For those with an interest in both cycling and physics, the Wikipedia article “Bicycle and motorcycle dynamics” is well worth reading. It is interesting to note that lateral movements of bicycles (basically, those involving turning) are so mathematically complex that they require “two coupled, second-order differential equations… to capture the principle motions” and that these equations cannot produce exact solutions.
That contrasts in an interesting way with the experience of making turns at speed on a bicycle, and the appreciation one gains for the relationship between body movements, bicycle movements, and the condition of the ground.
The article undermines the suggestion that an activity is as “easy as riding a bicycle”. The article is one of the longer articles I have read in Wikipedia.
I following a gradual downhill curve at higher speeds (say over 50 kph) , I have found it useful to turn the inner knee way out as a motorcyclists does a turn. This shift “turns” the bike, without the need to actually turn the handlebars. Examples of where this works are gradual going down mountain descents (eg. Seymour or Cypress Mountain, Duffy Lake Road). It does not work on hairpin turns where the only safe way to make the turn that I have found is to seriously decelerate.
Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions, functions that make the equation hold true. Only the simplest differential equations admit solutions given by explicit formulas. Many properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.