Questions tagged [euler-lagrange-equation]

In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.

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Trying to maximize $\int_a^b L(t,q(t),\dot{q}(t)) dt$ subject to $|\dot{q}(t)| = 1$

I am trying to find the differential equation which implies a smooth path $q:[a,b] \rightarrow \mathbb{R}^n$ subject to $|\dot{q}(t)| = 1$ (i.e. $q$ has unit speed) is a stationary point of $$ \int_a^...
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Identical vanishing of the variational derivative

I see the following regarding null Lagrangians in Courant and Hilbert's book, in the chapter on the Calculus of Variations: The Euler differential expression for an integrand $F(x, y, y', …)$ may ...
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Tilted rectangle falling down

Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from ...
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About Euler-Lagrange equation and Beltrami identity. What is the error in the process?

If $F=y+y'$, then Euler-Lagrange equation $\frac{\partial F}{\partial y}-\frac{d}{dx}(\frac{\partial F}{\partial y'})=0$ is not satisfied. Hence, we may conclude that there is no extremal for the ...
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Lagrange eq is the geodesic equation if and only if the curve is parametrized by constant speed $∥\dot\gamma (t)∥$

Let $(M, g)$ be a Riemannian manifold. The length of an admissible curve $\gamma : [a, b] \to M$ is defined by $L(γ) = \int_a^b ∥\dot \gamma (t)∥dt$. 1)Compute the Euler–Langrange equations for the ...
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Finding the ideal B-spline through data points using Euler-Lagrange: is it just too hard to do?

I am not even sure I have a question anymore (I will just give up)... in the past month or so I have been researching cubic Bézier curves. The idea was to find a fit through data points, using ...
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Problems differentiating four-vectors to find the equation of motion using Euler-Lagrange equations.

This post directly follows this question and is very similar in nature: Consider the following Lagrangian: $$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\...
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Convex Lagrangian prevents oscillations - why?

We are given an interval $I \subset \mathbb{R},$ the space of $q-$absolutely continuous functions $AC^q(I,\mathbb{R}^{n})$ which satisfy continuity of the function and the existence of its $q-$...
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How should I show the following stationary path with a natural boundary condition at $x=0$?

Let $a>0, b>0, c>0$ be constants. Show that the stationary path of the functional $$S[y]=\int_{0}^{a}(y'^2+2by)dx, \quad y(a)=0,$$ with a natural boundary condition at $x=0$ and subject to ...
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Functional for the prescribed mean curvature

Let $F:M\to\overline M$ be an immersion of a manifold $M$ into a Riemannian manifold $(\overline M, \overline g)$ and let $H\in C^\infty(M)$ be a given function. I would like to find a functional $I_H(...
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How should I find the general solution of the Euler-Lagrange equations for $(z_{1}, z_{2})$?

Let the two coupled Euler-Lagrange equations giving the stationary path of the functional $S[y_{1}, y_{2}]=\int [y_{1}'^2+2y_{2}'^2+(2y_{1}+y_{2})^2]dx$ be $y_{1}''-2(2y_{1}+y_{2})=0$ and $2y_{2}''-(...
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Deriving correct integral constraint equation for calculus of variations problem

I have the following calculus of variations problem: $$\mathcal{L}=-2X'(t)\ln{x'(t)}-2Y'(t)\ln{y'(t)}$$ where $x(t)$ and $y(t)$ are the functions I'm interested in, and $X(t)$ and $Y(t)$ are given as ...
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Derivation of Euler-Lagrange by Taylor's Formula

From Bruce van Brunts's book the derivation for the Euler-Lagrange equation uses the Taylor's theorem in the following way: Given $\hat{y}=y+\epsilon\eta$, $J(\hat{y})-J(y) = \int_{x_0}^{x_1} f(x,\hat{...
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Prove that a Lagrangian that Induces a Linear Elliptic Euler-Lagrange PDE has Unique Form

I am asking if existence, taken as an assumption for a solution $L$ to the linear operator equation $$\mathcal{E}L = F$$ with conditions on $F$, implies further conditions on $F$ and uniqueness of a ...
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How to take the gradient of this function similar to quadratic form?

This question is about Lagrangian multi-rigid body dynamics equation. forward kinematics is $$ x=G(q) $$ first-order differential forward kinematics equation is $$ \dot x= J(q) \dot q $$ Lagrangian ...
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Calculus of Variations no y' in function (?)

I'm a high school student who attempted to use calculus of variations for a project. I have this functional: $$T(\theta)=\int_{0}^{57}\left(\frac{1}{10\cos\left(\theta\right)+w\left(x\right)}+λ\frac{...
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How am I factorizing the $\frac{d}{dx}$ operator while trying to solve for the equation of a catenoid bubble?

