The phase space coordinate of any given particle can be written as a 6D "vector" consisting of (x, px, y, py, z, pz).
Assuming |p| is constant for every particle, you can make a canonical transformation to the variables a = px/|p|, b = py/|p|, and c = pz/|p|.
Note that these three new variables represent tangents of angles between the momentum vector and the three Cartesian coordinate axes.
Now your phase space vector is (x, a, y, b, z, c).
As each particle moves freely, Newton's first law says that they move along straight lines (angles fixed, coordinates changing).
You can express the transformation of the size phase space coordinates in matrix form:
[; \begin{bmatrix}
1 & L_1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & L_2 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & L_3 \\
0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix} ;].
You can see that this matrix has determinant 1, so it preserves the total phase space volume.