*Spiric sections*, formed by the intersections of a torus by planes parallel to its axis. Like the conics, they come in various forms, depending upon the distance of the plane from the axis of the torus (see Figure above). We examine how spiric curves may be found in the phase-space of a dynamical system.

## Posts Tagged 'Mechanics'

### Spiric curves and phase portraits

Published December 29, 2022 Occasional Leave a CommentTags: Mechanics, Topology

### Mercury’s Mercurial Orbit

Published November 10, 2022 Occasional ClosedTags: Astronomy, Mechanics, Relativity

The tiny deviation of the orbit of Mercury from a pure ellipse might seem to be of no consequence. Yet the minute precession of this planet was one of the factors leading to a revolution in our world view. Attempts to explain the anomaly in the context of Newtonian mechanics were unsatisfactory. It was only with the emergence of general relativity that we were able to understand the observed phenomenon. Continue reading ‘Mercury’s Mercurial Orbit’

### Dropping Pebbles down a Mine-shaft

Published October 27, 2022 Occasional ClosedTags: Geophysics, Mechanics

If you drop a pebble down a mine-shaft, it will not fall vertically, but will be deflected slightly to the East by the Coriolis force, an effect of the Earth’s rotation. We can solve the equations to calculate the amount of deflection; for a ten-second drop, the pebble falls about 500 metres (air resistance is neglected) and is deflected eastward by about 25 cm. The figure on the left shows the trajectory in the vertical xz-plane (scales are not the same).

We derive the equations after making some simplifying assumptions. We assume the mine-shaft is at the Equator; we assume the meridional or north-south motion is zero; we neglect variations in the gravitational force; we neglect the sphericity of the Earth; we neglect air resistance. We can still get accurate estimates provided the elapsed time is short. However, carrying the analysis to the extreme, we obtain results that are completely unrealistic. The equations predict that the pebble will reach a minimum altitude and then rise up again to its initial height a great distance east of its initial position. Then this up-and-down motion will recur indefinitely.

The swinging spring, or elastic pendulum, exhibits some fascinating dynamics. The bob is free to swing like a spherical pendulum, but also to bounce up and down due to the stretching action of the spring. The behaviour of the swinging spring has been described in a previous post on this blog [Reference 1 below].

### Mamikon’s Visual Calculus and Hamilton’s Hodograph

Published April 7, 2022 Occasional ClosedTags: Analysis, Mechanics

[This is a condensed version of an article [5] in *Mathematics Today*]

A remarkable theorem, discovered in 1959 by Armenian astronomer Mamikon Mnatsakanian, allows problems in integral calculus to be solved by simple geometric reasoning, without calculus or trigonometry. Mamikon’s Theorem states that *`The area of a tangent sweep of a curve is equal to the area of its tangent cluster’.* We shall illustrate how this theorem can help to solve a range of integration problems.

Continue reading ‘Mamikon’s Visual Calculus and Hamilton’s Hodograph’

### Émilie Du Châtelet and the Conservation of Energy

Published November 18, 2021 Irish Times ClosedTags: History, Mechanics

A remarkable French natural philosopher and mathematician who lived in the early eighteenth century, Émilie Du Châtalet, is generally remembered for her translation of Isaac Newton’s *Principia Mathematica*, but her work was much more than a simple translation: she added an extensive commentary in which she included new developments in mechanics, the most important being her formulation of the principle of conservation of energy [TM223 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Émilie Du Châtelet and the Conservation of Energy’

### Embedding: Reconstructing Solutions from a Delay Map

Published September 23, 2021 Occasional ClosedTags: Mechanics, Topology

M

In mechanical systems described by a set of differential equations, we normally specify a complete set of initial conditions to determine the motion. In many dynamical systems, some variables may easily be observed whilst others are hidden from view. For example, in astronomy, it is usual that **angles** between celestial bodies can be measured with high accuracy, while **distances** to these bodies are much more difficult to find and can be determined only indirectly.

Continue reading ‘Embedding: Reconstructing Solutions from a Delay Map’

### Cornelius Lanczos – Inspired by Hamilton’s Quaternions

Published July 16, 2020 Irish Times ClosedTags: Analysis, Applied Maths, Mechanics, Relativity

In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [TM191 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Cornelius Lanczos – Inspired by Hamilton’s Quaternions’

### A Ring of Water Shows the Earth’s Spin

Published February 13, 2020 Occasional ClosedTags: Geophysics, Mechanics

Around 1913, while still an undergraduate, American physicist Arthur Compton described an experiment to **demonstrate the rotation of the Earth** using a simple laboratory apparatus.

