On indeterminism in classical dynamics

Abstract

A simple result about the failure of determinism of classical particle dynamics is presented. This result is much more accessible than similar ones in the usual literature. Resumen. Se muestra un sencillo ejemplo de las limitaciones al determinismo de la dinámica clásica de partículas. Dicho ejemplo es mucho más accesible desde el punto de vista técnico que otros similares en la literatura usual.

A general theorem on indeterminism in classical particle dynamics: inflationary scenarios

“…The temporal inversion of this process describes a selfexcitation in which energy is created by means of a supertask. As in footnote 3, the easiest example of kinetic energy loss (by means of elastic collisions) to explain is Laraudogoitia (1997)s. Let us have infinitely many identical particles (of mass m, for instance) p i (i1) at rest at points x i =1/i and in the instant t=0 particle p 0 collides with p 1 (p 0 also has mass m, and approaches it from the right at unit velocity).…”

“…3 Until we have a more precise characterisation available of the idea of self-excitation (see section 3), we may anticipate intuitively here that an isolated system of particles self-excites in the instant T when in T there begins within it a sequence of collisions not 'required' by the movement of its particles in that instant. For instance, Laraudogoitia (1997) considers infinitely many particles p 1 , p 2 , p 3 ,...p n ,......all initially at rest. Suddenly, at a certain instant T (in which all of them are still at rest), there begins an infinite sequence of collisions as a result of which p 1 ends up being brought into movement by p 2 , p 2 by p 3 , p 3 by p 4 , and so on successively.…”

“…4 If a system of particles evolves in such a way that at the end all of them are at rest, the temporal inversion of that evolution implies the spontaneous self-excitation of the system and, therefore, indeterminism. Particles of identical mass provided the starting point for Laraudogoitia (1997) in obtaining the final state of relative rest. 5 The 'mechanism' for the violation of energy conservation by means of supertasks may be understood very intuitively with the models of Laraudogoitia (2007) and Atkinson (2007).…”

“…The total kinetic energy of the system of particles in t=0 is (m/2), but in t=1 it is null, because all the successive collisions (p 0 with p 1 , p 1 with p 2 , p 2 with p 3 , K etc) will have already occurred, leaving all the particles at rest. The problem with Laraudogoitia (1997) is that the total mass is also infinite, which means that the total kinetic energy is not well defined in all the inertial systems. This defect was corrected in Laraudogoitia (2007) and Atkinson (2007).…”