Contact Transformation Problem

[neelakash]

Homework Statement

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’.

Homework Equations

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?

 

[mighty2000]

Hello,

Thank you for your post. It seems like you have made good progress in finding the transformation for the given scenario. To answer your question about the significance of the term "contact transformation," this refers to a specific type of transformation in mathematics where the derivatives of a function are preserved. In this case, the transformation you are finding is a contact transformation because the derivatives of the original curve are preserved in the transformed curve. This type of transformation is useful in various applications, such as in differential geometry and physics.

Regarding your solution, it looks like you have a system of four equations with four unknowns, which means you should be able to find a unique solution. However, it is important to check that the equations are consistent and do not lead to any contradictions. It might also be helpful to graph the original and transformed curves to visualize the transformation and verify that it preserves the derivatives.

I hope this helps clarify the concept of contact transformation and provides some guidance for your solution. Good luck!