What is Euler lagrange equation: Definition and 43 Discussions
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.
Well, I started with the first equation of motion for the scalar field, but I'm really not sure if I'm doing it the right way.
\begin{equation}
\begin{split}
\frac{\partial \mathcal{L}}{\partial \varphi} &= \frac{\partial}{\partial \varphi} [(\partial_\mu \varphi^* -...
In the past, I have shown relatively easily that if we have a lagrangian of the form ##\mathcal{L}=\frac{1}{2}\dot{\mathbf{q}}^2-V(\mathbf{q})## simply plugging this into the EL equation gives us newtons second law: ##\ddot{\mathbf{q}}=-\frac{\partial V}{\partial \mathbf{q}}##. I am unfamiliar...
In polar coordinates, ##x=rcos(\theta)## and ##y=rsin(\theta)## and their respective time derivatives are
$$\dot{x}=\dot{r}cos(\theta) - r\dot{\theta}sin(\theta)$$
$$\dot{y}= \dot{r}sin(\theta)+r\dot{\theta}cos(\theta)$$
so the lagrangian becomes after a little simplifying...
I'm confused on how to derive the multidimensional generalization for a multivariable function. Everything makes sense here except the line,
$$
\frac{\delta S}{\delta \psi} = \frac{\partial L}{\partial \psi} - \frac{d}{dx} \frac{\partial L}{\partial(\frac{\partial \psi}{\partial x})} -...
Principle of stationary action allows us to find equations of motion if we plug appropriate lagrangian into Euler - Lagrange equation. In classical mechanics, this is the difference in kinetic and potential energy of the system.
However, how did Lagrange came to the idea that matter behaves...
Lagrangian mechanics is built upon calculus of variation. This means that we want to find out function which is a stationary point of particular function (functional) which in Lagrangian mechanics is called the action.
To know what this function is, action needs to be defined first. Action is...
In English, does the equation
have any standard name besides (generalization of) the Euler-Lagrange Theorem? I have seen the designation "Euler-Poisson Equation" used by the Russian mathematician Lev Elsholtz way back in 1956 repeated in recent Russian webpages, but am not sure whether this...
I need to vary w.r.t ##a_{\alpha \beta} ##
##\frac{\partial L}{\partial_{\mu}(\partial_{\mu}{a_{\alpha\beta}})}-\frac{\partial L}{\partial {a_{\alpha \beta}}}## (1)
I am looking at varying the term in the Lagrangian of ##\frac{1}{3}A^{\mu} \partial_{\mu}\Phi ##
where ##A^{\beta}=\partial_k...
It is the first time that I am faced with a complex field, I would not want to be wrong about how to solve this type of problem.
Usually to solve the equations of motion I apply the Euler Lagrange equations.
$$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...
Using the Einstein-Hilbert action for a Universe with just the cosmological constant ##\Lambda##:
$$S=\int\Big[\frac{R}{2}-\Lambda\Big]\sqrt{-g}\ d^4x$$
I would like to derive the equations of motion:
$$\Big(\frac{\dot a}{a}\Big)^2+\frac{k}{a^2}=\frac{\Lambda}{3}\tag{1}$$
$$2\frac{\ddot...
I am new to Lagrangian mechanics and I am unable to comprehend why the Euler Lagrange equation works, and also what really is the significance of the lagrangian.
Hello, I have been working on the three-dimensional topological massive gravity (I'm new to this field) and I already faced the first problem concerning the mathematics, after deriving the lagrangian from the action I had a problem in variating it
Here is the Lagrangian
The first variation...
Hi All,
Anyone willing to help out in explaining what eigenfreuqncy for this oscilatory system, would be? Also if anybody knows the equation to calulate this stuff please, if you're willing to share I'd be greatful!
Thanks, regards.
This image shows the equations.
I managed to almost get equation 5, but my partial derivative is not squared but instead multiplied by mu, and also I don't have a factor of 1/2.
Here is an image of the work I have. I'm sorry for any sloppiness. I tried to be as concise as possible when writing...
