A simple pendulum consists of a single point of mass attached to a rod of length and of negligible weight. We denote by the angle measured between the rod and the vertical axis. By applying the Newton’s law of dynamics we obtain the equation of motion

It can be simplified by putting

We simplify the equations by normalizing without loss of generality.

**LINEAR APPROXIMATION**

When the amplitude of is small enough, we can approximate . In this condition, the equation reduces to the equation of the well known harmonic oscillator

The general solution is

In the following, we will not use the linear approximation but rather study the nonlinear behavior of the pendulum.

**DIFFERENTIAL SYSTEM**

In practice, it is easier to study an ordinary differential equation as a system of equations involving only the first derivatives.

We can check that so the above system is really the same as the original equation.

**PHASE SPACE**

The variables and can be interpreted geometrically. Indeed, the angle corresponds to a point on a circle whereas the velocity corresponds to a point on a real line. Therefore, the set of all states can be represented by a cylinder, the product of a circle by a line. More formally, the phase space of the pendulum is the cylinder , its elements are couples (position,velocity).

Thus, at each point in the phase space, there is an attached vector . This can be geometrically represented as a vector field on the cylindrical phase space.

The vector field can also be interpreted as a velocity vector field. This means that a point in the phase space moves along a trajectory so that its velocity vector at each instant equals the vector of the vector field attached to the location of . Such a trajectory , also called an orbit, is simply the solution of an ordinary differential equation

where is the vector field defined by .

It is however more convenient to represent the trajectories on a plane instead of on a cylinder. This can be done by expanding the cylindrical phase space by periodicity onto a phase plane. The following diagram is called a phase portrait.

Alternatively, the following plot computed with Mathematica shows the vector field with a few trajectories in a more realistic way.

**EQUILIBRIUM POINTS**

A phase point is at equilibrium if the vector field cancels at this point, that is to say

These points are represented in purple color on the above phase portrait.

It is enough to study the two equilibrium points and since the others can be deduced by periodicity. The linearization is performed at by the jacobian matrix

In the case of this gives

As a consequence we have the linear approximation near

The behavior can be pictured schematically as follows.

There is a family of concentric periodic orbits around . This kind of equilibrium is called a center. We recognize this behavior on the phase portrait around for every integer .

In the case of this gives

As a consequence we have the linear approximation near

The behavior can be pictured schematically as follows.

The point is both attractive and repelling around . This kind of equilibrium is called a saddle. We recognize this behavior on the phase portrait around for every integer *k*.

In fact, these two equilibrium points correspond to the rod in vertical position, with angles and respectively.

**ENERGY**

If we multiply the equation by and integrate it then we arrive at a constant expression

This gives rise to the definition of an energy function

Such a system with a conserved quantity is called a conservative system. The trajectories are then contours of constant energy.

This somewhat optimistic article reminded of your post:

http://blog.wired.com/wiredscience/2009/04/newtonai.html

I have investigated nonlinear pendulum systems such as the analog phase-locked loop. The undamped system has analytical solutions, (for both rotary and oscillatory motion), in terms of Jacobian Elliptic functions, that with the phase portrait gives a total qualitative and quantatitive explanation. See SALVADORI and SCHWARZ, “Mathematical Problems in Engineeering”, circa 1958, for a good description of the Elliptic function solution from first principles. A solution exists for both non zero angular position and angular velocity initial conditions.

IAN NEWCOMBE

Hi, I am wondering what software produce that pendulum phase space picture? many thanks

Thanks for your interest. The realistic vector field was drawn with function “VectorPlot” provided by the software Mathematica (Wolfram). The trajectories were computed with function “NDSolve”.