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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.

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Euler-Lagrange equation alternative form

I have the following exercise: Let $a,b,A,B\in \mathbb{R},a<b,f\in C^2\left(\left[a,b\right]\times \mathbb{R}\right),$ and $J:\mathcal{M}\rightarrow\mathbb{R}$ given by $$J\left(y\right)=\int_a^bf\...
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Conserved quantities in Lagrangian Dynamics

I am given the Lagrangian for a particle of mass $m$ orbiting a black hole of radius $r_h$: $$L=\frac{1}{2} m\left\{-F(r) c^{2} \dot{t}^{2}+F(r)^{-1} \dot{r}^{2}+r^{2} \dot{\phi}^{2}\right\}, \quad F(...
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Why can the Euler-Lagrange equation be used to find the extremum of a functional?

The Euler-Lagrange equation is a differential equation. A functional is a function from some vector space to a real number. Functions that maximize or minimize functionals may be found using the Euler–...
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Deriving the Euler-Lagrange Equation using the Gateaux Derivative

Can anyone explain how the professor goes from line 4 to 5 of the derivation? In particular, how is: $$\frac{\delta L}{\delta u}h'=-\frac{d}{dx}\frac{\delta L}{\delta u'}h$$ The professor states ...
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Euler Lagrange Equation with co-/ contravariant Tensors

I am currently trying to show the following equation which, appears from the Euler Lagrange equation: $$\frac{\partial}{\partial \partial_{\nu} A^{\mu} } \left( \frac{1}{4} F_{\mu \nu} F^{\mu \nu} \...
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Derivative of a special double integrale in the calculus of variation

Let $u$ be a real function defined on $[0, T]$ and the functional $$V(u) = \iint\limits_{[0,T]^2}f(x,y)u(x)u(y)\mathrm dx\mathrm dy$$ and $f(x,y)$ is symmetric , continuous and can be writen on this ...
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showing a different form of the Euler Lagrange Equation provided $f\in{C^2}$ and $y'\ne{0}$

Suppose that $f(x,y(x),y'(x))$ is s.t $f\in{C^2}$ and $y'(x)\ne{0}.$ I am trying to show that the Euler-Lagrange equation $\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'}=0$ ...
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Euler-Lagrange Equation for Kantorovich Dual Problem

Given two probability measures $\mu$ and $\nu$, the Kantorovich Dual problem for quadratic cost is to $$ \text{minimize} \quad \int \phi(x)d\mu + \int \psi(y)d\nu $$ over pairs $(\phi,\psi)\in L^1(d\...
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How to Solve a Calculus of Variation Problem with Terminal Conditions

I've been struggling to solve calculus of variation problems with terminal conditions. The current textbook i'm using for my course seems to only tangentially touch upon the methodology. Here is one ...
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d’Alembert Lagrangian for second rank tensors

Consider the Lorentz scalar Lagrange density $$\mathcal{L}=\eta^{\mu\nu}\partial_\mu T^{\alpha\beta}\partial_\nu T_{\alpha\beta}$$ for a second rank tensor whose contravariant and covariant ...
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How to find a function having following properties?

Let the function be $f(x)$ such that it is monotonic and continuous function It has maximum area between $x=0$ and $x=1$ $f'(x)$ = $f(x)(1-f(x))$ $f''(0.5) =0$ thus, gradient is maximum at 0.5. ...
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Constants of motion for a system of two points moving on spherical surface with a force depending only on their relative distance

Consider two points that moves only on a spherical surface of radius $R$. There is only a force between them that has a potential $U(d)$ where $d$ is the distance between the two points. What is the ...
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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. ...
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Euler Lagrange equations, chain rule troubles

When considering the first integral the chain rule is used on $$F(y,y’,x)$$ When we do this why do we not consider it as $$F(y(x))$$ As y’ is a function of y But instead as y’ being a separate ...
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How to restrict only top range of ellipse function, and what is its domain?

