# 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. ### I The Euler-Lagrange equation and the Beltrami identity

This question is specifically about deriving the Beltrami identity. Just to give this question context I provide an example of a problem that is solved with Calculus of Variations: find the shape of a soap film that stretches between two coaxial rings. For the surface area the expression to be...
2. ### Determine whether ## S[y] ## has a maximum or a minimum

a) The Euler-Lagrange equation is of the form ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ##. Let ## F(x, y, y')=(y'^2+w^2y^2+2y(a \sin(wx)+b \sinh(wx))) ##. Then ## \frac{\partial F}{\partial y'}=2y' ## and ## \frac{\partial F}{\partial...
3. ### I Confused about applying the Euler–Lagrange equation

Hello! I have a Lagrangian of the form: $$L = \frac{mv^2}{2}+f(v)v$$ where ##f(v)## is a function of the velocity. I would like to derive the equation of motion in general, without writing down an expression for ##f(v)## yet. I have that ##\frac{\partial L}{\partial x} = 0##. However, what is...
4. ### Euler-Lagrange equation: pulley system

##m_{A} = 3 kg## ##m_{B} = 2 kg## ##y_{A} + y_{B} = c \Leftrightarrow y_{A} = c - y_{B}##, where c is a constant. ##\Rightarrow \dot{y_{A}} = -\dot{y_{B}}## The Lagrangian: $$L = T - V$$ ##T =\frac{1}{2}m_{A}\dot{y_{B}}^{2} + \frac{1}{2}m_{B}\dot{y_{B}}^{2}## ##V = m_{A}g(c - y_{B}) +...
5. ### Cartesian and polar coordinate in Simple pendulum, Euler-Lagrange

$$L = \frac {mv^2}{2} - mgy$$ It is clear that ##\dot{x}=\dot{\theta}L## and ##y=-Lcos \theta##. After substituting these two equations to Lagrange equation, we will get the answer by simply using this equation: $$\frac {d} {dt} \frac {∂L}{∂\dot{\theta}} - \frac {∂L}{∂\theta }= 0$$ But, What if...
6. ### I Exploring the Flexibility of Coordinates in Euler-Lagrange Equations

Hello all, so I’ve been reading Jennifer Coopersmith’s The Lazy Universe: An Introduction to the Principle of Least Action, and on page 72 it says: If I understand it right, she’s saying that in our Euler-Lagrange equation ## \frac {\partial L} {\partial q} - \frac {d} {dt} \frac {\partial L}...
7. ### 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. ### 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. ### I More rigorous Euler-Lagrange derivation

Sorry if there are other threads on this, but after a discussion with a friend on this (im in the mountains, so no books, and my googlefu isn't helping), I realize that my understanding of the variational principles arent exactly... great! So, maybe some one can help. Start with a functional...
10. ### I Discrete Euler-Lagrange equations

I want to derive the discrete EL equations $$\frac{d}{dt} \frac{\partial L}{\partial \dot \phi_a^{(i j k)}} - \frac{\partial L}{\partial \phi_a^{(i j k)}} = 0$$ We deal with a Lagrange density which only depends on the fields themselves and their first order derivatives. We discretize space...
11. ### B Significance of the solution of the Euler-Lagrange equation

I am new to Lagrangian mechanics and I have gone through basic examples of solving the Euler Lagrange equation for simple pendulums or projectiles and things like that. But I am unable to understand what we are exactly solving the equation for or what is the significance of the differential...
12. ### Hamiltonian of this system?

Let a mass m charged with q, attached to a spring with constant factor k = mω ^2 in an electric field E(t) = E0(t/τ) x since t=0. (Equilibrium position is x0 and the deformation obeys ξ = x - x0) What would the hamiltonian and motion equations be in t ≥ 0, in terms of m and ω?? Despise magnetic...
13. ### Euler-Lagrange equations and the principle of least action

Summary:: I am missing something in my integration by parts Consider the infinitesimal variation of the fields ##\phi_a (x)## $$\phi_a \rightarrow \phi_a + \delta \phi_a$$ The infinitesimal variation vanishes at the boundary of the region considered (ie. ##\delta \phi (x) = 0## at the...
14. ### What is the derivative of velocity with respect to position?

I'm reading a book on Classical Mechanics (No Nonsense Classical Mechanics) and one particular section has me a bit puzzled. The author is using the Euler-Lagrange equation to calculate the equation of motion for a system which has the Lagrangian shown in figure 1. The process can be seen in...
15. ### I Do 4-divergences affect the eqs of motion for nth order perturbed fields?

