A Beautiful SupertaskJON PEREZ LARAUDOGOITIAThroughout this article I will consider elastic collisions between pointparticles which move in one and the same unidimensional space, the axisX. As we will see, this restriction is not essential; we could have taken par-ticles of a finite size submitted to the restriction of moving in a straightline.Let us take a reference frame in which the particle A, of mass m, movesrightward and approaches the point 0 of the axis A'at a velocity v whileparticle B, also of mass m, moves to the left and also approaches 0 atvelocity v. Let them collide at 0. Since the collision is elastic, the totalkinetic energy of the particles will be conserved and, by symmetry, A willmove away to the left from 0 at velocity v and B will move away to theright from 0 also at velocity v. Let us describe this collision in a frame ofreference which moves to the right at velocity v with respect to the former.In it the initial situation corresponds to a particle A at rest which isapproached by B (whose mass is identical to A) at velocity 2v and movingtowards the left! After colliding, B remains at rest and A starts to movetowards the left at the velocity of B before the collision, that is 2v. This isa general well-known result in the classical mechanics of elastic collisions(valid even in relativistic mechanics) which we will make use of later.These preceding comments are intended to make this result more intu-itively plausible.Assume a reference frame in which there is an infinite number of par-ticles P,, each one of mass m, at rest in the points X,= j , (i e {1, 2, 3,4, ... }) while at the point X- 1, a particle Po of mass m and constantvelocity v is moving towards P,. After the collision, Po will remain at restin Xt = 2 , while P, will approach and collide with particle P2 at velocityv, P2 at rest in X7 - \ . After this second collision P, will remain at rest inX2 while P2 will approach P3 at velocity v, and so on. In general, for any i€ {1 ,2, 3,4,. .. }, P, will move during a certain interval of time at velocityv from X, = jj to Xi+I - y+r , but it will remain at rest in XM = yn. Thespace covered by P, is X-XHX = j.-^rn =T+i and the time taken toficover it is obviously t, = -fin . This is the time that transpires betweenthe two collisions to which P, is subjected (/ e {1, 2, 3, 4, ... }). Conse-quently, once Po and P, collide, all successive collisions will have takenplace after the period of time given by the sum of the seriesMind, Vol. 105 . 417 . January 1996 © Oxford University Press 199682 Jon Perez LaraudogoitiaIt is known that the sum of the series \ , jj , p , j? , ... is unity. So,the sum of p > p > p , ••• is j and, in consequence, the sum of (1)will be 2^ • Therefore after the collision between Po and P, and after anyperiod of time greater than or equal to ^ (therefore finite) has elapsed,the situation will be as follows: the particle P, will be at rest in the pointX,+l - yn (i € {0, 1, 2, 3, 4, .. .}), that is to say all the particles will beat rest.This is the supertask, an illustration of how the total initial energy of thesystem of particles \ mv2 can disappear by means of an infinitely denu-merable number of elastic collisions, in each one of which the energy isconserved. It will be clear too why the restriction to point particles is notessential. The discussion of the case would be formally analogous to theprevious one if we suppose for example that each P, is a sphere of radius2 ! ^ ( i e { 0, 1,2,3 ,4 , . ..}) .Nevertheless, the point at which the above supertask acquires specialtheoretical importance appears in connection with the problem of deter-minism in Newtonian particle mechanics. In discussing this problem,John Earman (1986) mentions several examples of particle systems(pointwise or not, and in finite or infinitely denumerable number), the evo-lution of which implies that Newtonian particle mechanics is not com-pletely deterministic. Clearly, he does not consider situations whichinvolve non-deterministic collisions between particles (pointwise or not),since these would trivialize the problem. The examples cited by Earmanare cases of indeterminism which involve the disappearance of particlesin the spatial infinity and the apparition of particles coming from spatialinfinity (obviously in finite time). So, all of these can be eliminated, asEarman himself recognises, imposing boundary conditions at infinity. Theinterest of the supertask presented in this article is to be found in its per-mitting us to find an example of indeterminism in Newtonian mechanics(which do not involve non-deterministic collisions) and for which theimposition of boundary conditions at infinity would not be effective.Let us consider the temporal inversion of the supertask describedabove. This inversion corresponds to a physically possible process, giventhe temporal symmetry of the laws of Newtonian mechanics (Earmanmakes use of the temporal symmetry of the laws of mechanics in the sameway in his examples). We take as base an initial configuration with pointmass particles P,, of mass m, at rest in points Xt = -y+i (i e {0, 1, 2, 3,4, ... }). The complete system may be spontaneously self-excited in sucha way that, after any interval of time afterwards which is greater than ^ ,we will have a system of infinite point mass particles P,, of mass m, at restin points X, = j , (i e {1,2,3,4,... }) together with a particle Po of massA Beautiful Supertask 83m moving rightward from X, = \ and following the rising values oiX atvelocity v. This form of self-excitation in the system is unforeseeable fromthe point of view of classical mechanics; it can take place at any instantand it may in fact repeat itself in time any number of times. Besides, theparticle Po which finally moves does so at a constant velocity v (which cantake any value), for which reason it will not disappear in spatial infinity.So, the strategy of imposing boundary conditions at infinity will not at