# Nonlinear Pendulum

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

$\displaystyle m l^2 \ddot{\theta} + m g l \sin \theta = 0$

It can be simplified by putting $\omega = \sqrt{g / l}$

$\displaystyle \ddot{\theta} + \omega^2 \sin \theta = 0$

We simplify the equations by normalizing $\omega = 1$ without loss of generality.

$\displaystyle \ddot{\theta} + \sin \theta = 0$

LINEAR APPROXIMATION

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

$\displaystyle \ddot{\theta} + \theta = 0$

The general solution is

$\displaystyle \theta ( t ) = \theta_0 \cos ( \phi )$

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.

$\displaystyle \begin{cases} \dot{x} = y \\ \dot{y} = - \sin x \end{cases}$

We can check that $\ddot{x} + \sin x = 0$ so the above system is really the same as the original equation.

PHASE SPACE

The variables $x$ and $y$ can be interpreted geometrically. Indeed, the angle $x = \theta$ corresponds to a point on a circle whereas the velocity $y = \dot{\theta}$ corresponds to a point on a real line. Therefore, the set of all states $( x , y )$ 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 $(\mathbb{R} / 2 \pi \mathbb{Z}) \times \mathbb{R}$, its elements are couples (position,velocity).

Thus, at each point $( x , y )$ in the phase space, there is an attached vector $( \dot{x} , \dot{y} ) = ( y , - \sin x )$. 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 $X$ 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 $X$. Such a trajectory $X(t)$, also called an orbit, is simply the solution of an ordinary differential equation

$\displaystyle \frac{d X}{dt} = F ( X )$

where $F$ is the vector field defined by $F ( x , y ) = ( y , - \sin x )$.

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 $X$ is at equilibrium if the vector field cancels at this point, that is to say

$\displaystyle \dot{X} = F ( X ) = 0$

$\displaystyle \begin{cases} y = 0 \\ \sin x = 0 \end{cases}$

$\displaystyle \begin{cases} y = 0 \\ x = k \pi \textrm{ with k integer } \end{cases}$

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

It is enough to study the two equilibrium points $( 0 , 0 )$ and $( \pi , 0 )$ since the others can be deduced by periodicity. The linearization is performed at $( x_0 , y_0 )$ by the jacobian matrix

$\displaystyle \left( \begin{array}{cc} 0 & 1 \\ - \cos x_0 & 0 \end{array} \right)$

In the case of $( 0 , 0 )$ this gives

$\displaystyle \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)$

As a consequence we have the linear approximation near $( 0 , 0 )$

$\displaystyle \begin{cases} \dot{x} = y \\ \dot{y} = -x \end{cases}$

The behavior can be pictured schematically as follows.

There is a family of concentric periodic orbits around $( 0 , 0 )$. This kind of equilibrium is called a center. We recognize this behavior on the phase portrait around $( 2 k \pi , 0 )$ for every integer $k$.

In the case of $( \pi , 0 )$ this gives

$\displaystyle \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$

As a consequence we have the linear approximation near $( \pi , 0 )$

$\displaystyle \begin{cases} \dot{x} = y \\ \dot{y} = x \end{cases}$

The behavior can be pictured schematically as follows.

The point is both attractive and repelling around $( \pi , 0 )$. This kind of equilibrium is called a saddle. We recognize this behavior on the phase portrait around $( \pi + 2 k \pi , 0 )$ for every integer k.

In fact, these two equilibrium points correspond to the rod  in vertical position, with angles $\theta=0$ and $\theta=\pi$ respectively.

ENERGY

If we multiply the equation $\displaystyle \ddot{\theta} + \sin \theta = 0$ by $\dot{x}$ and integrate it then we arrive at a constant expression

$\displaystyle \int \dot{x} ( \ddot{x} + \sin x ) = 0$

$\displaystyle \int \dot{x} \ddot{x} + \int \dot{x} \sin x = 0$

$\displaystyle \frac{1}{2} \dot{x}^2 - \cos x = \mathrm{constant}$

This gives rise to the definition of an energy function

$\displaystyle H ( x , y ) = \frac{1}{2} y^2 - \cos x$

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