### The “extraordinary talent and superior genius” of Sophie Germain

Published January 16, 2020 Irish Times ClosedTags: History, Mechanics, Number Theory

When a guitar string is plucked, we don’t see waves travelling along the string. This is because the ends are fixed. Instead, we see a standing-wave pattern. Standing waves are also found on drum-heads and on the sound-boxes of violins. The shape of a violin strongly affects the quality and purity of the sound, as it determines the mixture of standing wave harmonics that it can sustain [TM179 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘The “extraordinary talent and superior genius” of Sophie Germain’

In 1985, cosmonaut Vladimir Dzhanibekov commanded a mission to repair the space station Salyut-7. During the operation, he flicked a wing-nut to remove it. As it left the end of the bolt, the nut continued to spin in space, but every few seconds, it turned over through . Although the angular momentum did not change, the rotation axis moved in the body frame. The nut continued to flip back and forth, although there were no forces or torques acting on it.

Continue reading ‘The Intermediate Axis Theorem’### An Attractive Spinning Toy: the Phi-TOP

Published October 31, 2019 Occasional ClosedTags: Mechanics

It is fascinating to watch a top spinning. It seems to defy gravity: while it would topple over if not spinning, it remains in a vertical position as long as it is spinning rapidly.

There are many variations on the simple top. The gyroscope has played a vital role in navigation and in guidance and control systems. Many similar rotating toys have been devised. These include rattlebacks, tippe-tops and the Euler disk. The figure below shows four examples.

### The curious behaviour of the Wilberforce Spring.

Published August 22, 2019 Occasional ClosedTags: Mechanics, Physics

The Wilberforce Spring (often called the Wilberforce pendulum) is a simple mechanical device that illustrates the conversion of energy between two forms. It comprises a weight attached to a spring that is free to *stretch* up and down and to *twist* about its axis.

However, due to a mechanical coupling between the stretching and torsion, there is a link between stretching and twisting motions, and the energy is gradually converted from vertical oscillations to axial motion about the vertical. This motion is, in turn, converted back to vertical oscillations, and the cycle continues indefinitely, in the absence of damping.

The conversion is dependent upon a resonance condition being satisfied: the frequencies of the stretching and twisting modes must be very close in value. This is usually achieved by having small adjustable weights mounted on the pendulum.

There are several videos of a Wilberforce springs in action on YouTube. For example, see here.

Continue reading ‘The curious behaviour of the Wilberforce Spring.’

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.

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.

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

Published May 2, 2019 Irish Times ClosedTags: 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 ClosedTags: 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.

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?

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

Published March 14, 2019 Occasional ClosedTags: Applied Maths, Mechanics

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).

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

Published January 3, 2019 Irish Times ClosedTags: 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’

### Galileo’s Book of Nature

Published February 15, 2018 Irish Times ClosedTags: Astronomy, History, Mechanics

In 1971, astronaut David Scott, standing on the Moon, dropped a hammer and a feather and found that both reached the surface at the same time. This popular experiment during the Apollo 15 mission was a dramatic demonstration of a prediction made by Galileo three centuries earlier. Galileo was born in Pisa on 15 February 1564, just 454 years ago today [TM133 or search for “thatsmaths” at irishtimes.com].

### Slingshot Orbit to Asteroid Bennu

Published November 16, 2017 Irish Times ClosedTags: Astronomy, Mechanics

The *Voyager 1* and *Voyager 2* spacecraft have now left the solar system and will continue into deep space. How did we manage to send them so far? The *Voyager* spacecraft used gravity assists to visit Jupiter, Saturn, Uranus and Neptune in the late 1970s and 1980s. Gravity assist manoeuvres, known as **slingshots**, are essential for interplanetary missions. They were first used in the Soviet Luna-3 mission in 1959, when images of the far side of the Moon were obtained. Space mission planners use them because they require no fuel and the gain in speed dramatically shortens the time of missions to the outer planets.

Modern science is big: the gravitational wave detector (LIGO) cost over a billion dollars, and the large hadron collider (LHC) in Geneva took decades to build and cost almost five billion euros. It may seem that scientific advances require enormous financial investment. So, it is refreshing to read in *Nature Biomedical Engineering *(Vol 1, Article 9) about the development of an ultra-cheap centrifuge that costs only a few cents to manufacture [TM111 or search for “thatsmaths” at irishtimes.com].

