Euler-lagrange Definition and 122 Threads

Homework Statement The metric is: ds^{2} = y^{2}(dx^{2} + dy^{2}) I have to find the equation relating x and y along a geodesic.The Attempt at a Solution ds = \sqrt{ydx^{2} + ydy^{2}} - is this right? ds = \sqrt{y + yy'^{2}} dx F = \sqrt{y + yy'^{2}} So then I apply the Euler-Lagrange...

Hi, I am trying to minimize: \int_0^\infty{\exp(-t)(t\,f'(t)-f(t))^2\,dt} by choice of f, subject to f(0)=1 and f'(x)>0 for all x. The (real) solution to the Euler-Lagrange differential equation is: f(t)={C_1}t rather unsurprisingly. However, this violates f(0)=1. If...

If we have a functional J(y)=\int L(y,y',x)dx then the first variation is \delta J=\int\left(\frac{\partial L}{\partial y}\eta(x)+\frac{\partial L}{\partial y'}\eta'(x)\right)dx, where \eta(x) is the variation of the stationary solution. Now, if L is independent of y(x), then...

Homework Statement Show that the Lagrangian density: L=- 1/2 [\partial_\alpha \phi_\beta ][\partial^\alpha \phi^\beta ]+1/2 [\partial_\alpha \phi^\alpha ][\partial_\beta \phi^\beta ]+1/2 \mu^2 \phi_\alpha \phi^\alpha for the real vector field \phi^\alpha (x) leads to the field equations...

I have the following Lagrangian: \mathcal{L} = 1/2 \partial_{\mu} \varphi \partial^{\mu} \varphi - 1/2 b ( \varphi^{2} - a^{2} )^{2} , where a,b \in \mathbb{R}_{>0} and \varphi is a real (scalar) field and x are spacetime-coordinates. I calculated the Euler-Lagrange eq. and get...

I'm in dire need of help in understanding calculus of variations. My professor uses the Mathews and Walker text, second edition, entitled Mathematical Methods of Physics and, he has a tendency to skip around from chapters found towards the beginning of the text to those nearer the end. I...

I am stuck in trying to understand the derivation of the Euler-Lagrange equation. This mathematical move is really bothering me, I can't figure out why it is true. \frac{\partial f(y,y';x)}{\partial\alpha}=\frac{\partial f}{\partial y}\frac{\partial y}{\partial\alpha}+\frac{\partial...

Homework Statement If the integrand f(y, y', x) does not depend explicitly on x, that is, f = f(y, y') then \frac{df}{dx} = \frac{\partial f}{\partial y}y' + \frac{ \partial f } {\partial y' } y''Use the Euler-Lagrange equation to replace \partial f / \partial y on the right and hence show...

I'm trying to deduce the equations of motion in the form \frac{d}{dt} \frac{\partial \cal L}{\partial \dot{q}} - \frac{\partial \cal L}{\partial q} = 0 with little algebra directly from Hamilton's principle, like the geometric derivation of snell's law from the principle of least time. It...

I'm taking a Physics class at Stanford U. and I am having difficulty understanding how to mathematically understand or translate the Euler-LaGrange equations of motion in both Classical and Quantum Field Theory. Any sort of English translation, background or hinting as to what type of math I...

Homework Statement Let P be a rectangle , f_{0} : \partial P \rightarrow R) continuous and Lipschitz, C_{0} = \{ f \in C^{2}(P) : f=f_{0} \ on \ \partial P \}. and finally S : C_{0} \rightarrow R a functional: S(f) = \int^b_a (\int^d_c (\frac{\partial f}{\partial x})^{2}\,dy)\,dx +...

Homework Statement I'm asked to get Maxwell's equations using the Euler-lagrange equation: \partial\left(\frac{\partial L}{\partial\left\partial_{\mu}A_{\nu}\right)}\right)-\frac{\partial L}{\partial A_{\nu}}=0 with the EM Langrangian density...

Hi, I have some questions which I encountered during my thesis-writing, I hope some-one can help me out on this :) First, I have some problems interpreting coordinate-transformations ( "active and passive") and the derivation of the Equations of Motion. We have S = \int L(\phi...

Euler-Lagrange equations in QFT?? Hi, I have a problem with a Wikipedia entry::bugeye: http://en.wikipedia.org/wiki/Euler-Lagrange_equation The equations of motion in your quantized theory (2nd quantization) are d/dtF^=[F^,H^] i.e the quantized version of d/dtF={F,H}. My notation: F^ is the...

Can someone link me to a thorough online derivation of the Euler-Lagrange equation from the principle of least action?

This is a question that I'm asking myself for my own understanding, not a homework question. I realize that in most derivations of the Euler-Lagrange equations the coordinate system is assumed to be general. However, just to make sure, I want to apply the "brute force" method (as Shankar...

I'll just throw down some definitions and then ask my question on this one. In a conservative system, the Lagrangian, in generalised coordinates, is defined as the kinetic energy minus the potential energy. L=L(q_i,\dot{q}_i,t) = K(q_i,\dot{q}_i,t) - P(q_i,t). All q_i here being functions...

Euler-Lagrange equations for the Lagrangian density \mathcal{L} = V\psi \psi^* + \frac{\hbar^2}{2m}\frac{\partial \psi}{\partial x}\frac{\partial \psi^*}{\partial x} + \frac{1}{2}\left(i\hbar \frac{\partial \psi^*}{\partial t} \psi- i\hbar \frac{\partial \psi}{\partial t} \psi^*\right) gives...

This is not a homework question but one that is part of the course material and I can't really move on until I understand the basic calculus. I have a problem interpreting "d by dx of partial dF by dy' equals partial d by dy' of dF by dx" in the following question, which I set out and then...

So, I've been reading Thornton and Marion's "Classical Dynamics of Particles and Systems" and have gotten to the chapter on the calculus of variations. In trying the end of chapter problems, I find I'm totally baffled by 6-9: given the volume of a cylinder, find the ratio of the height to the...

Hi its me again, stuck once more. Sorry guys and gals :P Ok a problem I found on http://en.wikipedia.org/wiki/Action_%28physics%29 In a 1-D case how do we get from: \delta S = \int_{t_1}^{t_2} [L(x + \varepsilon, \dot{x} + \dot{\varepsilon})-L(x,\dot{x})]dt to: \delta S = \int_{t_1}^{t_2}...

hey, In my physics class we are now learinging beginging to learn about lagrange ion mechanics and I am a little stuck on the basics of it particularly fermat's principle (dealing with light travel) and applications of the Euler-Lagrange Equation, I can't seem to get many of the problems at the...