Setup
The phase space of the billiard flow $\left\lbrace \Phi^t \right\rbrace$ ($t \in \mathbb{R}$) is given by $\Omega = \mathcal{D} \times S^1$ where $\mathcal{D}$ is a planar billiard table in $\mathbb{R}^2$ and $S^1$ the circle.
Suppose $\mathcal{D}$ is bounded, then $\Omega$ is a torus whose cross section is $\mathcal{D}$. Specifying the location of the billiard ball by $q \in \mathcal{D}$ and its velocity vector $v \in S^1$, we call the natural projection $\pi_q(\Phi^t x), x \in \Omega$ the billiard trajectory on $\mathcal{D}$. This formulation is more or less a paraphrasing of that given on page 26 of the book Chaotic Billiards by Chernov and Markarian.
Question(s)
What does the billiard flow actually look like in $\Omega$ before we project it onto the table? Is it just some curve which 'flows' around in $\Omega$? Is information about the velocity $v$ lost in this projection?
Motivation
I am interested in how we might calculate, numerically (via simulation) the Lyapunov exponents for the flow of a billiard. While I realise this can be done via the billiard map (induced by the flow), I would like a more hands on approach. Such an approach (I assume) would amount to numerical calculations of the derivative of the projected flow $\pi_q(\Phi^t x)$ on $\mathcal{D}$.
Thanks!
The "natural projection" or the "dynamics on the base" corresponds simply to forget the velocity.
The projection on the base is a movement with constant velocity until it reaches the boundary where you need to specify some reflection rule. The canon would be to have equal speed but symmetric angle (at the boundary). See for example the drawings in https://en.wikipedia.org/wiki/Dynamical_billiards.
In order to study the Lyapunov exponents it is really better to consider only what happens on the boundary $\partial\mathcal D$.