Place particles in a field of arrows and watch direction become motion.
The rotation field (−y,x) has circular trajectories. The source field (x,y) scales both coordinates by e^t. The saddle field (x,−y) expands the x coordinate by e^t and contracts y by e^(−t). These analytic solutions satisfy the differential equations exactly; the animation samples them in time.
Arrow lengths in this view are normalized for clarity. Particle positions use the actual vector field, so the arrows do not display speed.
Arrow lengths are normalized to show direction; particles follow the actual field speed using exact solutions. Trails outside the displayed window are clipped. A replay returns to the initial unit circle.