What is Euler-lagrange: Definition and 129 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. W

    Confused about Euler-Lagrange Equations and Partial Differentiation

    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...
  2. T

    What Does Stationarity Mean in the Context of the Euler-Lagrange Equations?

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

    A question about the derivation of the Euler-Lagrange equation

    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...
  4. A

    Rigid bodies: generalized forces according to the Euler-Lagrange formalism

    I am trying to derive the dynamic equations of an aerial vehicle with 6 degrees of freedom (a quadrotor to be precise). I am using - two coordinate systems: the Earth frame and the body frame; - the Euler-Lagrange formalism: generalized coordinates {x,y,z,phi,theta,psi}, respectively, the...
  5. G

    Anyone know how to interpret the euler-lagrange differential equation?

    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...
  6. T

    Help with derivation of euler-lagrange equations

    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...
  7. M

    Help with the Euler-Lagrange formula for a geodesic

    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...
  8. C

    Maximizing a functional when the Euler-Lagrange equation's solution violates ICs

    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...
  9. D

    Simpler Euler-Lagrange equation

    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...
  10. B

    Deriving Field Equations for Real Vector Fields using Euler-Lagrange (Tensors)

    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...
  11. P

    Solution of Euler-Lagrange equation

    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...
  12. A

    Calculus of Variations Euler-Lagrange Diff. Eq.

    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...
  13. P

    Derivation of Euler-Lagrange Equation

    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...
  14. R

    Simplifying the Euler-Lagrange Equation for Explicitly Independent Functions

    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...
  15. D

    Simple Derivation of Euler-Lagrange Equations

    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...
  16. N

    How Can I Understand Euler-Lagrange Equations in Physics?

    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...
  17. N

    Euler-Lagrange Equation for Functional S

    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 +...
  18. N

    How Can Maxwell's Equations Be Derived Using the Euler-Lagrange Equation?

    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...
  19. haushofer

    Variations, Euler-Lagrange, and Stokes

    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...
  20. S

    Euler-Lagrange equations in QFT?

    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...
  21. E

    Where can I find a comprehensive derivation of the Euler-Lagrange equation?

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

    Verifying that the Euler-Lagrange equation uses generalized coordinates

    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...
  23. O

    Principle of Least Action & Euler-Lagrange Equations

    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...
  24. G

    Euler-Lagrange Equations for Schördinger Eq.

    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...
  25. O

    Simple calculus - interpretation Euler-Lagrange equation

    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...
  26. W

    Euler-Lagrange equation for paraboloid plane

    I have a classical mechanics question I couldn't conclude. The reason seems to be mathematical. It's this: There's a paraboloid shaped plane of mass M, which is standing on a frictionless surface and can slide freely. It's surface is y=ax^2. A point mass m is place on the plane. Solve the...
  27. T

    Minimum Surface Area Cylinder using Euler-Lagrange Equation?

    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...
  28. A

    Missing step: Euler-Lagrange equations for the action integral

    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}...
  29. M

    Applications of Euler-Lagrange Equation

    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...
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