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, br = 2 and bf = 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.
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
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, br = 2 and bf = 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.
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