Contact Transformation Problem

1. Oct 14, 2008

neelakash

1. The problem statement, all variables and given/known data

If an inﬁnitesimal segment of a curve (dy,dx) in a 2-D space, having a slope p, is transformed to (dY,dX) with a new slope P, ﬁnd the transformation to do this. Such a transformation should be called a ‘contact transformation’.

2. Relevant equations

3. The attempt at a solution

I assumed X=X(x,y) and Y=Y(x,y)

Then, dY=(∂Y/∂x)dx+(∂Y/∂y)dy and dX=(∂X/∂x)dx+(∂X/∂y)dy with (dY/dX)=P

Again, dy=(∂y/∂X)dX+(∂y/∂Y)dY and dx=(∂x/∂X)dX+(∂x/∂Y)dY with (dy/dx)=p

To specify the transformation,I need to specify the matrix elements: a11,a12,a21,a22
which are respectively,[the matrix is (dX,dY)=Matrix(dx,dy)]

(∂X/∂x),(∂X,∂y),(∂Y/∂x),(∂Y/∂y)

The above equations gave me four relations:

(∂Y/∂x)=P(∂X/∂x)...A

(∂Y/∂y)=P(∂X/∂y)...B

(∂y/∂X)=p(∂x/∂X)...C

(∂y/∂Y)=p(∂x/∂Y)...D

Do these relations (I notice 4 equations and 4 unknowns) give unique solutions?...Somehow, I am making mess with the solution part.Actually, though there are 4 equations,I am not sure if I can write the inverse of C and D as

(∂X/∂x)=p(∂X/∂y)...C'

(∂Y/∂x)=p(∂Y/∂y)...D'

Can anyone please suggest the significance of the "contact transformation" term?