To give a complete description of the state of a 3-dimensional N-particle system at any given time it is necessary to specify the 3N coordinates and 3N momenta. The 6N dimensional space of coordinates and momenta is called phase space (or Γ-space). As time progresses the phase point Γ, traces out a path which we call the phase space trajectory of the system. As the equations of motion for Γ are 6N first order differential equations, there are 6N constants of integration (they may be for example the 6N initial conditions Γ(0)). Rewriting the equations of motion in terms of these constants shows that the trajectory of Γ is completely determined by specifying these 6N constants. An alternate description of the time evolution of the system is given by the trajectory in the extended Γ'-space, where Γ' = (Γ,t). As the 6N initial conditions uniquely determine the trajectory, two points in phase space with different initial conditions form distinct non-intersecting trajectories in Γ'-space.
Figure 3.2 Phase Space Trajectory 6N-dimensional Γ-space. As time evolves the system traces out a trajectory in 6N-dimensional Γ-space.
To illustrate the ideas of Γ-space and Γ'-space it is useful to consider one of the simplest mechanical systems, the harmonic oscillator. The Hamiltonian for the harmonic oscillator is H = ½(kx2 + p2∕m) where m is the mass of the oscillator and k is the spring constant. The equations of motion are
(3.33)
and the energy (or the Hamiltonian) is a constant of the motion. The Γ-space for this system is 2-dimensional (x,p) and the Γ-space trajectory is given by
(3.34)
The constants x0 and p0 are the 2 integration constants written in this case as an initial condition. The frequency ω is related to the spring constant and mass by ω2 = k∕m. The Γ-space trajectory is an ellipse,
(3.35)
which intercepts the x-axis at ±(x20+p20∕m2ω2)½ and the p-axis at ±(p20+ m2ω2x20)½. The period of the motion is T = 2Π∕ω = 2Π(m∕k)½. This is the surface of constant energy for the harmonic oscillator. Any oscillator with the same energy must traverse the same Γ-space trajectory, that is another oscillator with the same energy, but different initial starting points (x0,p0) will follow the same ellipse but with a different initial phase angle.
The trajectory in Γ'-space is a elliptical coil, and the constant energy surface in Γ'-space is a elliptical cylinder, and oscillators with the same energy start from different points on the ellipse at time zero (corresponding to different initial phase angles), and wind around the elliptical cylinder. The trajectories in Γ'-space are non-intersecting. If two trajectories in Γ'-space meet at time t, then the two trajectories must have had the same initial condition. As the choice of time origin is arbitrary, the trajectories must be the same for all time.
In Γ-space the situation is somewhat different. The trajectory for the harmonic oscillator winds around the ellipse, returning to its initial phase point (x0,p0) after a time T. The period of time taken for a system to return to (or to within an ε-neighbourhood of) its initial starting phase is called the Poincaré recurrence time. For a simple system such as the harmonic oscillator the recurrence time is trivial to calculate, but for higher dimensional systems the recurrence time quickly exceeds the estimated age of the universe.