Posts Tagged 'Mechanics'

Billiards & Ballyards

In (mathematical) billiards, the ball travels in a straight line between impacts with the boundary, when it changes suddenly and discontinuously We can approximate the hard-edged, flat-bedded billiard by a smooth sloping surface, that we call a “ballyard”. Then the continuous dynamics of the ballyard approach the motions on a billiard.

SAMSUNG

Elliptical tray in the form of a Ballyard.

Continue reading ‘Billiards & Ballyards’

Boxes and Loops

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.

SimplePendulum-PhasePortrait-Colour

Phase portrait for a simple pendulum. Each line represents a different orbit.

Continue reading ‘Boxes and Loops’

Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph

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].

Hodograph-AB

Left: Elliptic orbit with velocity vectors. Right: Hodograph, with all velocity vectors plotted from a single point.

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

K3 implies the Inverse Square Law.

Kepler-DDR-Stamp-1971

Johannes Kepler. Stamp issued by the German Democratic Republic in 1971, the 400th anniversary of Kepler’s birth.

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.

Continue reading ‘K3 implies the Inverse Square Law.’

A Chirping Elliptic Rocker

Sitting at the breakfast table, I noticed that a small cereal bowl placed within another larger one was rocking, and that the period became shorter as the amplitude died down. What was going on? 

Rocking-Bowl

A small bowl with its handles resting on the rim of a larger bowl. The handles are approximately elliptical in cross-section.

Continue reading ‘A Chirping Elliptic Rocker’

The Kill-zone: How to Dodge a Sniper’s Bullet

Under mild simplifying assumptions, a projectile follows a parabolic trajectory. This results from Newton’s law of motion. Thus, for a fixed energy, there is an accessible region around the firing point comprising all the points that can be reached. We will derive a mathematical description for this kill-zone (the term kill-zone, used for dramatic effect, is the region embracing all the points that can be reached by a sniper’s bullet, given a fixed muzzle velocity).

Sniper-Killzone-1 Family of trajectories with fixed initial speed and varying launch angles. Two particular trajectories are shown in black. Continue reading ‘The Kill-zone: How to Dodge a Sniper’s Bullet’

Trappist-1 & the Age of Aquarius

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].

Artist’s impressions of the TRAPPIST-1 planetary system

Artist’s impression of the Trappist-1 planetary system. Image from https://www.eso.org/public/images/eso1805b/

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’


Last 50 Posts

Categories