Consider a simple pendulum without friction, swinging through a small angle. The state of its motion at any time is defined completely by its position and velocity at that time. If we know those values we may calculate its position and velocity for any time future or past. Past or future that is only until some outside influence disturbed or will disturb the motion. We could do this calculation because a pendulum as described follows such a simple law of nature that the differential equation describing it is solvable, yielding a manageable state function. What we will do instead of solving the differential equation for the pendulum is to use a mathematical model to get the state values.

In the first display in this section we will illustrate the way the position of the pendulum, measured by its angle from the vertical, varies as time passes. For now we keep the amplitude of the swings fairly small so that the actual horizontal displacement may be taken to be proportional to the angle the pendulum is moved off the vertical. The pendulum should be thought of as a weight hung on a rigid rod of negligible mass from a pivot without friction in a medium which offers no resistance to things moving through it. Admittedly this apparatus stretches reality but bear with me on this.

The displacement of the pendulum will be shown on a graph of position versus time. The position is measured as the angle from the vertical of the pendulum with displacement to the right of vertical being positive and that to the left being negative. The time axis of the graph is drawn vertically with time increasing in a downward direction. This presentation allows you to correlate visually the position of the pendulum with the graph. The position as a function of time for a pendulum meeting our restrictions is given by p=A*sin(t) where A is the amplitude. Run the Pendulum Position vs. Time display.

To better understand the motion of the pendulum we will look at both the position and velocity on the same graph. Run the Pendulum Position and Velocity vs. Time display.

Now if we relax some of our unrealistic requirements on our system and allow for friction in the pivot and air resistance as the pendulum moves we get a different sort of motion. Run the Pendulum Position and Velocity vs. Time, Damped display.

At this point let us consider a different way to analyze the motion of our pendulum. The pendulum we have been observing in the upper half of the past few displays represents a recognizable physical reality. Our experience with pendulums, and everything else for that matter, stems from observations in real space. This "real" space has dimensions of length, breadth and height measured in units of distance. The only movement we can directly observe is that of objects in this space. The study of dynamical systems, to be useful, has to produce understanding of what happens in this space.

The concept of the passage of time is also fundamental to real world observations. We recognize that things in our three dimensional world are different, depending on when we look at them. To really specify the location of an object relative to our observation point we need to identify how far up, how far left, how far away and when. This line of thinking leads to the proposition that time is a fourth dimension in real space. The past then might be taken to be things that happened behind us on the time axis. The future, things that are out in front of us.

We don't seem to be able to do much about the passage of time in the real world either to speed or slow our progress along the time axis but in the world of the mind we are free to jump around along the time dimension, reliving the past, imagining the future, slowing time in a mental experiment to more closely observe the evolution of a dynamical system and so on. In fact what we just did in looking at the graphs of position and velocity versus time was to mentally step completely out of the Space-time of the pendulum and watch as an independent observer, its motion.

Sometimes as an aid in understanding what is happening in a dynamical system it is useful to mentally create a new kind of space. This space has dimensions of the state variables for our dynamical system. In the case of the pendulum it would be one dimension measured in position and one dimension measured in velocity. Time might be considered a third dimension of this "phase space", just as it might be considered a fourth dimension of ordinary space.

We may mark the location of our pendulum in this phase space by marking the point measured so far along the position dimension and so far along the velocity dimension. This is just the way we would locate the pendulum bob in ordinary space, marking its distance left or right and its height. If the pendulum is hanging straight down and not moving, there is not much advantage to looking at it in phase space. The value of the phase space way of looking at things is in tracking the evolution over time of dynamical systems like our pendulum.

As the system moves, the point marking its location in phase space also moves. The trajectory traced out by this moving point is called a "phase space orbit". If the movement of the system along the time axis is not considered explicitly in our phase space, what we observe is a projection of this orbit on the plane spanned by position and velocity, the (p,v) plane. In the next display we will revisit the pendulum without friction and air resistance. The graph will be of the phase space orbit rather than position or velocity versus time. Run the Pendulum 2D Phase Space Representation display.

