- #1

Epimetheus

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## Homework Statement

Given the parabolic co-ordinate system defined, given Cartesian coordinates x and y, as

[tex]$\mu=2xy$[/tex]

[tex]$\lambda = x^2-y^2,$[/tex]

find the inverse transformation [tex]x(\mu, \lambda)[/tex] and [tex]y(\mu,\lambda)[/tex].

## Homework Equations

None

## The Attempt at a Solution

We compute [tex]\lambda/\mu^2[/tex] in order to obtain

[tex]x=\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}[/tex]

and

[tex]y=\frac{2\mu}{\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}}[/tex]

However, I don't think this is right ... I'm following a set of CM lecture notes, which immediately go on to claim [tex]x^2+y^2=\lambda^2+\mu^2[/tex] and

[tex]\dot x^2 + \dot y^2 = \frac14 \frac{\dot \lambda^2 + \dot \mu^2}{\sqrt{\lambda^2 + \mu^2}} [/tex]

which looks tantalizingly similar to spherical polars, but doesn't seem to follow from what I have ...

Any thoughts on inverting general "nonlinear" co-ordinate systems (other than cylindrical and spherical)? What about finding the inverse transformation for the stereographic projection defined by

[tex]\frac{x}{\xi} = \frac{y}{\eta} = \frac1{1-\zeta}[/tex]

so that [tex]x=\frac{\xi}{1-\zeta}; y=\frac{\eta}{1-\zeta} [/tex]?

It's hard to get rid of the "old" set of variables in this case.