Euler Equation

Snowdog2112

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Apr 30, 2006
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I am having trouble even getting started on this problem.

Change the independent variable to simplify the Euler equation, and then find a first integral of it.

∫y*sqrt(y'^2+y^2)dx with limits of integration x1 to x2

Any help is appreciated.
 
Snowdog2112 said:
I am having trouble even getting started on this problem.

Change the independent variable to simplify the Euler equation, and then find a first integral of it.

∫y*sqrt(y'^2+y^2)dx with limits of integration x1 to x2

Any help is appreciated.

Since the Lagrangian does not explicitely depend on x, you can simplify things by writing down the Hamiltonian and use that as a function of x it must be constant. We have

\(\displaystyle L(y, y') = y\sqrt{y'^2+y^2}\)

The conjugate momentum is:

\(\displaystyle p_{y}= \frac{\partial L}{\partial y'}= \frac{yy'}{\sqrt{y'^2+y^2}}\)

And the Hamiltonian is:


\(\displaystyle H = p_{y}y' - L = -\frac{y^3}{\sqrt{y'^2+y^2}}\)

Since we always have: \(\displaystyle \frac{dH}{dx}=-\frac{\partial L}{\partial x}\) and in this case \(\displaystyle L\) does not explicitely depend on \(\displaystyle x\), you can put \(\displaystyle H\) equal to a constant and solve the first order differential equation for y.
 
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