \phi) - m^2\phi ^2 \right] Now, to apply the...

Hi there, I'm having some problems trying to write down the Lagrangian of the following system: A uniform flexible chain of mass M and length L is hung under gravity on a frictional pulley of radius a and moment of inertia I whose axle is fixed at a point above the ground. Write down the...

Let \gamma^{\rho} \in M_{4}(\mathbb{R}) be the Majorana representation of the Dirac algebra (in spacetime signature \eta_{00} = -1), and consider the Majorana Lagrangian \mathcal{L} = \mathrm{i} \theta^{\mathrm{T}} \gamma^{0} (\gamma^{\rho} \partial_{\rho} - m) \theta, where \theta is a...

I've been tasked with showing that a Lagrangian under a set of transformations changes by a time derivative. All has gone well, except I'm left with two remaining terms, that I am completely confident, aren't there by mistake (as the 16 terms that should be expected have all popped out with the...

Is: \mathcal L=-\frac{m}{2} u^\alpha u_\alpha a correct Lagrangian for SR (assuming the parameter is proper time rather than world time)? It leads to the correct EOM when plugged into the Euler-Lagrange equation, m\frac{du^\alpha}{ds}=0 Or is this the correct Lagrangian: \mathcal L=-m...

Homework Statement Consider the following Lagrangian: L = \frac{m}{2}(x'^2+y'^2+z'^2) + \frac{q}{2}(xy'-yx') Where q denotes a charged particle. a) Find the equations of motion b) Find the solution for z c) Find the solution in the x-y plane, and prove that it corresponds to an oscillatory...

what is Lagrangian ? the Hamiltonian H = T + V represents the total energy of the system, and Lagrangian L = T-V, but what does it actually represents and what is the exact meaning of Lagrangian ? it represents excess energy or energy loss or some thing else ?

Write down the Lagrangian $\mathcal{L}(x_1,x_2,\dot{x}_1,\dot{x}_2)$ for two particles of equal masses, $m_1 = m_2 = m$, confined to the $x$ axis and connected by a spring with potential energy $U = \frac{1}{2}kx^2$. [Here $x$ is the extension of the spring, $x = (x_1 - x_2 - \ell)$ where $\ell$...

Homework Statement Find Scl for a particle under constant force f, that is: L = (m/2)v2 + fx Homework Equations S = ∫Ldt d(∂L/∂q^{.})/dt = ∂L/∂q The Attempt at a Solution Apologies if this belongs in the Introductory Physics section. Apologies for terrible formatting...

Homework Statement Hi guys. http://img189.imageshack.us/img189/5123/systemn.jpg The image shows the situation. A pointlike particle of mass m is free to move without friction along a horizontal line. It is connected to a spring of constant k, which is connected to the origin O. A...

Homework Statement A particle of mass m moves on the surface of a paraboloidal bowl with position given by r=rcosθi+rsinθj+\frac{r^{2}}{a}k with a>0 constant. The particle is subject to a gravitational force F=-mgk but no other external forces. Show that a suitable Lagrangian for the system is...

Homework Statement So we have started Lagrangian Mechanics in my class, and I really don't understand it at all. My teacher keeps doing the math on the board, but he hasn't really said what a Lagrangian is, and what an Action is. I really am lost from the start with these problems. Any help...

In the attached snip, the last few steps of the lagrangian equation is shown. I don't understand how the \frac{\delta V}{\delta\dot{q_j}}= 0. As an example let me take gravitational force. With change in velocity ( along the downwards direction obviously), there sure is a change in gravitational...

I understand how to use Hamiltonian mechanics, but I never understood why you construct the Hamilitonian by first constructing the Langrangian, and then performing a Legendre transform on it. Why can't you just construct the Hamiltonian directly? Does it have to do with the generalized...

Homework Statement Show that the free particle lagrangian is invariant to rotations in $$\Re^{3}$$, but I assume this means invariant up to a gauge term. $$L=m/2 [\dot{R^{2}} + R^{2}\dot{θ^{2}} +R^{2}Sin^{2}(θ)\dot{\phi^{2}}$$ Homework Equations I consider an aribtrary infinitesimal...

