Integration

Updating initial conditions...

Around 350 years ago or so, when mathematics was being done with a quill pen on parchment, Isacc Newton and some associates were highly motivated to find a short cut to finding the area under a curve. In principle it is a simple matter. One just divides the area up into 10,000 or so little strips, calculates the area of each strip and adds them together. The inventors of integral calculus found a way to replace thousands of trivial calculations with one complicated calculation called integration.

Now we have computers that like nothing better than doing simple math very fast. There is still a use for integtral calculus but we do not need to be concerned with that. In taking dynamical systems from an initial state to a future state we say we are integrating but we are really multiplying and adding a huge number of times.

The way it works is we total up the forces acting on the system's moving parts and divide by the appropriate mass to get the acceleration in accordance with Newton's second law. Multiplying that acceleration by a tiny amount of time, we get the change in velocity of the moving part. Multiplying that by the same tiny amount of time we get the change in position of the moving part during that time increment. We add the change in velocity to the initial velocity and the change in position to the initial position and viola, as the French say, we have the new state of the system at the end of the time increment. Finally we add the time increment to the initial time to advance the time parameter and go through the whole process again... and again... and again until we have advanced time to any desired value. We can make the time increment as small as needed to achieve any required accuracy.

When we break time up into little increments the distinction between integration and iteration becomes a bit fuzzy. One might think of the new state of a system after a time increment as the initial conditions for the next time increment so the dynamical system sort of iterates itself along the path of the moving parts as the time independent mathematical systems iterate their way from initial conditions to subsequent states.