change in velocity (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. 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.