Lagrangian Definition and 1000 Threads

Okay, so I've recently been reading through C. Pozrikdis' Introduction to Theoretical and Computational Fluid Dynamics, and came across an interesting exercise: "Discuss whether it is possible to label all point particles within a finite three-dimensional parcel using a single scalar variable...

We all know the proof, from Newtonian mechanics, that the motion of the center of mass of a system of particles can be found by treating the center of mass as a particle with all the external forces acting on it. I want to prove the same think, but within the framework of Lagrangian mechanics...

Homework Statement If (V,\omega) is a symplectic vector space and Y is a linear subspace with \dim Y = \frac12 \dim V show that Y is Lagrangian; that is, show that Y = Y^\omega where Y^\omega is the symplectic complement. The Attempt at a Solution This is driving me crazy since I...

I saw a proof recently that demonstrated that for F= ∫L(x, y(x), y'(x))dx, if y(x) is such that F is minimal (no other y(x) could produce a smaller F), then dL/dy - d/dx(dL/(dy/dx)) = 0. I understood the proof, and I was able to see that with a basic definition of Energy = (m/2)(dx/dt)^2...

Suppose we have a system with scleronomic constraints. Is the condition that ∂V/∂qj=0 for generalized coordinates qj a necessary condition for equilibrium? A sufficient condition? I managed to "prove" that the above condition is necessary and sufficient for any type of holonomic constaint...

Hi, guys, Why do we assume Lagrangian Density only depend on field variables and their first derivative? Currently, I am reading Ashok Das's Lectures on Quantum Field Theory. He says (when he is talking about Klein-Gordon Field Theory): "In general, of course, a Lagrangian density...

Hi, I have a quick question: Let's say I have a Lagrangian \mathcal{L} . From Hamilton's principle I find a governing equation for my system, call it N\phi=0 where N is some operator and \phi represents the dependent variable of the system. If \mathcal{L} has a particular symmetry, how...

Lagrangian sought for given conservation "law" Reading about the Lagrangian and conservation laws, I was wondering if, given the conservation law |\dot{x(t)}| = const where x is an n-dimensional vector, we can find the Lagrangian L(t, x(t), \dot{x(t)}) that produces this conservation law...

i'm just ready to start QM and I looked at the text and I turned to Shro eq to see if I could understand it and they mentioned Hamiltonian operator. It looked like the book assumed knowledge of H and L mechanics. Do I need to know this stuff? I wasn't told by others that I needed this. I was...

Homework Statement Homework Equations Euler-Lagrange equations of motion The Attempt at a Solution Part a): Particle must move on surface (one constraint). Number of generalised coordinates = 3N - K where N = number of particles and K = constraints. Therefore 2 generalised coordinates are...

I have two somewhat related questions. First, why would we care about the Lagrangian L = T - V (or K - U)? I understand with the Hamiltonian H = T +V, the total energy is conserved. But with the Lagrangian, what does it actually mean? Mathematically, it only inverts the potential energy...

If we considered some coordinate as being a generalized one, like when we are considering spherical coordinates-let us suppose that I chose theta and phi as generalized coordinates. After deriving the Lagrangian equation it turned out that the equation doesn't depend on phi. Which means that...

Homework Statement I want to derive the centrifugal and Coriolis forces with the Lagrangian for rotating space. The speed of an object for an outside observer is dr/dt + w x r, where r are the moving coordinates. So m/2(dr/dt + w x r)^2 is the Lagrangian. The Attempt at a Solution...

Homework Statement Derive the Non-Linear Schrödinger from calculus of variationsHomework Equations Lagrangian Density \mathcal{L} = \text{Im}(u^*\partial_t u)+|\partial_x u|^2 -1/2|u|^4 The functional to be extreme: J = \int\limits_{t_1}^{t_2}\int\limits_{-\infty}^{\infty}\...

Hi, I'm trying to work through something and it should be quite simple but somehow I've gotten a bit confused. I've worked through the Euler Lagrange equations for the lagrangian: \begin{align*} \mathcal{L}_{0} &= -\frac{1}{4}(\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu}) \\ &=...

Hi, Qualitatively: I am trying to decipher a method I've found in the literature, namely Whitham's method. It is a technique used to averaged out "fast variations" in the Lagrangian to then deduce governing equations for the system. I am trying to quantitatively deduce how accurate Whitham's...