Towering over O’Connell Street in Dublin, the Spire of Light, at 120 metres, is about three times the height of its predecessor [TM109 or search for “thatsmaths” at irishtimes.com]. The Spire was erected in 2003, filling the void left by the destruction in 1966 of Nelson’s Pillar. The needle-like structure is a slender cone of stainless steel, the diameter tapering from 3 metres at the base to 15 cm at its apex. The illumination from the top section shines like a beacon throughout the city.

Galileo noticed the regular swinging of a candelabra in the cathedral in Pisa and speculated that the swing period was constant. This led him to use a pendulum to measure intervals of time for his experiments in dynamics. Bu not all pendulums behave like clock pendulums.

### Which Way did the Bicycle Go?

Published October 8, 2015 Occasional ClosedTags: Analysis, Geometry, Mechanics

*“A bicycle, certainly, but not the bicycle,” said Holmes.*

In Conan-Doyle’s short story *The Adventure of the Priory School* Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal level of brilliance.

### Emmy Noether’s beautiful theorem

Published June 18, 2015 Irish Times ClosedTags: Algebra, Mechanics

The number of women who have excelled in mathematics is lamentably small. Many reasons may be given, foremost being that the rules of society well into the twentieth century debarred women from any leading role in mathematics and indeed in science. But a handful of women broke through the gender barrier and made major contributions. [TM070: search for “thatsmaths” at irishtimes.com ]

Continue reading ‘Emmy Noether’s beautiful theorem’The **ESA Rosetta Mission**, launched in March 2004, rendezvoused with comet 67P/C-G in August 2014. The lander Philae touched down on the comet on 12 November and came to rest after bouncing twice (the harpoon tethers and cold gas retro-jet failed to fire).

### Falling Bodies [1]: Sky-diving

Published November 13, 2014 Occasional ClosedTags: Fluid Dynamics, Geophysics, Mechanics

Aristotle was clear: heavy bodies fall faster than light ones. He arrived at this conclusion by pure reasoning, without experiment. Today we insist on a physical demonstration before such a conclusion is accepted. Galileo tested Aristotle’s theory: he dropped bodies of different weights simultaneously from the Leaning Tower of Pisa and found that, to a good approximation, they hit the ground at the same time.

### Light Weight (*)

Published October 30, 2014 Occasional ClosedTags: Applied Maths, Astronomy, Mechanics, Physics

Does light have weight? Newton thought that light was influenced by gravity and, using his laws of motion, we can calculate how gravity bends a light beam. The effect is observable during a total eclipse of the sun: photographs of the sky are compared with the same region when the sun is elsewhere and a radial displacement of the star images is found. But the amount predicted by Newton’s laws is only half the observed value.

*That’s Maths* in *The Irish Times* this week (TM050, or Search for “thatsmaths” at irishtimes.com) is about how a simple pendulum can demonstrate the rotation of the Earth.

Does quantum mechanics matter at everyday scales? It would be very surprising if quantum effects were to be manifest in a macroscopic system. This has been claimed for the problem of balancing a pencil on its tip. But the behaviour of a tipping pencil can be explained in purely classical terms.

### Clothoids Drive Us Round the Bend

Published April 17, 2014 Irish Times ClosedTags: Euler, Geometry, Mechanics, modelling

The article in this week’s *That’s Maths* column in the* Irish Times* ( TM043 ) is about the mathematical curves called clothoids, used in the design of motorways.

* * *

### Rollercoaster Loops

Published April 10, 2014 Occasional ClosedTags: Applied Maths, Mechanics, Physics

We all know the feeling when a car takes a corner too fast and we are thrown outward by the centrifugal force. This effect is deliberately exploited, and accentuated, in designing rollercoasters: rapid twists and turns, surges and plunges thrill the willing riders.

Many modern rollercoasters have vertical loops that take the trains through 360 degree turns with the riders upside-down at the apex. These loops are never circular, for reasons we will explain.

Continue reading ‘Rollercoaster Loops’

Remarkable progress in understanding the dynamics of the planets has been possible thanks to their relatively small masses and the overwhelming dominance of the Sun. The figure below shows the relative masses of the Sun, planets and some natural satellites, taking the mass of Earth to be unity.

