Let $\delta x$ be a line vector between two trajectories, could anyone tell me what is this transformation all about and why we would be interested in it?
$\delta z= \Phi(x,t) \delta x$
Reference: page 5, http://web.mit.edu/nsl/www/preprints/contraction.pdf
Thanks!
The idea seems to be that $z(x,t)$ is a set of new coordinates, for instance with the intention to map solutions to parallel lines or similar. Such a transformation might not be possible to construct in general, but one might be able to construct local approximations of such a transform. As in a local context the first interest is in the linear parts, one finds the new relative coordinates for $x+δx$ as $δz=\Theta(x,t)δx$. As, again, the intent is to make the close solutions parallel and equally spaced, a less time-dependent distance of the solutions can be defined via the Euclidean norm of $δz$, which induces a variable metric tensor in the $δx$ coordinates.
The rest of that section is concerned with translating back and forth between properties in $δz$ and $δx$ and what properties are desired in the construction of useful coordinate transformation matrices $Θ(x,t)$.