Question regarding the ergodic hypothesis or why does a trajectory in phase space come infinitely close to every allowed point in phase space According to the ergodic hypothesis a system of $N$ particles represented by a point in $6N$ dimensional phase space comes infenetly close to every point in that phase space.
It is argued that the trajectories in phase space cannot cross and therefore the trajectory of a system has to discover every point of the phase space if the energy is conserved. The trajectory would swipe over the whole hyper surface defined by the Hamiltonian $H(\vec{q},\vec{p})$ of the system and the Energy $E$ of the system.
$$H(\vec{q},\vec{p})=E$$
This sounds reasonable, but one could imagine a situation where a system spirals into one point of the phase space. Since infinitely many trajectories fit in any infenetly small phase space volume, I fail to see how such a system of $N$ particles has to discover the whole phase space.
 A: The ergodic hypothesis is not, in general, true. It is true for some systems, for some initial conditions.
The particular concern you raise is that there might be "sinks" in phase space - regions that trajectories enter, but never leave. For a finite-dimensional Hamiltonian system this is ruled out by Liouville's theorem, which is equivalent to the statement that the integral of the flow over any closed hypersurface is zero. However, a Hamiltonian system might have conserved quantities other than energy. In this case trajectories cannot explore the full energy hypersurface.
However, one can show that Hamiltonian systems do explore the entire region of phase space compatible with the values of all conserved quantities. The argument for this is almost tautological: For the path with initial condition $(q_0,p_0)$, we can define a function $f_{(q_0,p_0)}(q,p)$ which is $1$ if the path passes through $(q,p)$ and is $0$ otherwise. Then $f$ is conserved, and our path explores the whole region of phase space that satisfies the constraint $f = 1$.
The interesting question is that of finding complete sets of conserved quantities. If we can find a finite set of conserved quantities $\{g_i\}$ such that every $f_{(q_0,p_0)}$ can be written as a function of the $\{g_i\}$, then we can guarantee that the ergodic hypothesis is satisfied on the level sets of the $\{g_i\}$.
Examples:
For the simple harmonic oscillator, $\{E\}$ is a complete set of conserved quantities all by itself. Trajectories therefor explore all points on the appropriate energy hyposurface.
For the 2D simple harmonic oscillator, $\{E_x,E_y\}$ is a complete set. Trajectories therefore don't explore all points on the appropriate energy hyposurface.
