change in velocity[1] (acceleration) to other quantities of interest (force and mass) in physical systems.
We can think of a system of interest (for example, a box of gas) as being represented by a very large space of possible states that the system can take. For something like a box of gas, this space would be composed of points, each of which represents the specific position and velocity of each molecule in the system.[2] For Newtonian systems like gasses, this space is called a phase space. More generally, a space like this—where the complete state of a system at a particular time is represented by a single point—is called a configuration space or state space. Since DyST is concerned with modeling not just a system at a particular time (but rather over some stretch of time), we can think of a DyST model as describing a path that a system takes through its state space. The succession of points represents the succession of states that the system goes through as it changes over time.
Given a configuration space and a starting point for a system, then, DyST is concerned with watching how the system moves from its starting position. The differential equations describing the system give a kind of “map”—a set of directions for how to figure out where the system will go next, given a particular position. The configuration space and the differential equations work together as a tool-kit to model the behavior of the system in question over time. The differential
- ↑ Of course, velocity too is a dynamical concept that describes the change in something’s position over time. The Newtonian equation of motion is thus a second order differential equation, as it describes not just a change in a basic quantity, but (so to speak) the change in the change in a basic quantity.
- ↑ This means that for a system like that, the space would have to have 6n dimensions, where n is the number of particles in the system. Why six? If each point in our space is to represent a complete state of the system, it needs to represent the x, y, and z coordinates of each particle’s position (three numbers), as well as the x, y, and z coordinates of each particle’s velocity (three more numbers). For each particle in the system, then, we must specify six numbers to get a complete representation from this perspective.
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