# 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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### I have a Variation of Calculus, Euler Lagrange, Lagrange Multiplier problem and I don't know how to continue.

So I can't continue with my work. Don't know if I am just not seeing something under my nose or something. Given formula: $$I=\int\limits_0^{\pi/2}[(x'_2)^2-(x_2)^2]dt$$ with auxiliary constraints and ...
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### What is the most generic way to write a Lagrangian quadratic in velocities?

I'm trying to generalize the expression for a Lagrangian to a manifold that doesn't posses a metric. To be more clear, when the configurations space has a metric, we write the lagrangian quadratic in ...
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### Find curve that minimizes lenght, with integral constraint

I'm interested in finding the curve $q(t):[0,1] \rightarrow \mathbb{R}^+$ that satisfies the boundary conditions $q(0)=q(1)=0$, the integral condition $\int_0^1q(t)dt=a>0$, and that minimizes the ...
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### How to solve this functional $\int\int_\Omega \big((z_x)^2+(z_y)^2\big)dxdy$

Let $\Omega$ define the quadrant $0\leq x\leq L$, $0\leq y\leq L$. For $z=z(x,y)$ we want to solve Eulers equation for the functional: $\int\int_\Omega \big((z_x)^2+(z_y)^2\big)dxdy$ where $z=0$ along ...
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### Functional Poincaré Lemma for second-order PDE $\Delta u(x) + F(\nabla u(x))=0$?

Suppose I can write a second-order elliptic PDE for an unknown function $u:\mathbb R^n\to\mathbb R$ in the form $$\Delta u(x) + F(\nabla u(x))=0\qquad\forall x\in\mathbb R^n.$$ Under what conditions ...
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### Euler-Lagrange equation - cannot find mistake

I have a problem at hand that I'm trying to solve with the Euler-Lagrange equation, but I think I made a mistake somewhere. Supposed I have two functions on two real variables, $u(x, t)$ and $v(x,t)$. ...
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### Decide whether the extremal value of $I[y]$ is maximum or minimum

My question asks me to find $y(t)$ such that the integral $$I[y] = \int_0^1 (dy/dx)^2 dx.$$ is extremised subject to the constraint \int x(dy/dx) dx = 0 \...
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### Why $u=1$ is not a minimizer of $I(u)=\int_{0}^{1}u^2(2-u)^2dx$?
I'm trying to find the minimizer of the following functional: $$I(u)=\int_{0}^{1}u^2(2-u)^2dx,$$ where $u\in Y,$ and $Y=\{u\in C([0,1]):u(0)=1,u(1)=1\}.$ The Euler-Lagrange equation of this question ...