Euler-lagrange Definition and 122 Threads

Homework Statement A simple pendulum with mass m and length ℓ is suspended from a point which moves horizontally with constant acceleration a > Show that the lagrangian for the system can be written, in terms of the angle θ, L(θ, θ, t˙ ) = m/2(ℓ^2θ˙^2 + a^2t^2 − 2aℓtθ˙ cosθ) + mgℓ cos θ >...

First off, apologies if this is in the wrong forum, if my notation is terrible, or any other signs of noobishness. I just started university and I'm having a hard time with my first Lagrange problems. Help would be very much appreciated. 1. Homework Statement A body of mass m is lying on a...

Hello, So in the familiar case of non-relativistic particle Lagrangians/actions, we know the equations of motions are given by \frac{\partial \mathcal L}{\partial x^i} = \frac{\mathrm d }{\mathrm dt} \left( \frac{\partial \mathcal L}{\partial \dot x^i} \right) In the familiar case of...

Homework Statement a. Suppose two particles with mass $m$ and coordinates $x_1$, $x_2$ collides elastically, find the lagrangian and prove that the linear momentum is preserved. b. Find new coordiantes (and lagrangian) s.t. the linear momentum is conjugate to the cyclical coordinate. Homework...

Homework Statement [/B] So, I need to show Lorentz covariance of a Proca field E-L equation, conceptually I have no problems with this, I just have to make one final step that I cannot really justify. Homework Equations "Proca" (quotation marks because of the minus next to the mass part, I...

The title basically says it, if I want to use a potential that is time dependent (for example someone is amping up the electric field externally) and keep using the form ##L=T-V## with the standard E-L equations. Can one still use them or not? If no, why? I have seen two derivations of the E-L...

Homework Statement On very hot days there sometimes can be a mirage seen hovering as you drive. Very close to the ground there is a temperature gradient which makes the refraction index rises with the height. Can we explain the mirage with it? Which unit do you need to extremalise? Writer the...

I've attached the part from Landau & Lifschitz Mechanics where I got confused. "The necessary condition for S(action) to have a minimum (extremum) is that these terms (called the first variation, or simply the variation, of the integral) should be zero. " Why is this a necessary condition? If...

Homework Statement Homework Equations The Attempt at a Solution I have tried manipulating the equation a few different ways, but the Euler-Lagrange and the one I'm supposed to show for a) is so different that I just can't seem to work. Can someone please point me in the right...

I'm going through Zwiebach Chapter 6 on relativistic strings to try to solve a similar problem. I got all the way to my equation of motion \begin{eqnarray*} \delta S & = & [ p' \delta \theta]_{z 0}^{z 1} + \int_{z 0}^{z 1} d z \left( p - \frac{\partial ( p')}{\partial z} \right) \delta...

Homework Statement I'm trying to do a little review of Lagrangian Mechanics through studying the two-body problem for a radial force. I have the Lagrangian of the system L=\frac{1}{2}m_1\dot{\vec{r_1}}^{2}+\frac{1}{2}m_2\dot{\vec{r_2}}^{2}-V(|{\vec{r_1}-\vec{r_2}}|) . Now I'm trying to find...

Hey! I'm not sure if this belongs better here or in mechanics but while I was doing some mechanics problems I started wondering if Lagrange equations are true for any differential manifold. Obviously they work for pseudo-riemann ones (general relativity) but do they work for others (all)? I...

Homework Statement 6.20 ** If you haven't done it, take a look at Problem 6.10. Here is a second situation in which you can find a "first integral" of the Euler—Lagrange equation: Argue that if it happens that the integrand f (y, y', x) does not depend explicitly on x, that is, f = f (y, y')...

[SIZE="4"]Definition/Summary Also known as the Euler equation. It is the solution to finding an extrema of a functional in the form of v(y)=\int_{x_{1}}^{x_{2}} F(x,y,y') dx \ . The solution usually simplifies to a second order differential equation. [SIZE="4"]Equations...

For twice differentiable path x:[t_A,t_B]\to\mathbb{R}^N the action is defined as S(x) = \int\limits_{t_A}^{t_B} L\big(t,x(t),\dot{x}(t)\big) dt For a small real parameter \delta and some path \eta:[t_A,t_B]\to\mathbb{R}^N such that \eta(t_A)=0 and \eta(t_B)=0 the action for...

So I have this book that considers the problem of a flexible vibrating string, taking \phi(x,t) as the string's displacement from equilibrium. It then writes a Lagrangian density in terms of this \phi, takes \delta \mathcal{S} = 0, and eventually concludes that \frac{\partial}{\partial...