I am trying to graph the function of an ellipse that is: $$1=\frac{x^2}{49}+\frac{(y+1)^2}{9}$$. I want to make the horizontal ellipse's range $y \leq 0.838$. So, when I also have to write the domain ...
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Deriving Euler-Lagrange equation from minimization problem

Let $J[u]=\int_{x_1}^{x_2}g(x,u(x))\sqrt{1+(\frac{du}{dx})^2}dx$ where $g(x,u(x))$ is a smooth function. The following minimization problem is given: Find $u(x) $ subject to $u(x_1)=f_1$ and $u(x_2)=...
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Calculus of variation problem doesn't have a smooth solution

we define the minimization problem : $$\min J(x)=\int_{-1}^1x^2(t)[x'(t)-1]dt \quad \text{subject to }x(-1)=0 \quad \text{and} \quad x(1)=0$$ note that $0\leq J(x)$. and if we consider $x^*(t)=0 \...
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Differential Equation by Minimization

Suppose we want to solve $u + xu' = 0$, which has the general solution $u = \frac{C}{x}$, by minimizing the length squared of $u + xu'$. This should work due to the positive definite condition of ...
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Problem with Derivation of Motion Equations from Lagrangian

I intend to obtain the equations of motion of a sliding beam from a paper. I must obtain: \begin{multline} \frac{\partial }{\partial t} \frac{\partial L}{\partial \dot{q}_{i}} - \frac{\partial L}{\...
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Brachistochrone step from Advanced classical mechanics (Bagchi)

I'm not sure as to how the book got from: $dx=\frac{a+b}{2}\int{\sqrt{\frac{1+\cos(\theta)}{1-\cos(\theta)}}\sin(\theta)d\theta}$ to: $x=\frac{a+b}{2}(\theta-\sin(\theta))+constant$ where $a$ and $...
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Partial Derivatives of y and y'

In this application of the Euler-Lagrange equation, it is said that there is no $y$ in the function $\sqrt{1 + (y')^2}$. I see that the algorithm in progress treats $y'$ as unusually autonomous, as ...
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Find Extremum of Functional with No Initial Conditions

I have the following functional $ J(x)=\int\limits_{a}^{b}[{y(x)^2 + \dot{y}(x)^2 + 2 y(x) e^x}] dx $ and I want to find the extremum of it. So I compute the Euler-Lagrange Equation and I get the ...
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What to do when Euler Lagrange Equation is highly nonlinear ode

In $\mathbb{R}^3$, suppose there is a curve on X-Y plane $y(x)$ defined on $x\in [-a,a]$ satisfying: $y(x)\geqslant 0$; $y(-a)=y(a)=0.$ Rotate $y(x)$ along x-axis in $\mathbb{R}^3$ and get a solid ...
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Three dimensional example with variable endpoints

I'm studying calculus of variations and currently am trying to understand problems with variable endpoints in three dimensions. In two dimensions, my understanding is: For a functional $ I(y) = \...
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Computing Euler Lagrange Equation for a Certain Functional

Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by: $$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where ...
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Euler-Lagrange equations from a complex-valued Lagrangian

I've been looking without success for references describing a generalization of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” ...
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Calculus of variations with differential forms

I want to generalize calculus of variations with differential forms. Or better, I saw it somewhere some time ago, but now I cannot re-build it. Here is what I remember. Let be $(M, I, \Lambda)$ a ...
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Euler–Lagrange equation has no solutions

Show that the Euler–Lagrange equation for the functional: $$I(y) = \int_{0}^{1}y dx$$ subject to y(0) = y(1) = 0 has no solutions. Explain why no extremum for I exists. When forming the E-L ...
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Euler-Lagrange equations for dependent multiple functions

Find the extremals for the functional: $$ J(x) = \int_{0}^{1}\left[x\left(t\right)\dot{x}\left(t\right) + \ddot{x}^{2}\left(t\right)\right]\mathrm{d}t $$ where $x(0)=0$, $\dot{x}(0)=1$, $x(1)=2$, $...
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Differentiation of household’s lifetime utility

A household's lifetime utility is given by $$ V(\left\{ c_t , d_t\right\}^{\infty}_{t=0}) = \sum_{t=0}^{\infty} \beta^t \left( \log{c_t} + \gamma \log{d_t} \right) $$ The marginal rate of ...
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Converse of the Extremization Condition in Calculus of Variations

My understanding of the calculus of variations is as follows. Let $U=(a,b)\subset\mathbb{R}$ be an open interval and let $L:(C^2(U))^{2n} \times U\to C^2(U)$ be at least $C^2$-differentiable with ...
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Calculus of Variations (Gelfand & Fomin): Proof of Euler's Equation for Constrained Variation

I am in Section 2.12.1 of Calculus of Variations by Gelfand & Fomin. I am attempting to follow the proof of the Euler equation for Constrained Variation (Theorem 1, pg. 42). However, I'm confused ...
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Uniqueness of minimizer of Lagrangian

