 Each point of \(X\) represents a state of the ball (because it gives the ball’s position and momentum). Accordingly, the time evolution of the ball’s state is represented by a line in \(X\), a so-called phase space trajectory (from now on ‘trajectory’), showing where in phase space the system was at each instant of time. For instance, let us assume that at time \(t = 0\) the ball is located at point \(x_1\) and then moves to \(x_2\) where it arrives at time \(t = 5\). This motion is represented in \(X\) by the line segment connecting points \(\gamma_1\) and \(\gamma_2\). In other words, the motion of the ball is represented in \(X\) by the motion of a point representing the ball’s (instantaneous) state, and all the states that the ball is in over the course of a certain period of time jointly form a trajectory. The motion of this point has a name: it is the phase flow \(\phi_t\). The phase flow tells us where the ball is at some later time \(t\) if we specify where it is at \(t = 0\); or, metaphorically speaking, \(\phi_t\) drags the ball’s state around in \(X\) so that the movement of the state represents the motion of the real ball. In other words, \(\phi_t\) is a mathematical representation of the system’s time evolution. The state of the ball at time \(t = 0\) is commonly referred to as the initial condition. \(\phi_t\) then tells us, for every point in phase space, how this point evolves if it is chosen as an initial condition. In our concrete example, point \(\gamma_1\) is the initial condition and we have \(\gamma_2 = \phi_{t=5}(\gamma_1)\). More generally, let us call the ball’s initial condition \(\gamma_0\) and let \(\gamma(t)\) be its state at some later time \(t\). Then we have \(\gamma(t) = \phi_t (\gamma_0)\). This is illustrated in figure 2a.