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.

## Posts Tagged 'Astronomy'

### Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph

Published May 2, 2019 Irish Times Leave a CommentTags: Astronomy, Hamilton, Mechanics

The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of additional circles, or epicycles. He stuck with the circular model [TM162 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph’

### K3 implies the Inverse Square Law.

Published April 25, 2019 Occasional Leave a CommentTags: Astronomy, Mechanics

Kepler formulated three remarkable laws of planetary motion. He deduced them directly from observations of the planets, most particularly of the motion of Mars. The first two laws appeared in 1609 in Kepler’s *Astronomia Nova*. The first law (**K1**) describes the orbit of a planet as an ellipse with the Sun at one focus. The second law (**K2**) states that the radial line from Sun to planet sweeps out equal areas in equal times; we now describe this in terms of conservation of angular momentum.

The third law (**K3**), which appeared in 1619 in Kepler’s *Harmonices Mundi*, is of a different character. It does not relate to a single planet, but connects the motions of different planets. It states that the squares of the orbital periods vary in proportion to the cubes of the semi-major axes. For circular orbits, the period squared is proportional to the radius cubed.

### Trappist-1 & the Age of Aquarius

Published January 3, 2019 Irish Times Leave a CommentTags: Astronomy, Mechanics

The Pythagoreans believed that the planets generate sounds as they move through the cosmos. The idea of the harmony of the spheres was brought to a high level by Johannes Kepler in his book *Harmonices Mundi*, where he identified many simple relationships between the orbital periods of the planets [TM154 or search for “thatsmaths” at irishtimes.com].

Kepler’s idea was not much supported by his contemporaries, but in recent times astronomers have come to realize that resonances amongst the orbits has a crucial dynamical function. Continue reading ‘Trappist-1 & the Age of Aquarius’

### The 3 : 2 Resonance between Neptune and Pluto

Published December 13, 2018 Occasional 1 CommentTags: Astronomy

For every two orbits of Pluto around the Sun, Neptune completes three orbits. This 3 : 2 resonance has profound consequences for the stability of the orbit of Pluto.

Continue reading ‘The 3 : 2 Resonance between Neptune and Pluto’

### Gravitational Waves & Ringing Teacups

Published November 22, 2018 Occasional Leave a CommentTags: Astronomy, Relativity, Wave Motion

Newton’s law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of * gravitational waves* (GWs), and spiral inwards until they eventually collide.

### Johannes Kepler and the Song of the Earth

Published November 1, 2018 Irish Times 2 CommentsTags: Astronomy, Music

Johannes Kepler, German mathematician and astronomer, sought to explain the solar system in terms of divine harmony. His goal was to find a system of the world that was mathematically correct and harmonically pleasing. His methodology was scientific in that his hypotheses were inspired by and confirmed by observations. However, his theological training and astrological interests influenced his thinking [TM150, or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Johannes Kepler and the Song of the Earth’