# Aristotelian physics

Aristotle lived 384 BC – 322 BC and maybe was the first inventor of physical laws of motion. However, he got it wrong and the reason why he got it wrong can be found in friction. If we ride a bicycle, we need to pedal continuously to keep the speed constant, thus Aristotle thought, that if we apply a certain force on an object of mass $m$, the object will (instantaneously) move with a resulting constant velocity. According to Aristotle the (wrong) law of motion is \begin{align} F=m \cdot v \end{align} To consider the discrete dynamics of a system governed by this law, we first rewrite the system as a difference equation. \begin{align} F=m \cdot \dot x=m \frac{x(t+\Delta t) - x(t)}{\Delta t} \end{align} Rewriting this equation we get a discrete law that describes the motion of the system \begin{align} x(t+\Delta t)=x(t)+\frac{F \Delta t}{m}=x(t)+v \cdot \Delta t \end{align} with the velocity \begin{align} v =\frac{F }{m} \end{align} Notice in order to predict the future we just need to know, where we are now, thus one initial condition would be enough to predict the future. In continuous form we get for a negative force, which is proportional to the deviation from the origin (spring) \begin{align} F=-kx=m \cdot \dot{x} \end{align} the equation of motion \begin{align} x(t)=x_0 \cdot e^{-k/m \cdot t} \end{align} The constant $x_0$ is determined by the initial condition. Aristotle's law is irreversible in the sense that, once the origin is reached you can't retrodict where the particle came from. This is of course different in Newtons equations, that needs two initial conditions to predict the future - you also need to know where you came from.