I've been working on my Math IA for a while, its a project part of the IB course which requires me to write a ~20 page math paper investigating and exploring something within maths. I chose to write ...
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Calculus of Variations Weierstrass-Erdmann Gelfand and Fomin 3.17

The problem is 3.17 from Calculus of Variations by Gelfand and Fomin. It states Find the extremals of the functional $$ J[y]=\int_0^4(y'-1)^2(y'+1)^2dx,\;\;\;\;\;\;y(0)=0,\;y(4)=2$$ which have just ...
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Problem with variation of calculus for noether

In the variational calculus, we discuss general case variation where endpoints vary as well. (i.e $x_0$ and $x_1$ vary as well). By using some calculations, we end up with: $\delta J = \left[\frac{\...
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Optimizing the functional $\int_{x_0}^{x_1}(2xy+(y^{\prime\prime\prime})^2)dx$

Find the function $y$ with which the functional $$J=\int_{x_0}^{x_1}(2xy+(y^{\prime\prime\prime})^2)dx$$ has extremum. Here's what I've been suggested so far: we use $\delta J=0$ with any $\delta y$, ...
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Trouble with deriving the Euler-Poisson equation using Euler's approach

"A First Course in the Calculus of Variations" by Mark Kot, §2.2. Euler’s approach describes a method for deriving the Euler-Lagrange equation which I shall call "geometric". It is ...
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Finding Euler-Lagrange equation

Find the Euler-Lagrange equation for $\int_0^4(tx'-(x')^2)dt$. My attempt: $0 - \frac{d}{dt}(t-2x')=0$ From here, I am not sure what to do. Quite frankly, I don't even think I started the problem ...
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Is it possible to write Lagrangian mechanics for a catenary curve?

A catenary is formed when a rope or chain hangs freely in the effect of gravity supported by two ends. The shape is a result of optimisation of potential energy along the length of the rope. This kind ...
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Uniform Convergence of Lagrange function defined for $q-$absolutely continuous curves

In my course on Nonlinear PDEs, we investigate the integral functional $\int_{t_0}^{t_1} L(\dot{\gamma}(t), \gamma(t))dt$, defined on $q$-absolutely continuous curves $\gamma: I \rightarrow \mathbb{R} ...
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How should I show that the stationary path of the functional is given by the following with the given constant?

Let $a<b$ and let $f(x)$ be a continuously differentiable function on the interval $[a, b]$ with $f(x)>0$ for all $x\in [a, b]$. Let $A>0, B>0$ be constants. Show that the stationary path ...
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Transform a differential equation into Hamiltonian form

I am currently doing the following exercise in the book Modern Geometry - Methods and Applications Part I by Dubrovin, Fomenko, and Novikov. Exercise 33.4.1: Consider the differential equation \begin{...
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How to show that this functional has this stationary path by solving this second-order differential equation?

Let $n>1$ be a positive integer. Show that the functional $$S[y]=\int_{0}^{1}(y')^{n}e^{y}dx, \quad y(0)=1,\quad y(1)=A>1,$$ has a stationary path given by $$y=n \ln(cx+e^{1/n}),$$ where $$c=e^{...
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Is there a potential energy formula for a catenary curve?

I am trying to find the minimum energy point of a catenary curve. Can anyone please let me know whether there is a potential energy or energy formula for a catenary curve ? Especially I would like to ...
vbalaji21's user avatar
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How should I find this first-integral of a functional?

Let $0<a<b$. Consider the functional $$S[y]=\int_{a}^{b}x^5(y'^2-\frac{2}{3}y^3)dx$$ Prove that a first-integral of $S[y]$ is $$4x^5yy'+x^6(y'^2+\frac{2}{3}y^3)=c,$$ where $c$ is constant, ...
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Necessary optimality conditions: $\min_{\phi} \sum_\xi\int_{0}^{1} p(a,\xi) T(V_{w(a,\xi)}[\phi]) da$, where $V[\cdot]$ is an evaluation functional

I'm trying to set up a dynamic optimization problem as follows. Let $\mathcal{W} := [\underline{w},\overline{w}]$, and $w:[0,1]\times\{0,\dots,N\}\to \mathcal{W}$ \begin{align} \min_{\phi: \mathcal{W}\...
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Minimizing $\int_1^2\frac{1}{x}\sqrt{1+u^{\prime}(x)^2}dx$

I want to minimize $\int_1^2\frac{1}{x}\sqrt{1+u^{\prime}(x)^2}dx$ so that $u(1)=0$ and $u(2)=1$. Using the Euler-Lagrange-equation I obtain $u(x)=2-\sqrt{5-x^2}$. Futhermore, the second variation of $...
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Do vector calculus identities prove symmetries in Euler-Lagrange equations?

Let $\mathrm{S}$ be a differentiable manifold, and $T\mathrm{S}$ its tangent bundle. Let $\mathcal{L}:\mathbb{R}\times T\mathrm{S}\rightarrow \mathbb{R}$ be a Lagrangian functional $\mathcal{L}=\...
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A bar with a point sliding on it

Given a smooth bar of mass $M$ and of length $L$ attached to the ceiling in a horizontal position and given a point of mass $m$ on the bar placed in the point of attachment of the bar, find the ...
Denis D. Bavrin's user avatar
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Minimizing a functional subject to boundary conditions

I want to find a smooth function $y : [0,1] \to \mathbb{R}$ that minimizes $$ S = \int_0^1 (y(x)y'''(x) + 3 y'(x) y''(x))^2 dx $$ subject to the constraints / boundary conditions: (i) $y(0)=1$, (ii) $...
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Euler Lagrange Equation on special brachistochrone

I have to use the Euler Lagrange Equation on a special form of the brachistochrone, which includes gravity. So the formular would be: $$ T[y]=\int_{a}^{b}\tfrac{\sqrt{1+y'((x))^2}}{\sqrt{(y(x)g(x)}} $$...
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How should I derive the Euler-Lagrange equation by using the Gateaux differential?