Homework Statement Homework Equations The Attempt at a Solution I don't understand where the third term from the first equation of 5.192 come about.. as clearly L doesn't depend on x at all, so ∂L/∂x should be zero.

Hello.I recently discovered the Lagrangian approach on classical mechanics ptovlems, such as a spring pendullum, or even on particle physics problems, and i think it s a really smart way of getting results. I'd like to approach this method deeper and so my questions are the following: 1.What...

I've been doing some self-study in Peskin and Schroeder and been struggling a bit in Part III. Right now, I am stuck on the last two terms in 16.6 (Lagrangian for Yang-Mills). Presumably these come from (-1/4) (F^{a}_{\mu\nu})^2, but I am getting stuck on getting the indices to work out...

Homework Statement The Lagrange method does work for some velocity dependent Lagrangian. A very important case is a charged particle moving in a magnetic field. The magnetic field can be represented as a "curl" of a vector potential ∇B = ∇xA . A uniform magnetic field B0 corresponds to a vector...

I wanted to solve the problem of a cone rotating on its side over a table, around an axis that pass through it's apex, like in the figure. What I want to find is the angular speed ω, the spin of the solid, such that the cone "stands" over it's apex. I don't know how to set the condition...

Here and there I come across the following formula for the Lagrangian density of a real scalar field, but not a deriviation. \mathcal{L} = \frac {1}{2} [ \dot \phi ^2 - ( \nabla \phi ) ^2 - (m \phi )^2 ] Could someone show me where this comes from? The m squared term in particular...

Homework Statement Carry out the integration ψ = ∫[M(dr/r2)] / √(2m(E-U(r)) - (M2/r2)) E = energy, U = potential, M = angular momentum using the substitution: u = 1/r for U = -α/r Homework Equations The Attempt at a Solution This is as far as I've gotten: -∫ (Mdu) /...

Homework Statement 2 masses, m_{1} and m_{2} are fixed at the endpoints of a rigid rod of length l. mass m_{1} is attached to a horizontal bar so that it may move in the x direction freely, but not in the y direction. let θ be the angle the rod makes with the vertical, what is the...

Homework Statement The problem asked us to show that the Euler-Lagrange's equations are invariant under a point transformation q_{i}=q_{i}(s_{1},...,s_{n},t), i=1...n. Give a physical interpretation. Homework Equations \frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})=\frac{\partial...

Homework Statement We are given L = 1/2mv2 - mgz. a) Find the equations of motion. b) Take x(0) [vector] = 0; v(0) [vector] = v0 [vector] ; v0z > 0 and find x(τ) [vector] and v(τ) [vector], such that z(τ) = 0;  τ≠0. c) Find S. Homework Equations Euler-Lagrange equation and...

Homework Statement Show that the Lagrangian \mathcal{L}=\frac{m}{2}\vec{\dot{r}}^2 \, \frac{1}{(1+g \vec{r}^2)^2} is invariant under the Transformation \vec{r} \rightarrow \tilde{r}=\vec{r}+\vec{a}(1-g\vec{r}^2)+2g\vec{r}(\vec{r} \cdot \vec{a}) where b is a constant and \vec{a} are...

Homework Statement I am given a Lagrangian, which, per assignment text, describes a single degree of freedom: L= \frac{I}{2}(\dot{q}+\omega)^2-kq^2 I need to find the Hamiltonian. Now, what I am wondering, when performing the Legrende transform...

Homework Statement The pendulum of a grandfather clock consists of a thin rod of length L (and negligible mass) attached at its upper end to a fixed point, and attached at its lower end to a point on the edge of a uniform disk of radius R, mass M, and negligible thickness. The disk is free...

Hello, I run across the following Lagrangian, $$\mathcal{L} = m \dot{x}\dot{y} + \frac{1}{2} \gamma (x \dot{y} - \dot{x} y) $$ I can see how a variation with respect to $$ x, y $$ yields the (viscous) equations of motions $$ \ddot{x} + \dot{x} = 0 \quad, \quad \ddot{y} - \dot{y} = 0 $$...