Every time I try to read Peskin & Schroeder I run into a brick wall on page 15 (section 2.2) when they quickly derive the Euler-Lagrange Equations in classical field theory. The relevant step is this: \frac{∂L}{∂(∂_{μ}\phi)} δ(∂_{μ}\phi) = -∂_{μ}( \frac{∂L}{∂(∂_{μ}\phi)}) δ(\phi) + ∂_{μ}...

Every time I try to read Peskin & Schroeder I run into a brick wall on page 15 (section 2.2) when they quickly derive the Euler-Lagrange Equations in classical field theory. The relevant step is this: \frac{∂L}{∂(∂_{μ}\phi)} δ(∂_{μ}\phi) = -∂_{μ}( \frac{∂L}{∂(∂_{μ}\phi)}) δ(\phi) + ∂_{μ}...

Homework Statement Compute the Euler-Lagrange for: ∫y(y')2+y2sinx dx Homework Equations \frac{∂L}{∂y}-\frac{d}{dx} (\frac{∂L}{∂y'}) The Attempt at a Solution Usual computation by hand gives me y'2+2ysin(x) - 2yy'', but Mathematica says it's -y'2-2yy''. Am I doing something wrong?

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...

\[ \int_0^1yy'dx \] where \(y(0) = 0\) and \(y(1) = 0\). The first integral is \[ f - y'\frac{\partial f}{\partial y'} = c. \] Using this, I get \(yy' - y'y = 0 = c\) so ofcourse \(y(0)\) and \(y(1)\) equal \(0\) then but is this correct? It just seems odd.

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}...

I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for E-L. The whole notion of, and justification for, using 'pretend' differentials over a time interval of zero just isn't sinking in with me. And I notice...

I am pretty much confused with all the algebra of Christoffel symbols: I have an expression for infinitesimal length: F= g_{ij} \frac{dx^i dx^j}{du^2} and by using Euler-Lagrange equation (basically finding the shortest distance between two points) want to find the equation for geodesics...

Homework Statement Problem 1: Derive the Euler-Lagrange equation for the function ##z=z(x,y)## that minimizes the functional $$J(z)=\int \int _\Omega F(x,y,z,z_x,z_y)dxdy$$ Problem 2: Derive the Euler-Lagrange equation for the function ##y=y(x)## that minimizes the functional...

I am reading the book of Neuenschwander about Noether's Theorem. He explains the Euler-Lagrange equations by starting with J=\int_a^b L(t,x^\mu,\dot x^\mu) dt From this he derives the Euler-Lagrange equations \frac{\partial L}{\partial x^\mu} = \frac{d}{dt}\frac{\partial L}{\partial...

In this document, how do I get 3.2 on page 12? I assume it is the Euler-Lagrange equation given in 3.1 just rewritten. But how? Many thanks in advance

(sorry for my english :P) Hi, I need help with this problem: .The question is: Write the equations of motion using the Euler-Lagrange formalism, assuming small motions around the vertical position. clue:determine the work of the forces as a function of degrees of freedom.

I need to find the equation of motion of a double pendulum, as shown here: I've gotten as far as the two euler-lagrange differential equations, simplified to this: K1\ddot{θ}1 + K2\ddot{θ}2cos(θ1 - θ2) + K3\dot{θ}22sin(θ1 - θ2) + K4sin(θ1) = 0 K5\ddot{θ}2 + K6\ddot{θ}1cos(θ1 - θ2) +...

This is from a past paper (from a lecturer I don't particularly understand) Homework Statement a) {4 marks} Find the Euler-Lagrange equations governing extrema of I subject to J=\text{constant} , whereI=\int_{t_1}^{t_2}\text{d}t \frac{1}{2}(x\dot{y}-y\dot{x})=\int f(t,x,y,\dot{x},\dot{y})...

Homework Statement Hi. I am attempting to get the Euler-Lagrange equations of motion for the following Lagrangian: L(ψ^{μ}) = -\frac{1}{2} ∂_{μ} ψ^{\nu} ∂^{μ} ψ_{\nu} + \frac{1}{2} ∂_{μ} ψ^{\mu} ∂_{\nu} ψ^{\nu} + \frac{m^{2}}{2} ψ_{\nu} ψ^{\nu} Homework Equations So, I want to get...

Homework Statement We are given the ring Z/1026Z with the ordinary addition and multiplication operations. We define G as the group of units of Z/1026Z. We are to show that g^{18}=1. Homework Equations The Euler-phi (totient) function, here denoted \varphi(n) The Attempt at a Solution...

Homework Statement A bead of mass m slides without friction along a wire which has the shape of a parabola y=Ax² with axis vertical in the Earth's gravitational field g. a)Find the Lagrangian, taking as generalized coordinate the horizontal displacement x. b)Write down the Lagrange's equation...

