Phase space is a tool used by physicists to study systems, with each point representing a possible state. It is useful for analyzing mechanical systems, and can also be used for non-classical and non-deterministic systems. An example of a mechanical system is a mass on a spring, which produces an elliptical phase diagram. Orbits cannot intersect in mechanical systems, making phase space analysis useful.
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 a part thereof. These states are typically determined by the set of dynamic variables relevant to the evolution of the system. Physicists find phase space particularly useful for analyzing mechanical systems, such as pendulums, planets orbiting a central star, or masses connected by springs. In these contexts, the state of an object is determined by its position and velocity or, equivalently, its position and momentum. Phase space can also 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 suitable mechanical system to illustrate 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 gravity are ignored. Thus, the state of the system at any given moment is determined solely by the length of the spring and the velocity of the mass.
If someone pulls the mass down, the spring could stretch to a length of 10cm. When the mass is released, it is momentarily at rest, so its velocity is 25.4 in/s. The state of the system at this moment can be described as (0 inch, 10 inch/s) or (0 cm, 25.4 cm/s).
The mass first accelerates upwards and then slows down as the spring compresses. The mass may stop rising when the spring is 6 inches (15.2 cm) long. At that moment, the mass is again at rest, so the state of the system can be described as (6 inches, 0 inch/s) or (15.2 cm, 0 cm/s).
At the endpoints, the mass has zero velocity, so it’s no surprise that it moves fastest at the halfway point between them, where the length of the spring is 8 cm (20.3 inches). One might assume that the velocity of the mass at that point is 4 inches/s (10.2 cm/s). When passing the midpoint up, the system state can be described as (8 inches, 4 inches/s) or (20.3 cm, 10.2 cm/s). During the descent, the mass will move down, so the state of the system at that point is (8 inches, -4 inches/s) or (20.3 cm, -10.2 cm/s).
The graphical representation of these and other states experienced by the system produces an ellipse that represents the evolution of the system. Such a graph is called a phase diagram. The specific trajectory through which a particular system passes is its orbit.
If the mass had been lowered further in the beginning, the figure drawn in phase space would be a larger ellipse. If the mass had been released at the point of equilibrium, the point where the force of the spring exactly cancels out the force of gravity, the mass would remain in place. This would be a single point in phase space. Thus, it can be seen that the orbits of this system are concentric ellipses.
The example of mass on a spring illustrates an important aspect of mechanical systems defined by a single object: it is impossible for two orbits to intersect. The variables representing the state of the object determine its future, so there can be only one path in and one path out of each point on its orbit. Therefore, the orbits cannot cross each other. This property is extremely useful for analyzing systems using phase space.
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