### Robots & Biology

Published January 16, 2014 Irish Times ClosedTags: Applied Maths, Hamilton, Mechanics, modelling

The article in this week’s *That’s Maths* column in the* Irish Times* ( TM037 ) is about connections between robotics and biological systems via mechanics.

The application of mathematics in biology is a flourishing research field. Most living organisms are far too complex to be modelled in their entirety, but great progress is under way in simulating individual organs and modelling specific functions such as blood-flow and locomotion.

### White Holes in the Kitchen Sink

Published January 9, 2014 Occasional ClosedTags: Applied Maths, Fluid Dynamics, Mechanics, Physics, Wave Motion

A tidal bore is a wall of water about a metre high travelling rapidly upstream as the tide floods in. It occurs where the tidal range is large and the estuary is funnel-shaped (see previous post on this blog). The nearest river to Ireland where bores can be regularly seen is the Severn, where favourable conditions for these hydraulic jumps occur a few times each year.

But you do not have to leave home to observe hydraulic jumps. Continue reading ‘White Holes in the Kitchen Sink’

### Interesting Bores

Published January 2, 2014 Irish Times ClosedTags: Applied Maths, Fluid Dynamics, Geophysics, Mechanics, Wave Motion

This week’s *That’s Maths* column in the* Irish Times* ( TM036 ) is about bores. But don’t be put off: they are very interesting.

### The Antikythera Mechanism

Published November 21, 2013 Irish Times ClosedTags: Archimedes, Mechanics, Time measurement

The article in this week’s *That’s Maths* column in the* Irish Times* ( TM033 ) is about the Antikythera Mechanism, which might be called the First Computer.

**Two Storms**

Two storms, separated by 2000 years, resulted in the loss and the recovery of one of the most amazing mechanical devices made in the ancient world. The first storm, around 65 BC, wrecked a Roman vessel bringing home loot from Asia Minor. The ship went down near the island of Antikythera, between the Greek mainland and Crete. Continue reading ‘The Antikythera Mechanism’

### Sonya Kovalevskaya

Published September 19, 2013 Irish Times ClosedTags: Analysis, History, Mechanics

A brilliant Russian mathematician, Sonya Kovalevskaya, is the topic of the *That’s Maths* column this week (click *Irish Times:* TM029 and search for “thatsmaths”).

In the nineteenth century it was extremely difficult for a woman to achieve distinction in the academic sphere, and virtually impossible in the field of mathematics. But a few brilliant women managed to overcome all the obstacles and prejudice and reach the very top. The most notable of these was the remarkable Russian, Sonya Kovalevskya.

### New Estimate of the Speed of Light

Published September 12, 2013 Occasional ClosedTags: Mechanics, Physics, Recreational Maths

A team of German scientists have recently discovered a new method of measuring the speed of light using Einstein’s famous equation

*E* = *m c*^{2}

“I wonder if I shall fall right through the earth”, thought Alice as she fell down the rabbit hole, “and come out in the antipathies”. In addition to the author of the “Alice” books, Lewis Carroll – in real life the mathematician Charles L. Dodgson – many famous thinkers have asked what would happen if one fell down a hole right through the earth’s centre.

Galileo gave the answer to this question: *an object dropped down a hole piercing the earth diametrically would fall with increasing speed until the centre, where it would be moving at about 8 km per second, after which it would slow down until reaching the other end, where it would fall back again, oscillating repeatedly between the two ends.*

Continue reading ‘A Hole through the Earth’

Atmospheric pressure acting on a surface the size of a large dinner-plate exerts a force sufficient to propel a ten ton train! The *That’s Maths* column ( TM027 ) in the* Irish Times* this week is about the atmospheric railway.

Continue reading ‘The Atmospheric Railway’

Oscillations surround us, pervading the universe from the vibrations of subatomic particles to fluctuations at galactic scales. Our hearts beat rhythmically and we are sensitive to the oscillations of light and sound. We are vibrating systems.

An exhibition called *Oscillator* is running at the Trinity College Science Gallery and this week’s “That’s Maths” column ( TM015 ) gives a taste of what can be seen there.

** Fascinating Dynamics **

Continue reading ‘The Swingin’ Spring’

If you drop a slinky from a hanging position, something very surprising happens. *The bottom remains completely motionless* until the top, collapsing downward coil upon coil, crashes into it.

How can this be so? Continue reading ‘Falling Slinky’