Homework Statement Particle is moving along the curve parametrized as below (x,y,z) in uniform gravitational field. Using Euler- Lagrange equations find the motion of the particle. The Attempt at a Solution \begin{array}{ll} x=a \cos \phi & \dot{x}= -\dot{\phi} a \sin \phi \\...

I'm trying to understand the derivation of the Euler-Lagrange equation from the classical action. http://en.wikipedia.org/wiki/Action_(physics)#Euler.E2.80.93Lagrange_equations_for_the_action_integral" has been my main source so far. The issue I'm having is proving the following equivalence...

I read in hand and finch (analytical mechanics) that if you assume you have a lagrangian: L=(\phi,\nabla\phi,x,y,z) Then what does the euler lagrange equation look like in vector notation. I know that if you have a function with more than 1 independent variable then the euler-lagrange...

Suppose I have a particle of mass m in a uniform, downward gravitational field g, constrained to move on a frictionless parabola y = x^2 I get L = KE - PE = \frac {1}{2} m (\dot x^2 + \dot y^2) - mgy = \frac {1}{2}m \dot x^2 (1+4x^2) - mgx^2 \frac {\partial L}{\partial...

Hi What is the difference between Lagrange's equation of motion and the Euler-Lagrange equations? Don't they both yield the path which minimizes the action S? Niles.

Derivation of "first integral" Euler-Lagrange equation Homework Statement This is from Classical Mechanics by John Taylor, Problem 6.20: Argue that if it happens that f(y,y',x) does not depend on x then: EQUATION 1 \frac{df}{dx}=\frac{\delta f}{\delta y}y'+\frac{\delta f}{\delta...

Wikipedia: Euler Lagrange Equation defines a function L:[a,b] \times X \times TX \rightarrow \mathbb{R} \enspace\enspace\enspace (1) such that (t,q(t),q'(t)) \mapsto L(t,q(t),q'(t)) \enspace\enspace\enspace (2.) But (2) suggests that the domain of L is simply [a,b], thus: L:[a,b]...

Homework Statement Find the function F in J\left[y\right]={\displaystyle \int}_{a}^{b}F\left(x,y,y'\right)\ dx such that the resulting Euler's equation is f-\left(-\dfrac{d}{dx}\left(a\left(x\right)u'\right)\right)=0 for x\in\left(a,b\right) where a\left(x\right) and f\left(x\right)...

I have been trying to teach myself Lagrangian mechanics from a textbook “Lagrangian and Hamiltonian Mechanics” by MC Calkin. It has covered virtual displacements, generalised coordinates, d’Alembert’s principle, the definition of the Lagrangian, the Euler-Lagrange differential equation and how...

Homework Statement Given the the Lagrangian density L= \frac{1}{2}\partial_\lambda\phi\partial^\lambda\phi + \frac{1}{3}\sigma\phi^3 (a)Work out the equation of motion. (b)Calculate from L the stress tensor: T^{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial^\nu\phi -...

Below is the question: [PLAIN]http://img706.imageshack.us/img706/7549/42541832.jpg I don't even know where to start. Theres nothing about this topic in my notes & I can't remember doing it before. I've tried searching for the key words but that didn't help much. Does anyone have any...

I don't mean the actual definition of the Euler-Lagrange equation per-se, but rather a word definition that's slipping my mind. I remember that if you want to measure the shortest distance between two points, you have to minimize an integral of all possible paths or something. Is that thing...

I have a Lagrangian L = \frac{R^2}{z^2} ( -\dot{t}^2 +\dot{x}^2 +\dot{y}^2 +\dot{z}^2) and I want to find the Euler-Lagrange equations \frac{\partial L}{\partial q} = \frac{d}{ds} \frac{\partial L}{\partial \dot{q}} I'm fine with the LHS and the partial differentiation on the RHS, but when it...

What does it mean when it says "the integral of the Lagrange equation is stationary for the path followed by the particle"?

In the book Mathematical Methods for Engineers and Scientists 3, the derivation of the Euler-Lagrange equation starts roughly along the lines of this: In order to minimize the functional I=\int_{x_1}^{x_2}{f(x,y,y')dx}, one should define two families of functions Y(x) and Y'(x), where Y(x) is...

Hi, I am having a calculus class now and these days the instructor is introducing the Euler-Lagrange differential equation. I have no idea why the formula (general form) is like that way. Is anyone here know how to interprete the formula and help me to understand it? dF/df-(d/dx)dF/df'=0...

Hi, I am trying to follow a derivation of the euler lagrange equations in one of my textbooks. It says that \int ( f\frac{dL}{dx} + f'\frac{dL}{dx'}) dt = f\frac{dL}{dx'} + \int f ( \frac{dL}{dx} - \frac{d}{dt}(\frac{dL}{dx'}) ) dt where f is an arbitrary function and L is the...