3.1 Motivation and outline: What is causing motion?
Every now and then I make the experience that I sit in a train, reading a book. Then I look out of the window, realize that we are passing a train, feeling happy that we are further approaching my final destination; and then I realize that the train is moving and my train is still in the station. Indeed, the motion of objects in my compartment is exactly identical, no matter whether it is at rest or moves with a constant velocity; be it zero in the station, at 15 m/s in a local commuter train, or 75 m/s in a Japanese high-speed train. However, changes of velocity matter. I forcefully experience the change of speed of the train during an emergency break, and coffee is spilled when it takes too sharp a turn. Modern physics was born when Galileo and Newton formalized this experience by saying that bodies (e.g. the set of bodies in the compartment of a train) move in a straight line with a constant velocity as long as there is no net force acting on the bodies, and that the change of its velocity is proportional to the applied force.
Outline
In the first part of this chapter we will relate temporal changes of positions and velocities to time derivatives. Subsequently, we can formulate equations of motion that relate these changes to forces. The last part of the chapter deals with strategies to find solutions by making use of conservation laws.
| mass | $m$ |
| position | $ \mathbf q(t)$ |
| velocity | $\dot {\mathbf q}(t),\mathbf v(t) $ |
| acceleration | $\ddot {\mathbf q}(t)$ |
| forces | $\mathbf F_\alpha (\mathbf q,t)$ |
Table 3.1: Notations adopted to describe the motion of a particle. A single dot denotes the time derivative, and double dot the second derivative with respect to time.