How to prove that a hamiltonian system is not integrable?

To show that a system is integrable, we just need to find $N$ independent functions $f_j$ such that $\{ f_i, f_j \} = 0$.

But how to prove that such a set of functions do not exist? For example, how to do this for the three-body problem?

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Explicitly proving non-integrability of an arbitrary Hamiltonian system is an open problem.

For some classes of Hamiltonian systems (e.g systems on a plane) is possible to prove explicitly the non-integrability of the system, using theorems of Poincare, Burns, Ziglin and Yoshida (and generalizations).

For example there is a theorem of Poincare:

For a hamiltonian of the form:

1:    $H = \frac{p_x^2 + p_y^2}{2} + V(x,y)$

If the hamiltonian (1) can have an isolated periodic solution, then the system is not integrable (specificaly there does not exist a second integral of motion that is independent of the $H$)

For the meaning of isolated periodic solution in relation to Poincare's method see for example here and here

The theorem of Ziglin has more extensive applications:

If the Hamiltonian system (1) is integrable, and there is a monodromy matrix $\Delta$ from the monodromy group of the vertical variations equation, then any other monodromy matrix $\Delta'$ must commute with $\Delta$ or its eigenvalues must be $i, -i$

Yoshida's theorem involves hamiltonian systems with homogeneopus potentials (for example see here for a generalisation)

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

Furthermore there are approaches to integrability which involve Differential Galois theory (i.e Galois theory for differential equations) where one has the analogy solvability -> integrability. This approach can also unify various other approaches (e.g here and here)

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