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A phase space is an abstraction that physicists use to visualize and study systems; each point in this virtual space represents a single possible state of the system or one of its parts. These states are typically determined by the set of dynamic variables relevant to the system’s evolution. Physicists find phase space especially useful for analyzing mechanical systems, such as pendula, planets orbiting a central star or masses connected by springs. In these contexts, an object’s state is determined by its position and velocity or, equivalently, its position and momentum. Phase space also can be used to study non-classical — and even non-deterministic — systems, such as those encountered in quantum mechanics.
A mass moving up and down on a spring provides a concrete example of a mechanical system suitable for illustrating phase space. The motion of the mass is determined by four factors: the length of the spring, the stiffness of the spring, the weight of the mass and the velocity of the mass. Only the first and last of these change over time, assuming that minute changes in the force of gravity are ignored. Thus, the state of the system at any given time is solely determined by the length of the spring and the velocity of the mass.
If someone pulls the mass down, the spring might stretch to a length of 10 inches (25.4 cm). When the mass is let go, it is momentarily at rest, so its velocity is 0 in/s. The state of the system at this moment can be described as (10 in, 0 in/s) or (25.4 cm, 0 cm/s).
The mass accelerates upward at first and then slows down as the spring compresses. The mass might stop ascending when the spring is 6 inches (15.2 cm) long. At that moment, the mass is once again at rest, so the state of the system can be described as (6 in, 0 in/s) or (15.2 cm, 0 cm/s).
At the endpoints, the mass has zero velocity, so it is unsurprising that it moves fastest at the halfway mark between them, where the length of the spring is 8 inches (20.3 cm). One might assume that the mass’ speed at that point is 4 in/s (10.2 cm/s). When passing the midpoint on its way upward, the state of the system can be described as (8 in, 4 in/s) or (20.3 cm, 10.2 cm/s). On the way down, the mass will be moving in the downward direction, so the state of the system at that point is (8 in, -4 in/s) or (20.3 cm, -10.2 cm/s).
Graphing these and other states the system experiences produces an ellipse portraying the evolution of the system. Such a graph is called a phase plot. The specific trajectory through which a particular system passes is its orbit.
Had the mass been pulled down further at the beginning, the figure traced out in phase space would be a larger ellipse. If the mass had been released at the equilibrium point — the point where the force of the spring exactly cancels the force of gravity — the mass would stay in place. This would be a single dot in phase space. Thus, it can be seen that the orbits of this system are concentric ellipses.
The mass-on-a-spring example illustrates an important aspect of mechanical systems defined by a single object: it is impossible for two orbits to intersect. The variables representing the object’s state determine its future, so there can be only one path into and one path out of every point on its orbit. Therefore, orbits cannot cross each other. This property is exceedingly useful for analyzing systems using phase space.
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