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. Y

    Lagrangian and Euler-Lagrange Equation Problem

    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...
  2. N

    What Happens to Euler-Lagrange in Field Theories (ADM)?

    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...
  3. E

    Finding the Lagrangian for an elastic collision

    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...
  4. VintageGuy

    Tensor indices (proving Lorentz covariance)

    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...
  5. Coffee_

    Do the Euler-Lagrange equations hold for a time-dependent V?

    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...
  6. J

    Can Euler-Lagrange Equations Explain Mirages?

    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...
  7. A

    Question about derivation of Euler-Lagrange eq.

    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...
  8. Ascendant78

    Euler-Lagrange equation application

    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...
  9. E

    Euler-Lagrange equation (EOM) solutions - hairy lagrangian

    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...
  10. _Kenny_

    Euler-Lagrange Equations for Two Body Problem

    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...
  11. I

    Are Euler-Lagrange Equations Applicable to All Differential Manifolds?

    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...
  12. kq6up

    Euler-Lagrange First Integral

    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')...
  13. Greg Bernhardt

    What is the Euler-Lagrange equation

    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...
  14. J

    Euler-lagrange, positivity of second order term

    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...
  15. Xezlec

    Signs in the Field-Theoretic Euler-Lagrange Equation

    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...
  16. C

    Expanding delta in Field Theory Derivation of Euler-Lagrange Equations

    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}...
  17. C

    Expanding delta in Field Theory Derivation of Euler-Lagrange Equations

    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}...
  18. michael879

    Euler-Lagrange equations and symmetries

    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...
  19. S

    What is the Correct Euler-Lagrange for ∫y(y')2+y2sinx dx?

    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?
  20. Z

    Euler-Lagrange equation on Lagrangian in generalized coordinates

    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...
  21. D

    MHB Euler-Lagrange equation first integral question

    \[ \int_0^1yy'dx \] 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.
  22. K

    Fermat principle and Euler-Lagrange question

    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...
  23. S

    Lagrangian and Euler-Lagrange equation question

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  24. F

    Virtual differentials approach to Euler-Lagrange eqn - necessary?

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  25. R

    Proving Euler-Lagrange for constrained system

    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...
  26. L

    Euler-Lagrange and Christoffel symbols

    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...
  27. M

    Derivation of Euler-Lagrange equation?

    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...
  28. H

    Applying the Euler-Lagrange equations, a special case

    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...
  29. B

    Euler-Lagrange Equations with constraint depend on 2nd derivative?

    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...
  30. L

    Understanding the Chain Rule in Euler-Lagrange Equations

    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
  31. P

    Equations of Motion using Euler-Lagrange Formalism for Small Vertical Motions

    (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.
  32. A

    Solving an ODE-45 from Euler-Lagrange Diff. Eqn.

    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) +...
  33. J

    Calculus of variations: Euler-Lagrange

    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})...
  34. E

    Euler-Lagrange Equations and Derivatives

    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...
  35. A

    Applications of the Euler-Lagrange Equation

    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...
  36. H

    Commutative finite ring and the Euler-Lagrange Theorem

    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...
  37. fluidistic

    Equation of motion via Euler-Lagrange equation

    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...
  38. Z

    Differential equation after using Euler-Lagrange equations

    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 \\...
  39. J

    Euler-Lagrange Brachistochrone Problem in rotating system

    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...
  40. S

    Euler-Lagrange equation derivation

    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...
  41. W

    Euler-Lagrange equation in vector notation

    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...
  42. snoopies622

    Simple application of euler-lagrange equation

    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...
  43. N

    Comparing Lagrange's Equation of Motion and Euler-Lagrange Equations

    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.
  44. M

    Derivation of first integral Euler-Lagrange equation

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  45. Rasalhague

    What is the significance of the Euler-Lagrange Equation in variational calculus?

    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]...
  46. E

    Finding the Function F in the Euler-Lagrange Equation

    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)...
  47. andrewkirk

    Gauge invariance of Euler-Lagrange equations

    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...
  48. J

    Euler-Lagrange Field Theory Question

    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 -...
  49. M

    What are the applications of Euler-Lagrange equations?

    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...
  50. A

    Euler-lagrange definition slipping my mind

    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...
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