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.

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  1. B

    Equations of motion for Lagrangian of scalar QED

    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^* -...
  2. Hamiltonian

    Verifying that Newton's Equations are equivalent to the EL equations

    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...
  3. Hamiltonian

    Deriving ODEs for straight lines in polar coordinates for a given Lagrangian

    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...
  4. O

    Modifying Euler-Lagrange equation to multivariable function

    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})} -...
  5. Dario56

    I Principle of Stationary Action - Intuition

    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...
  6. Dario56

    I Action in Lagrangian Mechanics

    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...
  7. nomadreid

    I Standard designation for generalization of Euler-Lagrange?

    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...
  8. binbagsss

    A Euler-Lagrange Tensor Equations

    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...
  9. F

    Lagrangian for the electromagnetic field coupled to a scalar field

    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. $$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...
  10. J

    I Deriving vacuum FRW equations directly from action

    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...
  11. Hamiltonian

    I Lagrangian and the Euler Lagrange equation

    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.
  12. T

    Euler Lagrange equation and a varying 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...
  13. aligator11

    Engineering Homework problem - Pendulum oscillatory system

    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.
  14. dsilvas

    Algebra: How do I derive this equation given two other equations?

    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...
  15. sams

    A Question about Euler’s Equations when Auxiliary Conditions are Imposed

    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...
  16. W

    Geodesics and Motion in an EM Field

    I've also attached my attempt as a pdf file. My main issue seems to be I only get one A partial term. Any help would be appreciated.
  17. M

    The Lagrangian for a piece of toast falling over the edge of a table

    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...
  18. W

    Find the curve with the shortest path on a surface (geodesic)

    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...
  19. Runei

    I Total Derivative of a Constrained System

    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...
  20. C

    Proving Snell's law using Euler-Lagrange equations

    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...
  21. petterson

    A Maximization problem using Euler Lagrange

    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...
  22. Avatrin

    I Rigorously understanding chain rule for sum of functions

    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...
  23. A

    I Help a novice with EL equation derivation

    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...
  24. saadhusayn

    Pendulum oscillating in an accelerating car

    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...
  25. F

    I Equivalent Klein-Gordon Lagrangians and equations of motion

    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...
  26. joebentley10

    I Applying Euler-Lagrange to (real) Klein-Gordon 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...
  27. F

    Euler Lagrange equation issue with answers final form

    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...
  28. A

    B Euler-Lagrange equation for calculating geodesics

    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?
  29. hideelo

    Deriving Commutation of Variation & Derivative Operators in EL Equation

    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...
  30. B

    Euler Lagrange equation of motion

    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...
  31. T

    Understanding the Euler Lagrange Equation and Its Boundary Condition

    I am trying to derive it but I am stuck at the boundary condition. What is this boundary comdition thing such that the value must be zero?
  32. skate_nerd

    MHB Euler Lagrange equation of motion

    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...
  33. H

    Variational calculus Euler lagrange Equation

    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...
  34. H

    Euler Lagrange Equation Question

    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...
  35. W

    Help with Derivation of Euler Lagrange Equation

    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...
  36. B

    Euler Lagrange equation as Einstein Field Equation

    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 &...
  37. B

    Euler Lagrange Equation trough variation

    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...
  38. M

    Euler Lagrange equation - weak solutions?

    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...
  39. M

    How can I find the y(x) that minimizes the functional J?

    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...
  40. B

    Lagrangian mechanics - Euler Lagrange Equation

    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...
  41. S

    Euler lagrange equation and Einstein lagrangian

    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...
  42. O

    Euler lagrange equation, mechanics,

    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...
  43. E

    What is the Proof of the Euler Lagrange Equation?

    [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...