What do straight lines in a phase space diagram depict? This question surfaced up in my mid terms today .
 A: Phase space is used for simplifying and understanding some physical concepts/mathematical analysis. For example in phase space representing SHM or EM Wave diffraction/superposition, the length of the straight line represents the amplitude and the angle with the axis represents the phase or the initial phase of the system.
So there is a homework for you to figure out what is phase and what actually is the phasor. I hope this will guide you in right direction, as I can't give definite answers on this forum.
For more information please go through following.
https://en.m.wikipedia.org/wiki/Phase_space
A: The rectification theorem says that away from singular points, the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M, the dynamical system is integrable. The rectification theorem secures the local existence and uniqueness of solutions of ODEs. 
Research action-angle variables and integrability of ODEs.
