Supertasks, dynamical attractors and indeterminism

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Abstract

Atkinson [2006. Losing energy in classical, relativistic and quantum mechanics. in Studies in History and Philosophy of Modern Physics, 38 To appear.] has obtained some new results on supertasks and the violation of conservation laws. I discuss these results here, extending them from a different perspective, which, by focusing on considerations of indeterminism, facilitates the natural introduction in this context of the concept of dynamical attractor.

Section snippets

Classical mechanics, indeterminism and violation of the laws of conservation

In this section, I prove a result about the classical mechanics of systems of particles of finite total mass and finite spatial extension that evolve in time by means of determinist binary elastic collisions. The result is that systems of this type exist with determinist processes of evolution in which the total energy is not conserved and whose temporal inversion is not even a determinist process.

Let us consider an infinite system of particles (each one of finite mass, but of any value) P1, P2

Classical mechanics, determinism and non-violation of the laws of conservation

I shall now show that the indeterminism and non-conservation of energy found in the previous section are not characteristic of the type of systems considered there in the following precise sense: both can disappear simply by changing (even under the same initial conditions) the masses of the particles involved.

In the foregoing section, we considered the condition vn+1=vnn. An analysis of the slightly more general case vn+1=(1/α)vnn (α>0) is of interest. Now, Eq. (6) leads to((1-γn+1)/(1+γn

Relativistic mechanics, indeterminism and violation of the laws of conservation

In this section, I prove a result about the relativistic mechanics of systems of particles of finite total mass and finite spatial extension that evolve in time by means of determinist binary elastic collisions. The result is that systems of this type exist with determinist processes of evolution in which neither the total momentum nor the total energy is conserved and whose temporal inversion is not even a determinist process.

Following the strategy to prove the existence of indeterminism

Relativistic mechanics, determinism and non-violation of the laws of conservation

I shall now show that the indeterminism and the non-conservation of energy and momentum found in the previous section are not characteristic of the type of systems considered there in the following precise sense: both can disappear simply by changing (even under the same initial conditions) the masses of the particles involved.

In the preceding section, we considered the condition ε(vn+1)=ε(vn)=kn. As will become clear, it is interesting to analyse the case ε(vn)=2γn. From this and (25), ε(v

Conclusion

The results of the previous sections prove that, for the type of supertasks considered in Atkinson (2006), the energy may, or may not, be conserved both in classical and relativistic mechanics. The crucial difference between the two theories is to be found in the fact that, as Atkinson has proved, the loss of energy in a relativistic supertask (of the type mentioned) implies an equal loss of momentum, while the momentum is conserved in a non-relativistic supertask (limiting ourselves to finite

References (4)

  • D. Atkinson

    Losing energy in classical, relativistic and quantum mechanics

    Studies in History and Philosophy of Modern Physics

    (2006)
  • J. Banks et al.

    Chaos. A mathematical introduction

    (2003)
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