I have recently started calculus of variations and face the following task: Consider the Lagrangian $$L(y',y):=|y'|$$ on the space of curves $$U = \{y\in C^2([0;T];\mathbb R): y(0)=a, y(T=1)=b \}$$ ...
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Using Hamilton's principle to derive Newton's equations of motion in parabolic coordinates

I have recieved a very hard (optional) assignment on variational calculus, and I have not got a clue where to start other then stating the Euler-Lagrange equations. Here is the problem: According to ...
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In order to solve Euler Equation i get 1=0 what that's means

So I tried to understand what happened if the Lagrangian $L$ which minimize the action $S$ $$S=\int{L(x,u(x)) d x}$$ Doesn't depend on $ u'(x) $ and depend only on $u(x)$ For example let's looking ...
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How to derive the Euler Lagrange equation for geodesics?

In my book, it says a geodesic is associated to the functional $\int_0^l |\gamma'|^2$ , with a metric g. It then jumps to $\ddot{\gamma}^k + \Gamma^k_{ij}\dot{\gamma}^i\dot{\gamma}^j = 0$ where $\...
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Euler Lagrange and Geodesics

I’m trying to use the Euler Lagrange equations to derive the geodesic equations. I’ve assumed a lagrangian: $$ L = {1\over 2} g_{ij}\dot x^i \dot x^j $$ So one of the terms of the equation requires: ...
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Finding y(x) and extremals of functional with Euler equation

I want to use the Euler equation to find $y=y(x)$ to get the extremals for the following functional: $$\int_a^b \left[xy + 2\left(\frac{dy}{dx}\right)^2\right] \, \mathrm{d}x$$ I know that the Euler ...
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Euler-Lagrange for Anharmonic Oscillator

Suppose we are given some potential: $V(x)=\frac{1}{2}kx^2+\frac{1}{4}\lambda x^4$ where $k$ and $\lambda$ are constants. $\\$ i) I'm trying to find the Lagrangian and use Euler-Lagrange to find ...
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Interpreting Volterra Series Correctly

I was reading this and found on page 6 was a description of a generalization of Taylor series to linear functionals. Reproduced below as $$ F[\phi + \lambda] = \sum_{n=0}^{\infty} \left[\frac{1}{n!} ...
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Euler-Lagrange equations for maximization problem

Given $\Omega:=[x_a,x_b]\times[y_a,y_b]\subset\mathbb{R}^2$, consider the problem $$ \max_{f\in \mathcal{C}^2(\Omega, \mathbb{R})} \iint_\Omega \left[f_y(x,y)\int_{x_a}^{x}f(z,y)\,dz\right]\, dx\, dy $...
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Finding extremals of a functional with integral function as integrand

Assume you want to find the extremals for the functional $$ y \rightarrow \int_a^by(x)\left[\int_a^xy(\xi)\, d\xi\right]\, dx $$ where $[a,b]\subset \mathbb{R}$ and $y\in \mathcal{C}^1\left([a,b],\...
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How to apply d/dt to solve for equations of motion (Applying Lagrangian to Vibrations)

In my engineering vibrations course, I am encountering derivatives that are partial as well as total. I am a little rusty and am having a really hard time grasping why the differentiation with respect ...
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How to find system dynamic equation by using Lagrange's approach?

How can I find the dynamic equation using the Lagrange method?1
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How to find a Lagrange equation for below figure

How to find a Lagrange equation for below figure
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Lagrangian Equation for two masses moving on a incline plans [closed]

How to find equations of system dynamics using Lagrange’s approach
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Why $L(x,u,p)=\sqrt{\frac{1+p^2}{u}}$ does not depend on $x$, even when $u=u(x)$?

I'm reading an article which writes that a Lagrangian: $L(x,u,p)=\sqrt{\frac{1+p^2}{u}}$ does not depend on $x$, even when $u$ is function of $x$. So how does it not depend on $x$? Is it some ...
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Guaranteeing isoperimetry constraint for non-extremal functional in PDE.

First of all, hello and thank you for your time. Context I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[\mathbf{y}]=\int\...
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Solving the Euler-Lagrange equation for the brachistochrone problem with friction

This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence. I am asking for help understanding how ...
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euler-lagrange equation expansion

Euler-Lagrange equation $$\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'} = 0$$ Can also be written as $$f'_y-f''_{xy'}-f''_{yy'}y'-f''_{y'y'}y''=0$$ In my book it is ...