# Tagged Questions

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### Calculus of Variations

In the Calculus of Variations there is a passage from Euler's characteristic equation: $$\frac {\partial F}{\partial y} - \frac {d}{dx} \left(\frac {\partial F}{\partial y'} \right)=0$$ in ...
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### Calculus of variation: Reduce the order of a differential equation using a 1 parameter lie group adfmitted by it.

We are asked to reduce $y^"+y-y^{-3}=0$ using $X= \sin2x\frac{\partial}{\partial x}+y\cos2x\frac{\partial}{\partial y}$ I know we have to find the first prolongation of X and solve $X^1$F=0 using ...
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### Restricted function

Let $A=\{(x,y): x,y\in(-1,1)\}$. Is there a function $f:A\mapsto A$ such that $f(x,0)=(x,x^2)$ $f$ differentiable and bijective on $A$. I have tried a lot of constructions but the problem is in ...
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### Simplification of Euler-Lagrange equation when integral independent of y

I'm supposed to show, that if my function $f(y,y',x)$ is independent of $y$, then the Euler-Lagrange equation turns out to be $\partial f/\partial y' = const$. Now, the Euler-Lagrange equation is ...
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### Find the derivatives to transformed variables

Let $\theta \in \mathbb{R}$ and consider the rotational action $X = x \cos\theta - y \sin\theta$ ; $Y = x \sin\theta + y \cos\theta$. Find the transformed derivatives $Y'$ and $Y''$. How do I ...