chalkdust Lord Kelvin realised that, given some abstract computing elements, it is possible to solve diﬀerential equations using machines: a truly trailblazing achievement. Let us try to solve the diﬀerential equation representing simple harmonic motion (perhaps of a mechanical oscillator!), d2 y + ω2 y = 0, dt 2

(1)

by means of a clever setup consisting of integrators and other devices and using the technique developed by Lord Kelvin in 1876. We can write (1) more compactly as y¨+ω 2 y = 0, where the dots over the variables denote time derivatives. To simplify things a bit we will also assume that ω 2 = 1. Hence we rearrange (1) so that the highest derivative is isolated on one side of the equation, yielding

One of the most fascinating properties of an analogue computer is its extremely high degree of interactivity.

y¨ = −y.

(2)

Let us now assume that we already know what y¨ is (a blatant lie, at least∫for the moment). If we have∫some device capable of integration it would be easy to generate y˙ = y¨ dt + c0 and from that y = y˙ dt + c1 , with some constants c0 and c1 . Using a second type of computing element that allows us to change signs, it is therefore possible to derive −y from y¨ by means of three interconnected computing elements (two integrators and a sign changer). Obviously, this is just the right hand side of (2), which is equal to y¨, assumed known at the beginning. Now Kelvin’s genius came to the fore: we can set up a feedback circuit by feeding the first integrator in our setup with the output of the sign changing unit at the end. This is shown below in an abstract (standard) notation: this is how programs for analogue computers are wrien down. y¨

−y˙

y

−y

The basic circuit for solving y¨ = −y. From le to right we have two integrators and a summer (with each component inverting the sign).

The two triangular elements with the rectangles on their le denote integrators; while the single triangle on the right is a summer. It should be noted that for technical reasons all of these computing elements perform an implicit change of sign, so the lemost integrator actually yields −y˙ instead of y˙ as in our thought experiment above, while the summer with the one input y yields −y. However, if one sets up the two integrators and a summer as demonstrated above, the system would just sit there and do nothing, yielding the constant zero function as a solution of the diﬀerential equation (2): not an incorrect solution, but uerly boring. 55

spring 2016

Chalkdust, Issue 03

Popular mathematics magazine from UCL

Chalkdust, Issue 03

Popular mathematics magazine from UCL