“And” Time Draws Nigh

Posted in History, Poetry, The Universe and Stuff with tags , , , , , , , on November 30, 2020 by telescoper

It’s November 30th 2020, which means we have just three teaching weeks to go until the end of term. I am currently teaching two modules: Mechanics 1 and Special Relativity for first-year students and Vector Calculus and Fourier Series for second years. We’re now getting to the “and” bit in both modules.

I didn’t want to present the two topics mentioned in the title of the second year module as completely disconnected, so I decided to link them with a lecture in which I use the divergence theorem of vector calculus to derive the heat equation, the solution of which led Joseph Fourier to devise his series in Mémoire sur la propagation de la chaleur dans les corps solides (1807), a truly remarkable work for its time that inspired so many subsequent developments.

That gives me an excuse to repost the following “remarkable” poem about Fourier by William Rowan Hamilton:

In the first-year module I will be spending most of this week talking about potentials and forces before starting special relativity next week, at the proper time.

This day and age we’re living in
Gives cause for apprehension
With speed and new invention
And things like fourth dimension
Yet we get a trifle weary
With Mr. Einstein’s theory
So we must get down to earth at times
Relax relieve the tension
And no matter what the progress
Or what may yet be proved
The simple facts of life are such
They cannot be removed

As time goes by, the other thing drawing nigh is the loosening of Ireland’s current Level 5 Covid-19 restrictions which were imposed about six weeks ago though, judging by the crowds drinking in Courthouse Square on Saturday night, a lot of folks have thrown the rules out the window already.

I think it’s a dangerous time. The daily cases are still hovering around the 250-300 mark and will undoubtedly start climbing even before Christmas itself:

The chances of us getting back to anything resembling normality during the early part of next year are exceedingly slim.

Fourier, Hamilton and Ptolemy

Posted in History, Poetry, The Universe and Stuff with tags , , , , , , , on December 17, 2018 by telescoper

As we stagger into the last week of term I find myself with just two lectures to give in my second-year module on Vector Calculus and Fourier Series. I didn’t want to present the two topics mentioned in the title as disconnected, so I linked them in a lecture in which I used the divergence theorem of vector calculus to derive the heat equation, the solution of which led Joseph Fourier to devise his series in Mémoire sur la propagation de la chaleur dans les corps solides (1807), a truly remarkable work for its time that inspired so many subsequent developments.

Fourier’s work was so influential and widely admired that it inspired a famous Irish mathematician William Rowan Hamilton to write the following poem:

The serious thing that strikes me is not the quality of the verse, but how many scientists of the 19th Century, Hamilton included, saw their scientific interrogation of Nature as a manifestation of the human condition just as the romantic poets saw their artistic contemplation and how many poets of the time were also interested in science.

Anyway I was looking for nice demonstrations of Fourier series to help my class get to grips with them when I remembered this little video recommended to me some time ago by esteemed Professor George Ellis. It’s a nice illustration of the principles of Fourier series, by which any periodic function can be decomposed into a series of sine and cosine functions.

This reminds me of a point I’ve made a few times in popular talks about Astronomy. It’s a common view that Kepler’s laws of planetary motion according to which which the planets move in elliptical motion around the Sun, is a completely different formulation from the previous Ptolemaic system which involved epicycles and deferents and which is generally held to have been much more complicated.

The video demonstrates however that epicycles and deferents can be viewed as the elements used in the construction of a Fourier series. Since elliptical orbits are periodic, it is perfectly valid to present them in the form a Fourier series. Therefore, in a sense, there’s nothing so very wrong with epicycles. I admit, however, that a closed-form expression for such an orbit is considerably more compact and elegant than a Fourier representation, and also encapsulates a deeper level of physical understanding.

Fourier Series, Epicycles and Haemorrhoids

Posted in The Universe and Stuff with tags , , , , on August 13, 2015 by telescoper

My attention was drawn to this little video some time ago by esteemed Professor George Ellis. I don’t know why it has taken me so long to share it here. It’s a nice illustration of the principles of Fourier series, by which any periodic function can be decomposed into a series of sine and cosine functions.

This reminds me of a point I’ve made a few times in popular talks about Astronomy. It’s a common view that Kepler’s laws of planetary motion  according to which which the planets move in elliptical motion around the Sun, is a completely different formulation from the previous Ptolemaic system which involved epicycles and deferents and which is generally held to have been much more complicated.

The video demonstrates however that epicycles and deferents can be viewed as parts the construction of a Fourier series. Since elliptical orbits are periodic, it is perfectly valid to present them in the form a Fourier series. Therefore, in a sense, there’s nothing so very wrong with epicycles. I admit, however, that a closed form expression for such an orbit is considerably more compact and elegant than a Fourier representation and also encapsulates a deeper level of physical understanding.

It’s nore entirely relevant to the rest of this post but I discovered last week – by reading a book – that Johannes Kepler suffered so badly from haemorrhoids (piles) that he did all his calculations standing up. I just thought I’d share that with you.