Different initial conditions of position and velocity for our undamped (no energy losses) pendulum might result in an orbit in phase space of a different size but the shape would be the same. The collection of orbits representing all possible initial conditions that meet our restrictions on pendulum angle being small, would be a set of concentric ellipses. The whole set of possible orbits is sometimes called the phase space portrait of the system.

In the next display we look at the phase space orbit of the pendulum with damping (energy losses). Run the Pendulum 2D Phase Space Representation, Damped display.

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What we will do next is continue with our pendulum dynamical system to expand the way in which we perceive its phase space orbits. As a next step we will move the phase space orbit projection off to one side and turn it slightly to make room for the addition of a third axis, the time axis. Run the Pendulum in 3D Phase Space Projected on (p,v) display.

In the next two displays we again show the pendulum without and with energy losses. Now we turn time loose and let it run along the time axis so that the representations of the phase space orbits are no longer two-dimensional projections on the (p,v) plane but are three-dimensional. Of course the computer screen is still two-dimensional, which requires some imagination on the part of the observer to visualize the three-dimensional nature of the phase space orbits. We have tried to set up the graphics of the displays to assist this visualization but compromises are required. Run the Pendulum in 3D Phase Space display.

Now if we allow some energy loss from the pendulum, again the motion will be damped, eventually coming to a stop with the pendulum hanging straight down. Run the Pendulum in 3D Phase Space, Damped display.

In three dimensions, neither the undamped nor damped pendulum exhibits a closed orbit. This view of things may seem an unnecessary complication of your life for the moment but it will help us understand other dynamical systems less simple than our pendulum. Bear with me. Things are going to get worse before they get better.

How long does it take our pendulum to complete a cycle, that is to return to the maximum right hand position. Thinking of the phase space representation of the pendulum, it takes just long enough for the projection on the (p,v) plane of the state point, to complete a circuit. In other words one orbit period. Now imagine the time axis marked off in lengths of time equal to this period, so each mark is an additional cycle and at each mark there is a plane with dimensions of p and v. Run the Pendulum Phase Space Cycle Planes display to see an example.

Now run the Pendulum Phase Space Cycle Planes, Damped display.

How about this for a question? Where does it say that the coordinate axes of our phase space have to be straight lines? Well, nowhere as it turns out. In particular we want to examine the situation where the time axis is curved. In fact since we will be dealing with systems which may have some sort of periodicity associated with them as does the pendulum, we would like the time axis to curve back on itself in a circle. This in effect repeats the cycle after a certain number of orbits. The next two displays illustrate this. Run the Pendulum PSO Multi-Cycle display.

As demonstrated by the plot with no losses, having a closed loop time axis allows the orbits to just continue on around. In effect the initial conditions for the second pass are just the current conditions at the end of pass one. Notice that even though we allow a curved line for the time axis, we keep the axes orthogonal, that is perpendicular at the origin where they cross.

Next we will look at the damped pendulum in this way. Run the Pendulum PSO Multi-Cycle, Damped display.

The requirement that the coordinate axes be orthogonal as described above means that the positive position axis always points toward the outside of the closed time loop. Think about it. With that being true, the +p axis points to the right of the screen when it is at the right of the picture and toward the left of the screen when it is at the left.

Keep this orientation in mind as we look at the next few displays that depict a time loop exactly one orbit long. Run the Pendulum PSO Single Cycle display.

Now let's look at the damped pendulum in this way. Run the Pendulum PSO Single Cycle, Damped display.

The next lesson is a continuation of the simple pendulum story. Simple, by the way, refers to the pendulum, not the story. The material in the section you have just finished may have pushed you into unfamiliar territory. You might want to run through this section more than once. The ideas we introduced here are important in understanding the analysis of dynamical systems more complicated than the one-piece pendulum we have been studying. We call the next lesson "Pushing the Pendulum".

Are there any questions?