Lagrangian I have is little bit massy so I don't write in here. Like in ψψ(dagger) , or ψ∅ -> ψ∅, How can I calculate the differential cross section or total, or amplitudes?

What I know: L = T - V V = e^(x1-x2) + e^(x2-x3) for n = 3 and a mass = 1 What I believe: T = .5Ʃ (x'_i)^2 from 1 to n So let's say you have three bodys that can just be considered pints masses of 1 and on the same line: x1 x2 x3 They have an...

I'm a little confused about why the Lagrangian is Lorentz invariant and the Hamiltonian is not. I keep reading that the Lagrangian is "obviously" Lorentz invariant because it's a scalar, but isn't the Hamiltonian a scalar also? I've been thinking this issue must be somewhat more complex...

Hi. Let's say we have a complex scalar field \varphi and we separate it into the real and the imaginary parts: \varphi = (\varphi1 + i\varphi2) It's Lagrangian density L is given by: L = L(\varphi1) + L(\varphi1) Can you tell the argument behind the idea that in summing the densities of...

Homework Statement A particle of mass m slides frictionlessly down a smooth track defined by the function y=f(x)=((-x^3)/a^2) where a is a constant with units of length. The particle is also in a uniform gravitational field. Set the lagrangian up in cartesian coordinates x and yHomework...

I'm trying to express the classical gravitation Einstein-Hilbert lagrangian into some nice way, and I'm having a problem. It is well known that the Einstein-Hilbert action is the following (I don't write the constant in front of the integral, to simplify things) : S_{EH} = \int R \, \sqrt{-g}...

Homework Statement http://fotozrzut.pl/zdjecia/ad3bbdd9f6.jpg Mases are as stated on the picture, I is the moment of inertia of the pulley, angle marked is ω and a is a radius. Homework Equations L=T-V The Attempt at a Solution I try to formulate a lagrangian for this system...

Hello, Where can I find a good explanation (book) of the derivation via Noether's theorem of the three momentum and angular momentum operators of the usual maxwell lagrangian ? Thank you!

Homework Statement A cylinder on a inclined plane is rolling without slipping. Inclined plane is connected to wall with a spring and cylinder is connected to wall with a spring too. All frictions will be neglected, and all the given data has shown on the image below. As seen above, k2...

Homework Statement A thin rod of length 2b is suspended by 2 light strings both attached to the ceiling. Using x, y1, y2 as your generalized coordinates right down the lagrangian of the system. Where x is the longitudinal displacement of the rod and y1 and y2 are the horizontal displacements...

Homework Statement I included the problem as an attachment. My difficulty lies within understanding how to account for the applied torque within the Lagrangian and subsequent Euler-Lagrange equations, which is what I want to use to determine the equations of motion of the bead on the hoop...

A point of mass m, affected by gravity, is obliged to be in a vertical plan on a parabola with equation z = a.r^2 a is a constant and r is the distance between the point of mass m and the OZ vertical axis. Write the Lagrange equations in the cases that the plan of the parabola is : a) is...

Homework Statement Write the lagrangian equations for: A simple pendulum whose suspension point oscillates horizontally in its plan according to the law x = a.cos(ωt)My problem is trying to know which are the generalized coordinates. i considered :x (θ) = a.cos(ωt) + l.sinθ y (θ) = l.cos...

In Newton's problem,and other central force problems in Classical Mechanics, you can get with decreasing the center of mass movement to the lagrangian: L=1/2m(r' ^2+r^2 \varphi'^2)-V(r) because \varphi is cyclic, you can write: \frac{d}{dt}(mr^2 \varphi')=0 or, defining the angular...

I just read some basic concepts on General Relativity, and this idea pops up: I know we should use variations of metrics for gravitational field in the Lagrangian. But considering the resemblance of gravitational field(weak-field) to electromagnetic field, can we construct a 4-potential similar...

Probably, the essence of quantum theory (QT) is principle of uncertainty (HUP). The essence of QT is also the fact that Fourier transformation of wave function in phase(?) space gives wave function in momentum space. If one wave function is Gaussian (and so both ones) this gives HUP. Very...