Physics Notes

Abstract

Any system whose evolution from some initial state is dictated by a set of rules is called a dynamical system. When these rules are a set of differential equations, the system is called a flow, because their solution is continuous in time. When the rules are a set of discrete difference equations, the system is referred to as a map. The evolution of a dynamical system is best described in its phase space, a coordinate system whose coordinates are all the variables that enter the mathematical formulation of the system (i.e., the variables necessary to completely describe the state of the system at any moment). To each possible state of the system there corresponds a point in phase space. If the system in question is just a point particle of mass m, then its state at any given moment is completely described by its speed v and position r (relative to some fixed point). Thus, its phase space is two dimensional with coordinates v and r or p = mv and q = r, as in the common Newtonian notation. If instead we were dealing with a cloud of N particles, each of mass m, the phase space would be 2N-dimensional with coordinates p ₁, p ₂,..., p _n , q ₁, q ₂, ..., q _n . Note that N indicates the number of independent positions or momenta or the number of degrees of freedom.