Find the linear transformation between two coordinate systems on a two-dimensional flat surface that preserves the quantity $s^2 = x^2 − y^2$.
I can't figure out what it means to "preserve" this quantity and how to solve the question. Question from my classical mechanics class.
A hint by way of an analogy: If $r^2 = x^2 + y^2$ (the squared distance from the origin), then $r^2$ is preserved under any rotation, e.g. $$ R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}. $$
So if we define new coordinates $(X, Y)$ by $$ \begin{pmatrix} x \\ y \end{pmatrix} = R_\theta \begin{pmatrix} X \\ Y \end{pmatrix}, $$ then $$ r^2 = x^2 + y^2 = (X\cos\theta - Y\sin\theta)^2 + (X\sin\theta + Y\cos\theta)^2 = X^2 + Y^2 = R^2. $$
You can think of this geometrically as mapping a point on the circle of radius $r$ to a new point on the circle with the same radius.
For your exercise, replace the circles by hyperbolas. What transformation maps points on a given hyperbola $s^2 = x^2 - y^2$ to another point on the same hyperbola?