Hello. I have a question about when the Lagrangian can be used. In the textbook we are using it is shown that for constraint forces which do no work, the Lagrangian is of the form T-U. In this question though, it seems like rod would provide a normal force that is in the same direction as the...

Homework Statement Not really a homework question: just a general query. About half the time when working examples, I see a (1/2) thrown into a Lagrangian for use with Euler-Lagrange, but I can't seem to find out why. Is the (1/2) present (or not?) only for the case of a non-symmetric metric...

Homework Statement Consider a pendulum of mass m and length b in the gravitational field whose point of attachment moves horizontally x_0=A(t) where A(t) is a function of time. a) Find the Lagrangian equation of motion. b) Give the equation of motion in the case of small oscillations. What...

Homework Statement A Lagrangian for a particular physical system can be written as, L^{\prime }=\frac{m}{2}(a\dot{x}^{2}+2b\dot{x}\dot{y}+c\dot{y}^{2})-\frac{K% }{2}(ax^{2}+2bxy+cy^{2}) where a and b are arbitrary constants but subject to the condition that b2 -ac≠0.What are the...

Homework Statement A particle of mass m moves in 1D such that it has Lagrangian, L=\frac{m^{2}\dot{x}^{4}}{12}+m\dot{x}^{2}V(x)-V_{2}(x) where V is some differentiable function of x.Find equation of motion and describe the nature of motion based on the equation. The Attempt at a...

Disregard. I done figured it out.Homework Statement Find equations of motion for the metric: ds^2 = dr^2 + r^2 d\phi^2Homework Equations L = g_{ab} \dot{x}^a \dot{x}^b \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}^a} \right) = \frac{\partial L}{\partial x^a} The Attempt at a...

Homework Statement Consider a solid sphere of radius r to be placed at the bottom of a spherical bowl radius R, after the ball is given a push it oscillates about the bottom. By using the Lagrangian approach find the period of oscillation.Homework Equations The Attempt at a Solution Ok so this...

First, to make sure i have this right, lagrangian mechanics, when describing a dynamic system, is the time derivative of the positional partial derivatives (position and velocity) of the lagrangian of the system, which is the difference between the kinetic and potential energy of the system...

Homework Statement Consider a particle of mass m moving in a vertical x-y plane along a curve y = a*cos((2\pix)/\lambda). Consider its motion in terms of two coordinates x and y. Find Lagrange's equations of motion with undetermined multipliers. Homework Equations y = a*cos((2\pix)/\lambda)...

Homework Statement (i) A particle of mass m moves in the x - y plane. Its coordinates are x(t) and y(t). What is the kinetic energy of this particle? (ii) The potential energy of this particle is V (y). The actual form of V will remain unspecied, except that it depends only on the y...

Homework Statement I want to be able to plot a trajectory wrt time of a ball that rolls without slip on a curved surface. Known variables: -radius/mass/moment of inertia of the ball. -formula for the curvature of the path (quadratic) -formula relating path length and corresponding height...

Homework Statement Use the E-L equation to calculate the period of oscillation of a simple pendulum of length l and bob mass m in the small angle approximation. Assume now that the pendulum support is accelerated in the vertical direction at a rate a, find the period of oscillation. For what...

QUESTION: In the circular restricted 3-body problem, if we consider motion confined to the x-y plane and adopt units such that G(m1 + m2) = 1 [m1 and m2 are the masses of the two heavy bodies], the semimajor axis of the relative orbit of the massive bodies = 1, and n = 1 (n is mean motion...

I've been watching Sidney Coleman's QFT lectures (http://www.physics.harvard.edu/about/Phys253.html, with notes at http://arxiv.org/pdf/1110.5013.pdf), and I'm now on to the spin 1/2 part of the course. We've gone through all the mechanics of constructing irreducible representations D^{(s1,s2)}...

I attached a file that shows the free EM action integral and how it can be rewritten. I would like to know how to go from the first line to the second. I have to integrate by parts somehow, and I know surface terms get thrown out, but I do not know how the indices of the gauge fields should be...

I've heard it said that the Lagrangian of a free particle cannot possibly be a function of any position coordinate, or individual velocity component, but it is a function of the total magnitude of velocity. Why is this the case? I'd be grateful for any pointers in the direction of either a...

Homework Statement Essentially the problem that I am trying to solve is the same as in this topic except that it is for 3 springs and 3 masses https://www.physicsforums.com/showthread.php?t=299905 Homework Equations I have found similar equations as in the topic but I face a problem in...

