classical mechanics - Integrable vs. Non-Integrable systems

Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities ($n$ being the number of degrees of freedom), or $n$ whose Poisson brackets with each other are zero.

The way I understand it, these conditions correspond directly to us being able to do the Hamilton-Jacobi transformation, which is roughly equivalent to saying that the $2n-1$ conserved quantities are the initial conditions of the problem, which itself is a way of saying that the map from the phase space position at some time $t_0$ to that at time $t$ is invertible. But, if the last statement is right, why are there systems which are non-integrable at all? Shouldn't all systems' trajectories be uniquely determined by the equations of motion and initial conditions? Or is it that all non-integrable systems are those whose Lagrangians can't be written (non-holonomic constraints, friction etc)?

I've heard that Poincare proved that the gravitational three-body problem in two dimensions was non-integrable, but he showed that there were too few analytic conserved quantities. I don't know why exactly that means non-integrability, so if someone could help me there that would be great too.

Answer

Let there be given a $2n$-dimenional real symplectic manifold $(M,\omega)$ with a globally defined real function $H:M\times[t_i,t_f] \to \mathbb{R}$, which we will call the Hamiltonian. The time evolution is governed by Hamilton's (or equivalently Liouville's) equations of motion. Here $t\in[t_i,t_f]$ is time.

1. 
   

   On one hand, there is the notion of complete integrability, aka. Liouville integrability, or sometimes just called integrability. This means that there exist $n$ independent globally defined real functions

   $$I_i, \qquad i\in\{1, \ldots, n\},$$ (which we will call action variables), that pairwise Poisson commute, $$\{I_i,I_j\}_{PB}~=~0, \qquad i,j\in\{1, \ldots, n\}.$$
   
   


2. 
   

   On the other hand, given a fixed point $x_{(0)}\in M$, under mild regularity assumptions, there always exists locally (in a sufficiently small open Darboux$^1$ neighborhood of $x_{(0)}$) an $n$-parameter complete solution for Hamilton's principal function

   $$S(q^1, \ldots, q^n; I_1, \ldots,I_n; t)$$ to the Hamilton-Jacobi equation, where $$I_i, \qquad i\in\{1, \ldots, n\},$$
   are integration constants. This leads to a local version of property 1.
   
   

The main point is that the global property 1 is rare, while the local property 2 is generic.

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$^1$ A Darboux neighborhood here means a neighborhood where there exists a set of canonical coordinates aka. Darboux coordinates $(q^1, \ldots, q^n;p_1, \ldots, p_n)$, cf. Darboux' Theorem.