- Homotopy principle
In
mathematics , the homotopy principle (or h-principle) is a very general way to solvepartial differential equation s (PDEs), and more generallypartial differential relation s (PDRs). The h-principle is good forunderdetermined PDEs or PDRs, such as occur in theimmersion problem ,isometric immersion problem , and other areas.The theory was started by works of
Yakov Eliashberg ,Mikhael Gromov andAnthony V. Phillips . It was based on earlier results ofMorris W. Hirsch ,Nicolaas Kuiper ,John Forbes Nash , andStephen Smale .Rough idea
Assume we want to find a function f on Rm which satisfies a partial differential equation of degree k, in co-ordinates u_1,u_2,...,u_m). One can rewrite it as
:Psi(u_1,u_2,...,u_m, J^k_f)=0!,
where J^k_f stands for all partial derivatives of f up to order k. Let us exchange every variable in J^k_f for new independent variables y_1,y_2,...,y_N.Then our original equation can be thought as a system of:Psi^{}_{}(u_1,u_2,...u_m,y_1,y_2,...y_N)=0!,and some number of equations of the following type :y_j={partial y_iover partial u_k}.!,
A solution of :Psi^{}_{}(u_1,u_2,...u_m,y_1,y_2,...y_N)=0!,is called a non-holonomic solution, and a solution of the system (which is a solution of our original PDE) is called a holonomic solution.In order to check if a solution exists, first check if there is a non-holonomic solution (usually it is quite easy and if not then our original equation did not have any solutions).
A PDE "satisfies the h-principle" if any non-holonomic solution can be deformed into a holonomic one in the class of non-holonomic solutions.
Therefore, once you prove that an equation satisfies the h-principle, it is really easy to check whether it has solutions. It is surprising that most underdetermined partial differential equations satisfy the h-principle.
The simplest example
The position of a car in the plane is determined by three parameters: two coordinates x and y for the location (a good choice is the location of the midpoint between the back wheels) and an angle alpha which describes the orientation of the car. The motion of the car satisfies the equation
:dot x sinalpha=dot ycos alpha.,
A non-holonomic solution in this case, roughly speaking, corresponds to a motion of the car by sliding in 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). This last property is stronger than the general h-principle; it is called the C^0-dense h-principle.Ways to prove the h-principle
......
ome paradoxes
Here we list a few counter-intuitive results which can be proved by applying the h-principle:
1. Let us consider functions "f" on R2 without origin "f"("x") = |"x"|. Then there is a continuous one-parameter family of functions f_t such that f_0=f, f_1=-fand for any t, operatorname{grad}(f_t) is not zero at any point.
2. Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.
3.
Smale's paradox can be done using C^1 isometric embedding of S^2.Related theorems
*
Smale's paradox
*Nash embedding theorem
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