H-principle

In mathematics, the homotopy principle (h-principle) is a very general way to solve partial differential equations (PDE), and more generally partial differential relations (PDR). The h-principle is good for underdetermined PDE or PDR such as immersion problem, isometric immersions problem and so on.

The simplest example

A position of a car on the plane is determined by three parameters: two coordinates x and y for the location (best choice is the location of mid point of back wheels), and an angle lpha which describes the orientation of the car. The motion of the car satisfies the equation

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:dot x sinlpha=dot ycos lpha.,

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A non-holonomic solution in this case roughly speaking corresponds to a motion of a car by sliding on the plane. In this case the non-holonomic solutions are not only homotopic to holonomic ones but also can be arbitrarily well approximated by the holonomic ones (by going back and forth, like parallel parking in a limited space).

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This last property is stronger than the general h-principle: it is the so called C^0-dense h-principle.

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