Why isn't differential Galois theory widely used?

differential-geometry differential-algebra soft-question galois-theory integral-geometry

Ellis Kolchin developed differential Galois theory in the 1950s. It seems to be a powerful tool that can decide the solvability and the form of the solutions to a given differential equation.

Why isn't differential Galois theory widely used in differential geometry? It is plausible that we can solve some problems of differential/integral geometry using this theory.

So, what is the major pullback in this theory that prevents its wide application to other fields rather than discrete geometry (e.g., Diophantine geometry)?


Solution 1:

Among others, there is a nice concrete application differential Galois theory to the Non-Integrability of Hamiltonian Systems :

http://www.springer.com/us/book/9783034807203

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