The physics of the motion of bodies has been a source of mathematical problems and methods ever since Newton gave us his equations of motion. The availability of ultra-intense lasers, together with special relativity, has put a new wrinkle on this: one is now confronted with the mathematical problem of understanding the motion of particles suffering ultra-extreme accelerations, yet their velocities may not exceed the speed of light. Dynamical systems of this kind are (i) fundamental to laser physics, (ii) are simple yet nontrivial mathematically , and (iii) require methods from mathematics and physics which are both local and global. We shall illustrate these observations with two fundamental classes of dynamical systems: (1) those driven by pulsed and (2) those driven by steady lasers.

There exist three distinct paths that lead to different aspects of one's understanding of dynamical systems: 1. geometry, 2. physics, and 3. analysis. Each has its outstanding virtues, which we shall illustrate on the same laser-driven dynamical system. The geometrical way is expressed by curvature and geodesic deviation. The way of physics is expressed by parametrically coupled pendulums. And the analysis way is expressed by the phase plane approach of an autonomous system. A theoretical mathematician/physicist should be familiar with all three.