Question on finding Noethr's theorem and how to find the constant

[ppy]

Hi,

I have attached a file. I am stuck on question 1 on how to find Noether's constant. The solutions are provided however I do not see what they have done. It states that η$^{x}$=1 and η$^{y}$=-1 I do not understand how we know this. I can see that $\xi$=0 because else the -x$^{2}$-y$^{2}$-2xy term will not cancel.


Thanks,
Melissa Poole

 

[Sunfire]

It states that η$^{x}$=1 and η$^{y}$=-1 I do not understand how we know this.

Hello,
in the general case we seek invariance under all coordinate transformations of this type:
$\tilde{t} = t + \epsilon \xi$
$\tilde{x} = x + \epsilon \eta_x$
$\tilde{y} = y + \epsilon \eta_y$

which in this particular problem is given as
$\tilde{t} = t$, thus $\xi = 0$
$\tilde{x} = x + \epsilon$, thus $\eta_x = 1$
$\tilde{y} = y - \epsilon$, thus $\eta_y = -1$