Editing Homotopy principle (section)

==Rough idea==

Assume we want to find a function <math>f</math> on <math>\mathbb{R}^m</math> which satisfies a partial differential equation of degree <math>k</math>, in coordinates <math>(u_1,u_2,\dots,u_m)</math>. One can rewrite it as

:<math>\Psi(u_1,u_2,\dots,u_m, J^k_f)=0</math>

where <math>J^k_f</math> stands for all partial derivatives of <math>f</math> up to order&nbsp;<math>k</math>. Exchanging every variable in <math>J^k_f</math> for new independent variables <math>y_1,y_2,\dots,y_N</math> turns our equations into

:<math>\Psi^{}_{}(u_1,u_2,\dots,u_m,y_1,y_2,\dots,y_N)=0</math>

and some number of equations of the type 
:<math>y_j={\partial^k f\over \partial u_{j_1}\ldots\partial u_{j_k}}.</math>

A solution of

:<math>\Psi^{}_{}(u_1,u_2,\dots,u_m,y_1,y_2,\dots,y_N)=0</math>

is called a '''non-holonomic solution''', and a solution of the system which is also solution of our original PDE is called a '''holonomic solution'''.

In order to check whether a solution to our original equation exists, one can first check if there is a non-holonomic solution. Usually this is quite easy, and if there is no non-holonomic solution, then our original equation did not have any solutions.

A PDE ''satisfies the <math>h</math>-principle'' if any non-holonomic solution can be [[homotopy|deformed]] into a holonomic one in the class of non-holonomic solutions. Thus in the presence of h-principle, a differential topological problem reduces to an algebraic topological problem. More explicitly this means that apart from the topological obstruction there is no other obstruction to the existence of a holonomic solution. The topological problem of finding a ''non-holonomic solution'' is much easier to handle and can be addressed with the [[obstruction theory]] for topological bundles.

While many underdetermined partial differential equations satisfy the h-principle, the falsity of one is also an interesting statement. Intuitively this means that the objects being studied have non-trivial geometry which can not be reduced to topology. As an example, embedded [[Lagrangian submanifold|Lagrangians]] in a [[symplectic manifold]] do not satisfy an h-principle, to prove this one can for instance find invariants coming from [[Pseudoholomorphic curve|pseudo-holomorphic curves]].