Intuitively, I'd say that adding a 4-divergence to the Lagrangian should not affect the eqs of motion since the integral of that 4-divergence (of a vector that vanishes at ∞) can be rewritten as a surface term equal to zero, but... In some theories, the addition of a term that is equal to zero...
16. ### A Lagrangian to the Euler-Lagrange equation

Hello all, I understand the formation of the Lagrangian is: Kinetic Energy minus the potential energy. (I realize one cannot prove this: it is a "principle" and it provides a verifiable equation of motion. Moving on... One inserts the Lagrangian into the form of the "Action" and minimizes it...
17. ### Euler-Lagrange equation

My question : I am wondering about definition of a function. when ##y_x = (\frac{b+y}{a-y})^2## Why in this book is defined solution ##y = y(x)## in from ## y = y(θ(x))## . And have a relationship in the form ## y = \frac{1}{2} (a-b) - \frac{1}{2} (a+b) cosθ ## ...
18. ### I Apply Conservation Law to Euler-Lagrange Equation

In my most recent thread, I discussed the conservation law involving the 4-velocity vector: gab(dxa/dτ)(dxb/dτ) = -c2 Now, I've read that you can apply this law to the Euler-Lagrange equation in order to get some equations that are apparently equivalent to the geodesic equations. Now here is...
19. ### Finding the Lagrangian Matrix for Two-Spring Systems

Homework Statement The problem is attached. I'm working on the second system with the masses on a linear spring (not the first system). I think I solved part (a), but I'm not sure if I did what it was asking for. I'm not sure exactly what the question means by the... L=.5Tnn-.5Vnn. Namely, I'm...
20. ### I Derivation of the Euler-Lagrangian

I have a question about a very specific step in the derivation of Euler-Lagrangian. Sorry if it seems simple and trivial. I present the question in the course of the derivation. Given: $$\begin{split} F &=\int_{x_a}^{x_b} g(f,f_x,x) dx \end{split}$$ Thus...
21. ### Null geodesics given line element

Hi, I'm the given the following line element: ds^2=\Big(1-\frac{2m}{r}\large)d\tau ^2+\Big(1-\frac{2m}{r}\large)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2) And I'm asked to calculate the null geodesics. I know that in order to do that I have to solve the Euler-Lagrange equations. For...
22. ### Euler-Lagrange Equation for Several Dependent Variables

Homework Statement [/B] Homework Equations $$f_u- \frac{d}{dx} \left(f_{u'} \right) = 0$$ $$f_v- \frac{d}{dx} \left(f_{v'} \right) = 0$$ The Attempt at a Solution So I calculated the following, if someone could check what I've done it would be greatly appreciated, but I'm not convinced...
23. ### I Origin of the half factor in Euler-Lagrange for geodesics

I was wondering where does the 1/2 factor come from in the Euler-Lagrange equation, that is: L = \sqrt{g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu} implies that \partial_\mu L = \pm \frac{1}{2} (\partial_\mu g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu ) I'm not sure I entirely understand where it comes...
24. ### I Does a covariant version of Euler-Lagrange exist?

Hello everyone. I've seen the usual Euler-Lagrange equation for lagrangians that depend on a vector field and its first derivatives. In curved space the equation looks the same, you just replace the lagrangian density for {-g}½ times the lagrangian density. I noticed that you can replace...
25. ### 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...
26. ### A Euler-Lagrange Equations: EM Field Term

This problem is about one small step in the derivation of Maxwell's equations in free space from the field Lagrangian. The Lagrangian contains a term proportional to ##\partial \mu A_\nu \partial^\mu A^\nu - \partial \nu A\mu \partial ^\mu A^\nu## where A is the four-vector potential. The...
27. ### Solving Geodesic Equations with Euler-Lagrange and Noether's Theorem

Homework Statement Homework Equations There are 5 equations we can use. We have the fact that Lagrangian is a constant for an affinely parameterised geodesic- 0 in this case for a light ray : ##L=0## And then the Euler-Lagrange equation for each of the 4 variables. The Attempt at a Solution...

31. ### 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...
32. ### I Euler-Lagrange Equation Q: Where Does 2mr˙r˙θ Come From?