In the Classical Dynamics of Particles and Systems book, 5th Edition, by Stephen T. Thornton and Jerry B. Marion, page 220, the author derived Equation (6.67) from Equation (6.66) which is the following:
Equation (6.67):
$$\left(\frac{\partial f}{\partial y} − \ \frac{d}{dx}\frac{\partial...
First of all, disclaimer: This isn't an official assignment or anything, so I'm not even sure if there is a resonably simple solution.
Consider the following sketch.
(Forgive me if it isn't completely clear, I didn't want to fiddle around for too long with tikz...)
Let us assume that we can...
Homework Statement
Let ##U## be a plane given by ##\frac{x^2}{2}-z=0##
Find the curve with the shortest path on ##U## between the points ##A(-1,0,\frac{1}{2})## and ##B(1,1,\frac{1}{2})##
I have a question regarding the answer we got in class.
Homework Equations
Euler-Lagrange
##L(y)=\int...
Hi all,
I was working on a problem using Euler-Lagrange equations, and I started wondering about the total and partial derivatives. After some fiddling around in equations, I feel like I have confused myself a bit.
I'm not a mathematician by training, so there must exist some terminology which...
Homework Statement
Prove that snell's law ## {n_1}*{sin(\theta_1)} ={n_2}*{sin(\theta_2)} ## is derived from using euler-lagrange equations for the time functionals that describe the light's propagation, As described in the picture below.
Given data:
the light travels in two mediums , one is...
Hi,
I'm trying to solve the following problem
##\max_{f(x)} \int_{f^{-1}(0)}^0 (kx- \int_0^x f(u)du) f'(x) dx##.
I have only little experience with calculus of variations - the problem resembles something like
## I(x) = \int_0^1 F(t, x(t), x'(t),x''(t))dt##
but I don't know about the...
In my quest to understand the Euler-Lagrange equation, I've realized I have to understand the chain rule first. So, here's the issue:
We have g(\epsilon) = f(t) + \epsilon h(t). We have to compute \frac{\partial F(g(\epsilon))}{\partial \epsilon}. This is supposed to be equal to \frac{\partial...
Hello everyone,
Reading Landau and Lifshitz Course of Theoretical Physics Volume 1: Mechanics (page 3) I got suck in the following step (and I cite in italics):
The change in S when q is replaced by q+δq is
\int_{t_1}^{t_2} L(q+δq, \dot q +δ\dot q, t)dt - \int_{t_1}^{t_2} L(q, \dot q, t)dt...
We have a car accelerating at a uniform rate ## a ## and a pendulum of length ## l ## hanging from the ceiling ,inclined at an angle ## \phi ## to the vertical . I need to find ##\omega## for small oscillations. From the Lagrangian and Euler-Lagrange equations, the equation of motion is given...
Suppose one starts with the standard Klein-Gordon (KG) Lagrangian for a free scalar field: $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}$$ Integrating by parts one can obtain an equivalent (i.e. gives the same equations of motion) Lagrangian...
I'm currently studying Quantum Field Theory and I have a confusion about some mathematics in page 30 of Mandl's Quantum Field Theory (Wiley 2010).
Here is a screenshot of the relevant part: https://www.dropbox.com/s/fsjnb3kmvmgc9p2/Screenshot%202017-01-24%2018.10.10.png?dl=0
My issue is in...
Homework Statement
For the following integral, find F and its partial derivatives and plug them into the Euler Lagrange equation
$$F(y,x,x')=y\sqrt{1+x'^2}\\$$
Homework Equations
Euler Lagrange equation : $$\frac{dF}{dx}-\frac{d}{dy}\frac{dF}{dx'}=0$$
The Attempt at a Solution...
Hello I am little bit confused about lagrange approximation to geodesic equation:
So we have lagrange equal to L=gμνd/dxμd/dxν
And we have Euler-Lagrange equation:∂L/∂xμ-d/dt ∂/∂x(dot)μ=0
And x(dot)μ=dxμ/dτ. How do I find the value of x(dot)μ?
