Global Interaction in Classical Mechanics

Abstract

In this paper, an example is presented for a dynamic system analysable in the framework of the mechanics of rigid bodies. Interest in the model lies in three fundamental features. First, it leads to a paradox in classical mechanics which does not seem to be explainable with the conceptual resources currently available. Second, it is possible to find a solution to it by extending in a natural way the idea of global interaction in the context of what is called interaction by impenetrability. Third, the solution presented throws light on a problem posed and discussed in the recent literature in connection with the mass conservation principle.

Notes

[1] If we simply laid down the conservation of kinetic energy as a condition we would not be able to avoid the indeterminacy that arises from the possibility that the kinetic energy of translation varies at the expense of the kinetic energy of rotation. If we imposed the conservation of kinetic energy of translation as a condition directly, we would be excluding a priori arbitrarily precisely that possibility. What I am saying is that the condition of elastic impact (ε = 1) eliminates both the indeterminacy and the arbitrariness.

[2] Since elasticity (ε = 1) implies that the degrees of freedom of rotation are not excited, the excitation of the degrees of freedom of rotation implies that there is no elasticity. This is natural. In general, non‐elasticity implies a violation not of the conservation of the energy but of the kinetic energy of translation, precisely because a part of this latter is invested in energy of the internal degrees of freedom of the system. In our case, the excitation of the degrees of freedom of rotation would be an excitation of the internal degrees of freedom, which would make the collision inelastic not because the total kinetic energy would not be conserved but because the kinetic energy of translation would not be conserved.

[3] X∗ can just be negative at most because, if it moves, D can do so only leftwards, and the laws of dynamics make it impossible for it to remain at X ₀ = 0 after t = 1. On the other hand, it is an elementary algebraic exercise to check that in region X < 0, D could never simultaneously collide with more than one ring, and so we are justified in appealing, as we have done in the text, to the result of the transmission of velocity obtained in Section 2. The circumstances in which physical objects (usually particles) disappear from space are not unknown in the literature. Examples can be found in Xia (1992), Pérez Laraudogoitia (1997), Angel (2001), and others. Obviously, disappearances are never obtained as solutions to the differential equations of motion (they could not be obtained this way, as the equations always provide positions in space as a function of time). It would be interesting to have a broader concept of ‘solution’ that would enable us, in particular, to specify the circumstances in which disappearances (and appearances, if we assume temporal reversibility) take place. Unfortunately, none of this is available to us at present, and we have to make do with justifying specific ‘solutions’ of disappearance for specific cases. In all cases (at least in all those I know of), these justifications take the form of reductio ad absurdum arguments. For example, for Xia (1992), the reductio makes use of the continuity of the particle world lines involved. As we have seen, the reductio in the case analysed in the present paper makes use of the previously established character of the (potential) collisions between the rings A_i and the disk D. In Pérez Laraudogoitia (1997), the reductio is of the same type as in Xia (1992). This underlines the fact that the model presented in this paper is not a minor variation with respect to Pérez Laraudogoitia (1997), which the following observation makes explicitly clear: neither the paradox presented here nor the consequences of the solution that I propose can be manifested in the model introduced in the former paper. The nature of this essential difference can be seen to lie in the different form of the reductios involved. With respect to Angel (2001), see Section 5. The fact that a solution of disappearance is justified by a reductio ad absurdum enables us to understand why, when accepting solutions of disappearance, we are not opening the door to all kinds of monsters. For example, with regard to the model in the present paper, disk D will not disappear before t = 1, and none of the A_i rings will ever disappear. Any reductio posited to justify such disappearances would have to be based on specially introduced ad hoc hypotheses and therefore would have no better justification than those hypotheses.

[4] It should be clear that I am assuming that continuity guarantees identity throughout time, not that continuity may be used to determine identity throughout time. Consider the case of the collision of two identical point particles. The continuity of all the world‐line particles involved here guarantees that the particles are the same after the collision as before the collision but obviously does not allow us to determine which is which (in intuitive terms, do the particles bounce off each other or do they pass through each other?).

[5] Of course it is not required that all the particle world lines be defined at the same instants of time, but if there were no instant of time at which all the particle world lines are defined, then it would in principle be possible that the body world tube of a connected body C would not be path‐connected (for instance, one half of the particle world lines of C could be defined at time interval I ₁, and the other half at I ₂, without I ₁ and I ₂ having any adherent points in common). In any case, the requirement that all the particle world lines of a body world tube should be defined for at least one same instant in time implies the introduction of a criterion of individuation for body world tubes (and therefore for material bodies), which will be crucial to the solution of the paradox.

[6] DA ₁ is uniquely specified at each instant of time by the set union of the set of particle world lines that uniquely specify disc D with the set of particle world lines that uniquely specify ring A ₁.

[7] It goes without saying that DA ₁ also interacts globally with A ⁻ and does not interact individually with any of the A_i (i ≥ 2). This explains why the finite momentum transmitted from DA ₁ to A ⁻ does not affect the kinematic state of any of the A_i (i ≥ 2), given the infinite mass of A ⁻.

Additional information

Notes on contributors

Jon Pérez Laraudogoitia

Jon Pérez Laraudogoitia is Professor in the Department of Logic and Philosophy of Science at Universidad del País Vasco, Spain.