On the following post, where it says q=θ for the Euler-Lagrange equation where does the 2mr˙r˙θ come from? https://www.physicsforums.com/threads/variable-length-pendulum.204840/
33. ### MHB Derivation of Euler-Lagrange equations w/ Hamilton's equations

I've got a problem that asks us to derive the Euler-Lagrange equations by only using Hamilton's equations and the definition of the Hamiltonian in terms of the Lagrangian. Here's what I tried: The Hamiltonian is defined as \begin{align*} \mathcal{H} = \dot{q}_ip_i - \mathcal{L} \end{align*}...
34. ### A Generalising the Euler-Lagrange equation for scalar fields

The Euler-Lagrange equation obtained from the action ##S=\int\ d^{4}x\ \mathcal{L}(\phi,\partial_{\mu}\phi)## is ##\frac{\partial\mathcal{L}}{\partial\phi}-\partial_{\mu}\big(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\big)=0##. My goal is to generalise the Euler-Lagrange equation...
35. ### 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?
36. ### I Derivation of Euler-Lagrange equation with change of coordinates

Why isn't ##\frac{\partial L}{\partial t}\frac{\partial t}{\partial \dot{q_m}}## included in (5.41), given that ##L## could depend on ##t## explicitly?
37. ### I Help with Euler-Lagrange Equation

I have begun teaching myself Lagrangian field theory in preparation for taking the plunge into quantum field theory ( it's just a hobby, not any kind of formal course ). When working through exercises, I have run across the following issue which I don't quite understand. I am being given a...
38. ### Lagrangian mechanics: Bar connected to a spring

Homework Statement Mass 1 can slide on a vertical rod under the influence of a constant gravitational force and and is connected to the rod via a spring with the spring konstant k and rest length 0. A mass 2 is connected to mass 1 via a rod of length L (forms a 90 degree angel with the first...
39. ### Minimisation Problem (Euler-Lagrange equation)

Homework Statement http://i.imgur.com/BV5gR8q.png Homework Equations d/dx ∂F/∂y'=∂F/∂y The Attempt at a Solution I have no problem with the first bit, but the second bit is where I get stuck. Since the question says the speed is proportional to distance, I have taken v(x)=cx where c is some...
40. ### I Any inflexion-point solutions to Euler-Lagrange equation?

The following pages use Euler-Lagrange equation to solve for the shortest distance between two points and in the last paragraph mentions: "the straight line has only been proved to be an extremum path". I believe the solution to the Euler-Lagrange equation gives the total length ##I## a...
41. ### I Euler-Lagrange equation even if the action isn't stationary?

It seems like I could get the Euler-Lagrange equation for any function that allows symmetry of second derivatives even when the action is not stationary. Suppose ##L=L(q_1, q_2, ... , q_n, \dot{q_1}, \dot{q_2}, ... , \dot{q_n}, t)##, where all the ##q_i##'s and ##\dot{q_i}##'s are functions of...
42. ### Euler-Lagrange question about strange differentiation

I'm watching Susskind's Classical Mech. YouTube lecture series and am really confused about something he's doing where otherwise I've followed everything up until this point without a problem. In Lecture 3 he's dealing with the Euler-Lagrange equation applied to minimizing the distance between...
43. ### Initial Conditions Applied to a Lagrangian

Homework Statement The scenario is a pendulum of length l and mass m2 attached to a mass of m1 which is allowed to slide along the horizontal with no friction. The support mass moves along in the X direction and the pendulum swings through the x-y plane with an angle θ with the vertical. After...
44. ### Calculus of variations

Hello, here is my problem.http://imgur.com/VAu2sXl'][/PLAIN] [Broken] http://imgur.com/VAu2sXl My confusion lies in, why those particular partial derivatives are chosen to be acted upon the auxiliary function and then how they are put together to get the Euler-Lagrange equation? My guess is...
45. ### Euler Lagrange Derivation (Taylor Series)

Mod note: Moved from Homework section 1. Homework Statement Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way:  L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] +...
46. ### Sloppy Derivation of Euler-Lagrange.

Just wondering how much validity there is to this derivation, or if it's just a convenient coincidence that this works. We have a Lagrangian dependent on position and velocity: \mathcal{L} (x, \dot{x}) Let's say now that we've perturbed the system a bit so we now have: \mathcal{L} (x +...
47. ### The point of the Euler-Lagrange equation?

When trying to come up with the geodesic equation for a sphere I came across this equation. My question, is this equation just a short cut so we don't have to integrate and differentiate with two variables. Here is the equation...
48. ### Finding the Euler-Lagrange equation for a special pendulum

Homework Statement Homework Equations Euler - Lagrange equation: ##\frac{d}{dt}(\frac{\partial L}{\partial \dot\theta})=\frac{\partial L }{\partial \theta}## Hamilton's equations: ##\frac{\partial H}{\partial \theta}=-p_{\theta}\text{ and }\frac{\partial H}{\partial...
49. ### Lagrangian and Euler-Lagrange of a Simple Pendulum

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 >...
50. ### MHB How do I solve the Euler-Lagrange equation for this functional?

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...