ThatsMaths

We will describe some generic behaviour patterns of dynamical systems. In many systems, the orbits exhibit characteristic patterns called boxes and loops. We first describe orbits for a simple pendulum, and then look at some systems in higher dimensions.

Libration and Rotation of a Pendulum

The simple pendulum, with one degree of freedom, provides a valuable model for a wide range of physical phenomena. The pendulum is constrained to move in a plane, and has two essentially different modes of behaviour:

• In libration, the bob oscillates about the suspension point.
• In rotation, the bob moves in a full circle.

We can illustrate the behaviour of the pendulum using a phase portrait. We plot the angular velocity (or angular momentum) against the angular deflection. For low energy levels, the pendulum bob is confined to move within an arc . It oscillates or librates back and forth and the sign of the angular velocity changes for every half-cycle. The trajectory has the form of an oval. This is shown in the blue area of the phase portrait (see Figure above).

For higher energy levels, the bob continues to rotate in a circular orbit, repeatedly overtopping the circle. The angular velocity does not change sign. The orbits are the wavy lines in the red regions of the figure. The rotation may be anti-clockwise (top red region) or clockwise (bottom red region).

In many dynamical systems with more than one degree of freedom, there are analogues of these two distinct behaviour patterns. We describe a few examples below.

Stellar Motion in a Globular Cluster

A globular cluster is a collection of stars, typically on the order of a hundred thousand stars. We can examine the motion of a single star as it is influenced by the gravitation attraction of the cluster, which can be regarded as static over a reasonable period of time.

In such stellar systems, two distinct types of orbit are found. Since the force is not central, the angular momentum is not conserved. If we consider motions in the symmetry plane perpendicular to one axis, with differing frequencies about the other two directions, we can distinguish two possibilities.

• Box orbits
• Loop orbits.

In a box orbit, a star oscillates independently about the two axes as it moves along its orbit. As a result of this motion, it fills in a simply connected region of space that includes the centre and that, for small amplitude, approximates a rectangle. The star is free to come arbitrarily close to the centre of the system. If the frequencies with respect to the axes are rationally related, the orbit will be closed. It will then resemble a Lissajous curve. The angular momentum takes both positive and negative values.

In a loop orbit, the angular momentum about a perpendicular to the orbital plane remains of one sign. The orbit fills a region limited by two approximately elliptic curves, and is bounded away from the centre. We illustrate the two orbit types in the figure above.

Billiards on an Elliptical Table

On an elliptical billiard table, the trajectories fall into two classes. Once again, we find box orbits and loop orbits. If the ball is struck in such a way that it passes between the foci, the orbit passes repeatedly between the foci. It never crosses the major axis outside the foci. The figure below, from the book of Hugo Steinhaus, illustrates this behaviour.

If the ball passes once across the major axis outside the inter-focal segment, then it can never pass between the foci. It continues, with angular momentum never changing sign, to rotate in a single sense about the two foci.

Generically, box orbits are bound by an ellipse and a hyperbola, while loop orbits fall between two confocal ellipses. The figure below shows that the orbits on the billiard table are homologous to the trajectories found for a star in a globular cluster.

There are even more exotic orbits when three-dimensional motion is considered. However, it will require another article to do them justice.

Sources

Binney, James and Scott Tremaine, 2008: Galactic Dynamics Princeton Univ.~Press, Princeton and Oxford. 885pp.

Steinhaus, Hugo, 1969: Mathematical Snapshots., Oxford Univ.~Press, 311pp.