2.3 Phase space

In the case of evolutionary processes which can be analysed quantitatively, we can record the modification of particular quantities as a function of time. These variations must stem from specific causes. If the state is defined completely at a given instant by n variables Xi,... ,xu, the evolution of the process is described by a system of n ordinary differential equations

 

 

 

 

 

(2.3.1)

 

 

The n time-dependent variables represent physical quantities such as location, velocity, temperature, pressure etc. Not only in physics, but also in other branches

of science such as biology or chemistry, many processes in temporal evolution can be described by systems of ordinary differential equations. If we formally introduce the column vectors

 

 

 

 

(2.3.2)

we can write the system equ. (2.3.1) with a compact notation in the form (see also

equ. (2.2.10))

 

 

(2.3.3)

where n is the number of equations defining the whole system. The fact that the system consists exclusively of first-order differential equations is not a restriction since any system of ordinary differential equations of higher order can be trans­formed into a system of first order by the introduction of additional variables.

We wish to stress once again that the system equ. (2.3.1) respectively (2.3.3) is autonomous since the right-hand side does not depend explicitly on the indepen­dent variable t. This is not a restriction either, as each non-autonomous system of equations can be transformed into an autonomous one by the trick of an additional variable :z'n+ i = t and the trivial relationship .x'rl+ i = 1.

It is very useful to represent the temporal evolution of a system in an abstract space, the phase space. It is n-dimensional and is spanned by the state variables or coordinates Xi, £2,..., xn. In the phase space, the state of the system at a given time is represented by a point. This point moves with time and its velocity is specified by the vector F. Since the velocity F is known from equ. (2.3.3), the velocity field can be represented immediately in the phase space (see fig. 2.3.1). The aggregate of the directions specify a direction field, reminiscent of the stream lines in a liquid. A point chosen arbitrarily at the instant t = to describes a trajectory which at each point runs tangentially to the vector field of the velocity. The graph of the motion in the phase space is called trajectory or orbit and all the possible motions taken as a whole are denoted phase flow 4>t- The autonomous set of first-order equations (2.3.1) allows us to portray the velocity field without integration directly in the phase space. We thus already obtain a first impression of the form of the solution.

Through each point of the phase space runs one trajectory only. Physically speak­ing, this means that if a state is known at a particular instant, both the future and the past are determined by integration. This also means that trajectories which represent a unique solution can never intersect. In the special case of the mechanics of a particle where the dynamic state of a mass point is specified in three-dimensional space by its position (three space coordinates) and by its veloc­ity (three velocity components), the phase space is six-dimensional. For a single- degree-of-freedom oscillator, the phase space degenerates to the phase plane with the components deflection and velocity (fig. 2.2.1).

/

 

t --

Fig. 2.3.1

Trajectory and velocity field in the three-dimensional phase space