Phase Space on Poincare Maps Practical Aspects

Visualizing Phase Space on Poincare Maps Practical Aspects 

To visualize some of the effects described in the previous section, Poincare showed that the behavior of two degree-of-freedom nonlinear systems can be profitably studied by mapping the dynamics onto a well-chosen plane. This is because the conservation of energy requires all trajectories to wander on a three-dimensional hypersurface. In his honor, these maps are often referred to as Poincare maps. The plane chosen to map the dynamics onto is referred to as a surface of section. 

Often the plane is chosen to be a simple function, such as a constant value of one of the phase-space coordinates. Knowledge of any combination of three phase-space coordinates determines the fourth (within a sign) because energy is conserved. For example, if the system is of the form of Eq. [29] and (p2, qj, 2) are given, pj is simply 

Therefore, we could define a useful surface of action by choosing a constant value of 1 = q f such that each trajectory passes it every time it undergoes an oscillation in the q direction (a good practical choice for q f would probably be the value of q for which the potential energy is a minimum). Each time a trajectory passes q f with pi 0 (to remove the sign ambiguity), we record (p2, ) 100,122 Jhis means that all points on our Poincare map will lie on the surface 

If we carry out such a calculation for a large number of trajectories at the same fixed energy , we obtain a unique map of the global dynamics at that energy. The map is unique because each point on it uniquely specifies a single trajectory in phase space, and trajectories in phase space do not intersea at any given instant in time. If they did intersect, they would have to arise from identical initial conditions, and then claissical mechanics would no longer be a causal theory  

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