Lagrangian Definition and 1000 Threads

Homework Statement Find the equations for the utility maximizing values for x and y U(x,y) = x^2 + y^2Homework Equations Budget constraint: I = PxX +Pyy L(x,y,\lambda ) x^2 + y^2 + \lambda (I - PxX - PyY) The Attempt at a Solution I got the three partial derivatives and set equal to zero...

In Goldstein, the action is defined by I=\int L dt. However, when dealing with constraints that haven't been implicitly accounted for by the generalized coordinates, the action integral is redefined to I = \int \left( L + \sum\limits_{\alpha=1}^m \lambda_{\alpha} f_a \right) dt. f is...

Hello, I've started a course on QFT and I'm having some troubles trying to find the solution of this exercise: Write the action of a non-relativistic spineless free particle in a manifestly hermitian way The problem should be simple but I'm a bit lost in the hermitian way part... What does it...

Hi, the attached picture shows a derivation of what I can only assume to be the property that the lagrange equations are invariant under a transformation of the coordinates. But I have some trouble understanding how you go from the term pointed out the rear of the arrow to the point pointed...

Homework Statement See attached picture for problem. Homework Equations Lagranges equation. The Attempt at a Solution So I have found the lagrangian to be: L = ½ma2(θ'2+\omega2sinθ) - mgacosθ I think this is correct but I have some questions on the further solution of the...

Hey guys. I'm trying to gather some tips that people have acquired that helps them write the Lagrangian for a system. Obviously, the classic examples are drilled into our heads over and over, but just when you think you can tackle any problem the professor throws at you, there is that tricky one...

A lot of conservation laws are derived from the lagrangian in my book. However, I fail to see why the Lagrangian incorporates action-reaction. Since it works for an arbitrary amount of particles and linear momentum can be show to be conserved from translational invariance it must do so. But...

Homework Statement Show that if the potential in the Lagrangian contains-velocity dependent terms, the canonical momentum corresponding to the coordinate of rotation θ, is no longer the mechanical angular momentum but is given by: p = L - Ʃn\bulletri x ∇viU Homework Equations...

I have seen the lagrange equations derived from Newtons laws in the special case, where forces were derivable from a potential. Now with the introduction of hamiltons principle, I think my book wants to say this: We can always find a lagrangian such that the principle of least action holds...

So in my internet readings on Lagrangian mechanics I started researching applications with non-potential and/or non-conservative forces and came across this page: http://farside.ph.utexas.edu/teaching/336k/Newton/node90.html This page is fascinating but I'm having a bit of difficulty...

So the integral of the lagrangian over time must be stationary according to hamiltons principle. One can show that this leads to the euler lagrange equations, one for each pair of coordinates (qi,qi'). But my book has now started on defining a generalized lagrangian where lagrangian...

Hi all, Doing some self-study on Lagrangian uses on the internet and I'm getting it pretty well thus far, but I'm just not sure how external forces fit in exactly. Up until now I've only tackled problems with gravity and constraints involved but intuitively I know that kinetic and potential...

Homework Statement A uniform thin disk rolls without slipping on a plane and a force is being applied at its center parallel to the plane. Find the lagrangian and thereby the generalized force.Homework Equations Lagranges equation.The Attempt at a Solution This is my first ever exercise of this...

Homework Statement Right I've got a relativistic particle in D dimensional space interacting with a central potential field. Writing out the entire lagrangian is a bit complicated on this but I'm sure you all know the L for a free relativistic particle. The potential term is Ae-br where r is...

This issue has always bothered me, and I would like to hear a logical resolution. The classical prescription for finding it is L=T-V. From the LaGrangian, the equations are motion are then deduced using the Euler-LaGrange eqs. But - the equations are motion are required in order to determine T...

This seems like such a simple question that I fully expect its solution to be embarrassingly easy, but try as I might I can't get the answer. Consider some system which can be described by N generalized coordinates q_1,...,q_N and a Lagrangian L(q_i,\dot{q}_i,t). (I'll just use q_i as a stand...

On page 13 in Landau-Lifgarbagez Mechanics, the total time time derivative of the Lagrangian of a closed system is given to be, \frac{d L}{d t} = \sum_i \frac{\partial L}{\partial q_i} \dot{q_i} + \sum_i \frac{\partial L}{\partial \dot{q_i}} \ddot{q_i} Why does this stop here? I mean, why...

My friend and I have been getting all confused about the following problem with a Lagrangian. It comes from David Tong's online notes on QFT, but given it is about the Lagrangian, I figure it does well in this section. Ok, Tong is talking about Noether's theorem, and using the example of...

Let be L=(x^2 + y^2)x* +2xyy* where x* = dx/dt and y* = dy/dt. Which physical system is referred it to? why? Thanks

Recently i am reading A.Zee's Quantum Field Theory in a Nutshell 2nd Edition. there is a equation that i can not derive by myself. I suspect its correctness. <k1k2|exp(-iHT)|k3k4>=<k1k2|exp(i∫dxL(x))|k3k4>, where the L(x) on the RHS is an operator function of space-time. This equation...

After finding the equations of motion of a pendulum in an accelerating cart: \ddot{\phi} + \frac{acos\phi +gsin\phi}{l}=0 ,the method that Taylor uses in Prob 7.30 for finding the small angle frequency, is to rewrite \phi as \phi_{0}+\delta \phi. Then you can use a trig identity in the...

This is not a homework or test or textbook question or exercise. I am asking purely out of curiosity. Please do not tell me to post this in homework help or give me another infraction. I have gathered that the Lagrangian approach will follow an individual particle to record some streamline...

Preparing for classical prelim, just wondering if this solution is correct. Homework Statement A particle with mass m and charge q moves in a uniform magnetic field \boldsymbol{B}=B\boldsymbol{\hat{z}}. Write a Lagrangian describing the motion of the particle in the xy plane that gives the...

Any advice on deriving the lagrangian for a particle in a magnetic field? L=\frac{1}{2}mv^{2} + \frac{q}{c}\mathbf{v}\mathbf{.} \mathbf{A} I've been searching through Griffith's, Jackson, and google to no avail. Can we start from the lorentz force and work backwards? Thanks for the help.

Since it is based on the kinetic energy less the potential energy, what does the Lagrangian actually represent? Is there some intuitive way to understand why it is defined so and why it is such a fruitful concept using the principle of least action?

In texts about Lagrangian mechanics,at first Lagrangian is defined as below: L=T-V T and V being kinetic and potential energy respectively But when you proceed,they say that for some forces like magnetic forces Lagrangian is as such and can't be obtained by the above formula but it doesn't...

In some texts about Lagrangian mechanics,its written that the generalized coordinates need not be length and angles(as is usual in coordinate systems)but they also can be quantities with other dimensions,say,energy,length^2 or even dimensionless. I want to know how will be the Lagrange's...

Hello, how do we apply the idea of the Lagrangian to a Brownian motion? I guess what I mean is what is the Lagrangian functional form for a Brownian motion? Thanks

I am wondering, how does lagrangian of such system look like? Will it be: L=\frac{m_{1} \cdot \dot{y}^2}{2} + \frac{m_{2} \cdot \dot{x}^2}{2} +\frac{m_{3} \cdot (\dot{y'}^2+\dot{x'}^2)}{2} + \frac{I \cdot \dot{ \alpha }^2}{2} - mgy - mgy' where: y'=\frac{l}{2}sin(\alpha)...

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...

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}) \\ &=...

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 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 (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...