In the calculus of variations and classical mechanics, the Euler-Lagrange equations is a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero.
In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations. In classical mechanics, it is equivalent to Newton's laws of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.
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
>...
Find the Euler-Lagrange equation for the functional $F(u)=\displaystyle\int_{\Omega} \left(\dfrac{1}{2}A\nabla u(x)\cdot \nabla u(x)-f(x)u(x)\right)dx$ where $\Omega$ is an bounded domain in $\mathbb{R}^n$ and $A$ is an symmetric matrix.Hello MHB! I Need help for this problem :). I have clear...
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')...
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.
Equations
F_{y}-D_{x}F_{y'} \ = \ 0...
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}...
I'm hoping this is a really simple question, but I can't seem to find a definitive answer anywhere!
If the action is invariant under some symmetry transformation, do the equations of motion need to be invariant as well?
The trouble I'm having relates to SU(N) yang-mills theories where I'm...
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.
Hello,
The Fermat principle says that
(***) Δt = (1/c) ∫ μ(x,y) √1+y'2 dt
Say, we are studying a GRIN material where the refraction index is μ = μ(x,y) and want to figure out the shape of the ray trajectory y=y(x).
Here is what I know (this is not a homework question) but am unsure if...
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...
Homework Statement
Given two Euler-Lagrange systems with generalized coordinates ##r_1## and ##r_2,## and input ##u_1## and ##u_2##. Suppose now that a constraint is placed on them such that ##r_1 = f_1(q)## and ##r_2 = f_2(q)##.
Propose a Lagrangian for the constrained system and show that...
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...
In the proof to Theorem 7.3 from this paper on FNTFs, the authors invoke the so-called "Langrange equations." I assume they mean the Euler-Lagrange equations. (But maybe not...?) Unfortunately I'm not at all familiar with the Euler-Lagrange equations, and in reading what they are, I have no...
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...
(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
Find and describe the path y = y(x) for which the integral \int\sqrt{x}\sqrt{1+y^{' 2}}dx (the integral goes from x1 to x2... wasn't sure how to put that in. sorry) is stationary.
Homework Equations
\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial...
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 \\...
Homework Statement
Bead slides on a wire (no friction) shaped as r = r(\theta) in the Oxy plane. The Oxy plane and the constraining wire rotate about Oz with \omega = const
r, \theta is the rotating polar frame; r, \phi is the stationary frame.
Find the trajectory r = r(\phi) in the...
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 -...