Fox and Rabbit - the movie

Here, the computer's two analogue switches were used to switch scope's x and y inputs rapidly between fox's and rabbit's (x,y) position (the switches controlled from a separate function generator)...voila, two spots of light on the oscilloscope screen. The outer one is the fox...

All integrator capacitors were set to 0.1 μF for some fast-paced action. Everything else as per previous post.

Fox and Rabbit - part the second

Here the fox and rabbit are free to move in two dimensions [1], thus:

where r is the Euclidean distance between fox and rabbit. This requires all eight integrators (and all four summers) of the computer:

In this patching, the mass of the rabbit is unity, the fox is twice as massive as the rabbit, and b_r = 1 and b_f = 0.1. Also, γ = -1 giving the required inverse square relationship.

To get the reciprocal of the square of the distance (r) squared, I used ln(x) and exp(x) functions. Since these actually give 3ln(x) and exp(x/2)-10 respectively (to make best use of the computer's range), I have to include the -2/3 between them, and then add ten volts, to give the required 1/r².

Fox and rabbit wiring. Analogue computer enclosure nearly complete.

Initial conditions were as per reference [1]. Results below.

Comments: the fox / rabbit trajectories start off similar to those in [1], but decrease in amplitude - I think this is probably due to leakage in the integrators. Also, the trajectories are sensitive to any offsets - particularly in the integrators. Also, some of the integrators / summers are clipping.

Reference

[1] 'Anti-Newtonian dynamics', J. C. Sprott, Am. J. Physics, 77 (9), September 2009, pp. 783 - 787.

Fox and Rabbit - part the first

Anti-Newtonian dynamics, based on reference [1].

Rabbits and foxes just never seem to get along....

When the fox exerts a force on the rabbit, the rabbit simultaneously exerts a force equal in magnitude and in the same (not-opposite) direction, on the fox, i.e. the rabbit runs away.

In one dimension we have:

The right hand side terms of the above allow for a drag which is linearly dependent on velocity (i.e. corresponding to fox and rabbit moving through a fluid (vegetation?) at relatively slow speeds, with no turbulence (whatever that would be - something anthropogenic perhaps e.g. man with gun?)).

The distance between the fox and rabbit is given by

where parameter γ controls the flavour of the force (as function of separation). Setting γ = 0 (as per reference [1]) makes the force between fox and rabbit constant in magnitude, and with a sign which is negative if the fox is chasing to the left (i.e. rabbit is further left than fox), and positive if the fox is chasing to the right (i.e. rabbit is further right than fox).

The above equations converted into the following rather compact patching diagram, using the values for the various parameters given in [1] (i.e. mass of rabbit and fox both equal unity, b_r = 2 and b_f  = 1):

(The r / |r| was dealt with directly, using the computer's divider - I also tried using the x > y function on the computer to create sign(r), which gave the same results.)

A voltage proportional to elapsed time (to connect to plotter's horizontal axis) was obtained by integrating a unit voltage (not shown).

The initial conditions given in [1] gave the following plots for position of fox and rabbit:

Evidently, blue fox chases and overshoots red rabbit, both decelerate and the chase resumes in opposite direction. And so on...

Reference

[1] 'Anti-Newtonian dynamics', J. C. Sprott, Am. J. Physics, 77 (9), September 2009, pp. 783 - 787.