An orthogonal transformation preserves a symmetric bilinear form. A symplectic transformation can be defined as a linear transformation that preserves a skew-symmetric bilinear form on a $2n$-dimensional vector space. Is there a similar definition for a contact transformation? What I'm looking for is a definition of a contact transformation that does not rely on any differential structure.
See http://en.wikipedia.org/wiki/Contact_geometry for the background info.
In Darboux coordinates the contact form is $$ \boldsymbol{\omega}=dz-p_i\,dq^i\,. $$ A prime example of a transformation that leaves $\boldsymbol{\omega}$ invariant is the Legendre transformation $$ P_i=q^i\,,\quad Q^i=-p_i\,,\quad Z=z-p_i\,q^i\,. $$ Proof. $$\require{cancel} dZ-P_i\,dQ^i=dz-p_i\,dq^i\cancel{-q^i\,dp_i+q^i\,dp_i}=\boldsymbol{\omega}\,. $$ This is mentioned in the Historical Remarks section of the Wikipedia article. It is a bit astonishing that the simple proof is not spelled out there.