Find the Gateaux differential for the functional $ S[y]=y(0)+\frac{1}{2}\int_{0}^{1}(4y'^2+x^2y^2)dx, y(1)=2 $ and use it to derive the Euler-Lagrange equation. Be sure to specify all boundary ...
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Equivalent Formulations of Variational Problems

This post is supposed to collect some Theorems and techniques which can be used to analyse variational problems by (a) finding a related variational problem s.t. their optimal values are the same or ...
Small Deviation's user avatar
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Euler Lagrange equation of a functional on the space of traceless symmetric matrices

Let $\mathcal{S}_0:\{Q\in \mathbb{R}^{3\times 3}:\text{tr}Q=0 \hspace{5pt}\text{and}\hspace{5pt} Q_{ij}=Q_{ji} \hspace{5pt} \text{for any $i,j=1,2,3$}\}$ and $$ \widetilde{ \mathcal{E}}(Q) =\int_{\...
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Proof of Identity for Lagrangians

My textbook states (in a derivation of the Euler-Lagrange equation) the following which I would like to understand: \begin{equation*} S[\bar{x}+\delta x] =\int_{t_a}^{t_b} L(\dot{x}+\delta \dot{x}, x+\...
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Test whether the extremal is unique or not

If $y(t)$ is a stationary function of $$J[y]=\int_{-1}^1(1-x^2)y'^2\,dx$$ with B.C. $y(-1)=1, y(1)=1$ subject to $\int_{-1}^1y^2=1$. Then which is correct ? (A) $y$ is unique. (B) $y$ is always a ...
Empty's user avatar
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The doubts about variational calculus [closed]

Forgive me my maybe-non-understanding questions, but this topic seemed quite hard to grasp and the only choice I think I got is ask it as if what I think I believe and you can point me where I'm wrong....
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Confusion over equation of motion from lagrangian

I'm currently reading Topological Solitons by Manton & Sutcliffe and am having a bit of trouble with deriving an equation of motion. Suppose $M$ is a smooth manifold of dimension $D$ and $\mathbf{...
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Can a coupled Lagrangian produce uncoupled EOM's?

If we have a set of N-dimensional vectors $A_i$, and a Lagrangian with a set of NxN square matrices $M_{ij}\neq 0$: $$L = \sum_{ij} A_i M_{ij} A_j\tag{1}$$ That couples these vectors, can this produce ...
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Euler-Lagrange equations for a double integral

Consider a functional $$ J[f] = \int_{a}^{b} \int_{a}^{y} \frac{e^{f(y) - f(x)}}{f'(x)} g(x,y) \, dx \,dy, $$ where the constants $a$, $b$ and the function $g$ are given. I am looking for a function $...
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Pendulum with Oscillating Fulcrum with Newton's Laws?

I was given an honors project to solve for the equations of motions of a pendulum with an oscillating fulcrum. I (somewhat) understand the procedure on how to solve it with lagrangian mechanics and ...
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solution of Brachistochrone Problem with friction

from https://mathworld.wolfram.com/BrachistochroneProblem.html I found the EL equation (29) and the parametric solution equations $~(32)~,(33)~$. Eq. (29) \begin{align*} &{~(1+y'^2)\,(1+\mu\,y')+...
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Minimising $\int_{-\infty}^{x_0}\cos\left(C-f(x)\right)e^{x-x_0}\mathop{\mathrm dx}$

For an analytic and bounded $f:\mathbb{R}_{\leq x_0}\rightarrow\mathbb{R}$, consider the integral \begin{equation} I=\int_{-\infty}^{x_0}\cos\left(C-f(x)\right)e^{x-x_0}\mathop{\mathrm dx}, \end{...
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Derivative of a curve on tangent bundle $TQ$ and second-order equations (where is the acceleration?)

This is very basic question that I should have resolved long ago but didn't and it still plagues me. Let $TQ$ be the tangent bundle of some (configuration) manifold, $Q$, and let $(\pmb{q},\pmb{v}) = (...
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Check the second variation is positive for brachistochrone problem.

When studying the brachistochrone problem, I was asked to check the second variation and verifying the cycloid is, indeed, a minimal path. I know the second variation is to check that: $$\frac{\...
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Minimizer over set of $C^1$ functions

Exercise (1.4) of Renardy and Rogers: "An introduction to partial differential equations" asks to Show that there is an infinite family of minimizers of $$ J(u) = \int_0^1 (1-u'(t)^2)^2\,...
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