Analogies to simpler systems@


Analogies to simpler systems: The motion of gravitational bodies can be thought of as a mass swinging from a pendulum. While the motion itself won’t be the same, its qualitative behavior will be.


2-body motion is regular like the simple pendulum. 3-body motion, however, is chaotic like the double pendulum.

Can other systems imitate orbital mechanics?@

This is a natural question to ask yourself — maybe the idea of an eternal gravitational dance isn’t yet completely comfortable with you, so it’d be nice to imagine something else that resembles it. But is it possible?

The answer is yes.

The simple pendulum@

As we will see, the (classical, unperturbed) 2-body problem falls into two main categories of motion: bounded and unbounded. Each corresponds to a region of phase space having qualitatively different behaviors distinguished by a separatrix.

  • Bounded motion will be periodic, meaning it repeats itself after some amount of time.

  • Unbounded motion will not be periodic, or aperiodic

In the 2-body problem, bounded motion is completely analogous to the pendulum swinging back and forth having an amplitude and period.

In terms of mathematics, the pendulum can be described by

\[\ddot{\theta} + \sin\theta = 0.\]

Without a solution to the above equation, its qualitative behavior can still be fully understood using its Hamiltonian

\[H = \frac{\dot{\theta}^2}{2} - \cos\theta - 1.\]

Where did this come from? Why is this useful? We’ll get to these questions later! Just appreciate for the moment that level sets of \(H\) outline solutions in the \((\theta, \dot{\theta})\) phase space shown below.

Phase portrait of the simple pendulum.

Fig. 1 Phase portrait for the pendulum.@

Particularly, \(H=0\) (the thick black line connecting the two red circles) is the separatrix between qualitatively different behaviors.

  • \(H<0\) (elliptic): “Back-and-forth” motion

  • \(H=0\) (parabolic): “Top-to-top” motion

  • \(H>0\) (hyperbolic): “Over-the-top” motion

We’ll see how these three cases (elliptic, parabolic, and hyperbolic) relate to the different cases in Keplerian orbital mechanics.

The important point is that these are the only behaviors available, and the parabolic case is the only one of zero measure. This means that, for a given Hamiltonian \(H\), motion with a slightly different Hamiltonian \(H + \epsilon\), where \(\epsilon\) is very small, will generally be the same class as \(H\) itself.

The double pendulum@

In contrast to the simple pendulum, the double pendulum offers much more complicated dynamics. So much more complicated in fact that a phase diagram like the one above can’t be drawn for it!

However, we can numerically compute “snapshots” of the flow through its phase space using Poincaré surfaces. In the case of the double pendulum (and the planar 3-body problem), we can project the flow from 4D to 3D, giving a visualisation of what’s happening in phase space.

The most recognisable difference between the simple and double pendulum is the emergence of chaos. “What is chaos?” you may ask. That’s a fair question, and one that’s been studied in detail.

  • The overall essence of chaos is the theme of sensitive dependence upon initial conditions.

This quality in general poses serious philosophical ramifications, but here we’re only concerned with its mathematical consequences.

Connecting the analogies@

In terms of orbital dynamics, the difference between the simple and double pendula is similar to the difference between the 2-body and 3-body problems.

  • 3-body motion requires a high level of mathematics to analyze

This shouldn’t scare anyone off, however. In fact, the opposite. The rewards to be had at the bottom of this bigger toolbag are beautiful and great. But they’ll come in time — for now, we provide a review of 2-body motion next.