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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.

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. Reference: Wikipedia.

It was developed by Swiss-Russian mathematician Leonhard Euler and French-Italian mathematician Joseph-Louis Lagrange in the 1750s.

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On Euler-Lagrange equations

Let's say I have the functional $F(x(\cdot))=\int_a^b \sqrt{f(\dot{x}(s),x(s))}\,ds$ where $x(t):(a,b)\rightarrow\mathbb{R}^n$ and let's consider $G(x(\cdot))=\int_a^b f(\dot{x}(s),x(s))\,ds.$ Then, under what conditions are the minimun of $F$ and…
miraunpajaro
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Computing the Euler-Lagrange Equation for a Lagrangian in Two Variables

$$\mathcal L \left( t, x, y, u(x,y,t),\frac{\partial u}{\partial t}, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right) = −\frac{1}{2} \left( \frac{\partial u}{\partial t} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial…
Mathscat
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Special forms of the Lagrangian

One of my textbooks includes the following question. Find the most general Lagrangian function of the form ... $$ L(t,x,x ̇)=f(t,x)x ̇^2 ̇+g(t,x) x ̇^3+h(t,x)x ̇^4 $$ ... for which the extremals are straight lines. Now superficially it doesn't look…
Bardfast
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How to solve the Euler-Lagrange to minimise the time taken to traverse this path

Let this image be in the x,y(x) plane. I am trying to compute the minimum time taken to get from A to B. I have determined that, using Euler-Lagrange methods,$$t=\sqrt{\frac{R}{g}}\int_{x_A}^{x_B}\frac{\sqrt{1+y'^2}}{\sqrt{R^2-x^2-y^2}}dx,$$ where…
ODP
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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: $$ T[y]=\int_{a}^{b}\tfrac{\sqrt{1+y'((x))^2}}{\sqrt{(y(x)g(x)}} $$ I then have the…
Bastian Sommerfeld
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Functional Minimization with constraints

Suppose you have a multi-variable functional given as $$ \int_{t_0}^{t_1} L(t,x,x', x'' ..., y, y', y'', ... ) \ dt $$ That you wish to to optimize [i.e find an $x$ and $y$ that maximize] but subject to a constraint $$ \Omega(t,x,x', x'', ... y,y',…
Sidharth Ghoshal
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Derivation of the Euler-Lagrange Equation and the Principle of Least Action

. Hello, everyone I have some trouble with the calculus in the derivation of the Euler-Lagrange equation for the principle of least action. Specifically: $\newcommand{\lagr}{\mathcal{L}}$ $\delta S = \delta \int_{t_1}^{t_2}\lagr(q,\dot{q},t)dt=0 $…
euler132
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Lagrangian & conservation of energy

I'm working through Garrity's "E & M for Mathematicians." An exercise is to show that if the Lagrangian $L = T - U$, where $T$ is a function of $x'$, $y'$ and $z'$, and $U$ a function of $x$, $y$, and $z$ (i.e., $T$ and $U$ both not explicit…
JoeJeffrey
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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')+2(y-\mu\,x)\,y''=0}\tag 1 \end{align*} Eq.…
Eli
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Mechanics: generalized kinetic energy

I have recently started reading “ Classical Dynamics of Particles and Systems “ by Marion and Thornton. Page 258 gives us a general expression for kinetic energy. The end result of the derivation is given by: Kinetic energy =$ \sum_{a} \sum_{i,j,k}…
user927859
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Euler-Lagrange equation and local maximum

I am struggling to prove that my EL equation gives a local maximum. (if possible, I want to show it is a global maximum as well). It satisfies the Legendre necessary condition for maximum, but I cannot show for sufficient conditions. I checked…
Anonymously lost student
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Equations of Motion to follow an trajectory

There are two points $\vec{x}(t)$ and $\vec{u}(t)$ that change over time. We are interested in the time interval $t \in [0, T]$. The positions and velocities of both points at time $t=0$ are known. The entire trajectory of the second point…
Aleksejs Fomins
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Obtaining the Lagrangian Formulation for a Linearized System

I have a system of one NLS equation and one Poisson equation: $$ i \dfrac{\partial u}{\partial Z} + \dfrac{1}{2}\nabla^2 u + 2 \ \Phi \ u = 0 \\ \nu \nabla^2 \, \Phi - 2 \, q \, \Phi + 2 \mid u \mid^2 \: = \: 0\\ $$ Where u is an envelope…
Cassandra Chaya Khan
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Euler-Lagrange equation for a given functional

I have a certain functional, very similar to the following: $\mathcal{L}=\int \limits_{\Omega} \left\{- b(x)\ c(x)\ + \int \limits_{\Omega '} x x' [c(x)-c(x')] \ \mathrm{d}x'\right\} \mathrm{d}x =\int \limits_{\Omega} \Lambda \ \mathrm{d}x$ where…
Riccardo
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parametrize curve on 2d plane using angle and apply euler lagrange equation

I want to solve the problem: Find the curve satisfies following conditions. Minimize the functional $J$ The coordinates of the start/end points are given Direction(tangential vector) at the start/end points are given The length of the curve is…
damhiya
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