Homework Statement a) the lagrangian for a system of one degree of freedom can be written as. L= (m/2) (dq/dt)2sin2(wt) +q(dq/dt)sin(2wt) +(qw)2 what is the hamiltonian? is it conserved? b) introduce a new coordinate defined by Q = qsin(wt) find the lagrangian and hamiltonian...

Homework Statement A hollow semi-cylinder of negligible thickness, radius a and mass M can rotate without slipping over the horizontal plane z=0, with its axis parallel to the x-axis. On its inside a particle of mass m slides without friction, constrained to move in the y-z plane. This...

Hi all! Long story short, my QFT class recently covered gauge equivalence in QED, and this discussion got me thinking about more general gauge theory. I spent last weak reading about nonabelian symmetries (in the context of electroweak theory), and I like to think I now have a grasp on the...

Homework Statement I apologize if this is not the right place to put this. If it is not please redirect me for future reference. 11. Consider a uniform thin disk that rolls without slipping on a horizontal plane. A horizontal force is applied to the center of the disk and in a direction...

Homework Statement This isn't strictly a homework question as I've already graduated and now work as a web developer. However, I'm attempting to recover my ability to do physics (it's been a few months now) by working my way through the problems in Analytical Mechanics (Hand and Finch) in my...

Homework Statement A bead of mass m slides on a long straight wire which makes an angle alpha with, and rotates with constant angular velocity omega about, the upward vertical. Gravity acts vertically downard. a)Choose an appropriate generalized coordinate and find the Lagrangian. b)Write...

In Landau and Lifgarbagez, Vol. 1, it says "the component of angular momentum along any axis (say the z-axis) can be found by differentiation of the Lagrangian: M_{z} = \Sigma_{a} \partial L/\partial \dot{\varphi_{a}} where \varphi is the angle of rotation about the z axis. This is evident...

Homework Statement Greetings! This is an example problem at the end of Chapter 1 in Mechanics (Landau): A simple pendulum of mass m whose point of support oscillates horizontally in the plane of motion of the pendulum according to the law x=acos(\gamma t) . Find the Lagrangian...

Homework Statement A mass m is attached to one end of a light rod of length l. The other end of the rod is pivoted so that the rod can swing in a plane. The pivot rotates in the same plane at angular velocity \omega in a circle of radius R. Show that this "pendulum" behaves like a simple...

In QFT, Lagrangian is often mentioned. While in QM, it's the Hamiltonian, Is the direct answer because in QFT "position" of particle is focused on and Lagrangian is mostly about position and coordinate while in QM, potential is mostly focus on and Hamiltonian is mostly about potential and...

Hey, in general relativity, essentially I am asking how any metric (I.e. schwarzschild metric) was found. are the metrics derived or are they extrapolated from the correct lagrange equations of motion? If there is a derivation available, please provide a link. thanks

http://en.wikipedia.org/wiki/Lagrangian#Explanation I am trying to prove pV=nRT and in order to do so one need to get lagrangian (not the math formula it seems) Here is an explanation http://en.wikipedia.org/wiki/Lagrangian#Explanation why is \frac{\delta S}{\delta \varphi _i}=0...

I was wondering why when we derive the euler lagrange equations and when we use them we treat x and x dot as independent quantities?

So my text (Ryder 2nd edition, page 252) is defining the "pure gauge-field Lagrangian" as: G_{\mu \nu}\equiv \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}-ig\left[ A_{\mu},A_{\nu}\right] \mathcal{L} = -\frac{1}{4}Tr G_{\mu \nu} G^{\mu \nu} Dumb question: Isn't G_{\mu \nu} G^{\mu...

Homework Statement Hello. I have a problem with setting up the lagrangian for a system here. The problem is stated at page 8 problem 2.3 with a diagram at the following -->link<--- 2. The attempt at a solution I used two generalized coordinates corresponding to the angle between...

Does anybody know what interpretation the invariant corresponding to the global U(1) invariance of the Dirac Lagrangian is? I have always had it in my head that it's charge, but then I realized that uncharged free particles such as neutrinos satisfy this equation too! Any thoughts much...

Say we have a Lagrangian \mathcal{L}=\bar{u}i\kern+0.15em /\kern-0.65em Du+\bar{d}i\kern+0.15em /\kern-0.65em Dd-m_u\bar{u}u-m_d\bar{d}d, where u and d are fermions. In Peskin&Schroeder p. 667 it says that if m_u and m_d are very small, we can neglect the last two terms of the Lagrangian. I'd...