A symplectomorphism is defined as a function $f$ fulfilling
$$ f^* \omega = \omega \quad ,$$
where $\omega$ is the canonical form on some even-dimensional space. In particular, the jacobian of any symplectomorphism $ T:(q,p) \mapsto (x,y)$ is a symplectic matrix, i.e.
$$ \left[ \frac{\partial(x,y)}{\partial(q,p)} \right]^T \mathcal{J} \left[ \frac{\partial(x,y)}{\partial(q,p)} \right] = \mathcal{J} \quad , $$
where ${\,\,\,}^T$ denotes the matrix transpose and $\mathcal{J}$ is the canonical symplectic matrix. I am working with a class of systems for which a specific coordinate transformation is very promising to simplify a lot of stuff, but unfortunately it is not symplectic. Its jacobian obeys
$$ \left[ \frac{\partial T(q,p)}{\partial(q,p)} \right]^T \mathcal{J} \left[ \frac{\partial T(q,p)}{\partial(q,p)} \right] = g(q,p) \, \mathcal{J} \quad , $$
such that the correct hamiltonian vector fields can be extracted just as if the mapping were symplectic, but substituting $\imath_X \omega$ by $g(q,p) \imath_X \omega$ in Hamilton's equations.
I remember reading somewhere about these symplectomorphisms with a multiplier, but I could never find the reference again. Does anyone have any information about these objects?
