Why are Hamilton systems almost never integrable?

How can you prove that a Hamilton system cannot be integrated?

The explicit proof that a any Hamilton system is an open problem.

For some classes of Hamilton systems (e.g. systems in one plane) it is possible to use the Non-integrability of the system using the theorems of Poincare, Burns, Ziglin, and Yoshida (and generalizations) explicit to prove.

For example there is a theorem from Poincare:

For a Hamiltonian of the form:

1: H. = p2x + p2y2 + V. (X, y)

If the Hamilton operator (1) is a isolated periodic solution , is the system not integrable (in particular there it no second integral of motion that is independent of H.)

On the importance of isolated periodic solution regarding the Poincare method, see for example here and here

Ziglin's theorem has more extensive uses:

If the Hamilton system (1) can be integrated and there is a Monodromy matrix thereΔ from the monodromy group of vertical equation of variation , then each other Monodromic matrixΔ 'must oscillate with Δ or its eigenvalues ​​must be me, -i

Yoshida's theorem includes Hamilton systems with homogeneous opus potentials (see here for a generalization)

Related approaches include the Painleve properties and the characterization of the equations of motion (e.g. here, here and here).

In addition, there are approaches to integrability which include the differential Galois theory (ie the Galois theory for differential equations), whereby one uses the analogy Solvability -> Has integrability . This approach can also unify various other approaches (e.g. here and here).