A # phase space

Phase space is the mathematical space of all possibilities in a given situation. A motion is then described by a path, trajectory, or orbit in this space. This not the usual kind of path laid out on the ground, but a series of locations in phase space, describing motion or change over a period of time. The terms do, however, that recall the origins of qualitative dynamics in Henri Poincaré's study of planetary motion. The dimension of the phase space is the number of initial conditions needed to uniquely specify a path and is equal to the number of variables in the dynamical system. The temporal behavior of the system is viewed as the succession of states in the system's state space. In the case of a simple pendulum, for example, the instantaneous configuration is given by just two numbers – the position of the pendulum bob and its velocity – which completely describe the system's state. For more complex systems, such as a chain of n pendulums coupled together, the state of the system is much larger. It requires, in this case, 2n numbers to specify the state of the entire system. This collection of all possible configurations is the phase space.