I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality
δ(dq/dt) = d(δq)/dt
Where q is some coordinate, and δf is the first variation in...
Homework Statement
Find the equations of motion for both r and \theta of
Homework Equations
My problem is taking the derivative wrt time of
and
\dfrac{\partial\mathcal{L}}{\partial\dot{r}}=m \dot{r} \left( 1 + \left( \dfrac{\partial H}{\partial r}\right)^2 \right)
The Attempt at a...
I have a system with one generalized coordinate, x. In the potential energy part of the lagrangian, I have some constants multiplied by the absolute value of x. That is the only x dependence the lagrangian has, so when I take the partial derivative of the lagrangian with respect to x (to get the...
I am trying to understand an example from my textbook "applied finite element analysis" and in the variational calculus, Euler lagrange equation example I can't seem to understand the following derivation in one of its examples
∫((dT/dx)(d(δT)/dx))dx= ∫((dT/dx)δ(dT/dx))dx= ∫((1/2)δ(dT/dx)^2)dx...
Homework Statement
Consider the function f(y,y',x) = 2yy' + 3x2y where y(x) = 3x4 - 2x +1. Compute ∂f/∂x and df/dx. Write both solutions of the variable x only.
Homework Equations
Euler Equation: ∂f/∂y - d/dx * ∂f/∂y' = 0
The Attempt at a Solution
Would I first just find...
Hello all,
I am having some frustration understanding one derivation of the Euler Lagrange Equation. I think it most efficient if I provide a link to the derivation I am following (in wikipedia) and then highlight the portion that is giving me trouble.
The link is here
If you scroll...
I want to prove that Euler Lagrange equation and Einstein Field equation (and Geodesic equation) are the same thing so I made this calculation.
First, I modified Energy-momentum Tensor (talking about 2 dimension; space+time) :
T_{\mu\nu}=\begin{pmatrix} \nabla E& \dot{E}\\ \nabla p &...
Homework Statement
"Vary the following actions and write down the Euler-Lagrange equations of motion."
Homework Equations
S =\int dt q
The Attempt at a Solution
Someone said there is a weird trick required to solve this but he couldn't remember. If you just vary normally you get \delta...
Hello there,
I was wondering if anybody could indicate me a reference with regards to the following problem.
In general, the Euler - Lagrange equation can be used to find a necessary condition for a smooth function to be a minimizer.
Can the Euler - Lagrange approach be enriched to cover...
Hello there,
I am dealing with the functional (http://en.wikipedia.org/wiki/First_variation)
J = integral of (y . dy/dx) dx
When trying to compute the Euler Lagrange eqaution I notice this reduces to a tautology, i.e.
dy/dx - dy/dx = 0
How could I proceed for finding the y(x) that...
Euler Lagrange Equation : if y(x) is a curve which minimizes/maximizes the functional :
F\left[y(x)\right] = \int^{a}_{b} f(x,y(x),y'(x))dx
then, the following Euler Lagrange Differential Equation is true.
\frac{\partial}{\partial x} - \frac{d}{dx}(\frac{\partial f}{\partial y'})=0...
Dear everyone
can anyone help me with the euler lagrange equation which is stated in d'inverno chapter 11?
in equation (11.26) it is said that when we use the hilbert-einstein lagrangian we can have:
∂L/(∂g_(ab,cd) )=(g^(-1/2) )[(1/2)(g^ac g^bd+g^ad g^bc )-g^ab g^cd ]
haw can we derive...
Could somebody explain to me how lagrange multipliers works in finding extrema of constrained functions? also, what is calculus of variations and lagrangian mechanics, and can somebody explain to me what the lagrangian function is and the euler-lagrange equation. And, i read something about...
[SOLVED] Euler Lagrange Equation
Hi there ,
I am missing a crucial point on the proof of Euler Lagrange equation , here is my question :
\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{df}{dy^{'}}\right)=0 (Euler-Lagrange equation)
If the function "f" doesn't depend on x explicitly...