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# TIME IRREVERSIBILITY BY A SIMPLE VARIATION OF THE MATHEMATICAL PENDULUM EXPERIMENT

Датум креирања: понедељак, 16 јун 2014

Vladan Panković, Darko Kapor, Vojislav Božić - Sremac

Department of Physics, Faculty of Sciences, 21000 Novi Sad, Trg Dositeja Obradovića 4., Serbia , Ова адреса ел. поште је заштићена од спамботова. Омогућите JavaScript да бисте је видели.

Abstract

In this work we consider a simple variation of the mathematical pendulum experiment dynamics and demonstrate that this dynamics is time irreversible. In the usual mathematical pendulum experimental set-up there are one vertical stative, string and sphere whose dynamics is time reversible. We add a new vertical stative with ring, through which mentioned string and sphere can be dragged. Sphere touches an inclined plane under the ring and it can move toward upper or down inclined plane end. Initially string is fixed in the ring in such way that part of the string between two statives has a non-stretched U form, while other part of the string under ring has a stretched | form, so that initial form of the string can be simply denoted by {U , | ). When fixation becomes cancelled sphere becomes to move toward down part of the inclined plane so that finally part of the string between two statives has stretched ― form, while other part of the string under ring has a stretched \ form, so that final form of the string can be simply denoted by {― , \ ). All this can be video recorded and by realization of this video record backward we can formally simulate time reversal of the system dynamics that implies the following transition from initial {― , \ ) into final {U , | ) string form. However, this simulated time reversal is principally different from real motion of the sphere (with corresponding speed toward inclined plane upper end) from {― , \ ) initial string form into {― , ι) final string form. In this way it is simply demonstrated that dynamics of the suggested simple variation of the mathematical pendulum is time irreversible. All this can be very interesting for a better understanding of the basic principles of the classical mechanical dynamics (for rigid and deformable bodies) as well as time reversal transformation.

PACS numbers: 45.20.D-
Key words: Time reversal, time arrow

1. Introduction

As it is well-known all Newton's basic classical mechanical dynamical laws and gravity law (with strictly deterministic character) are symmetric in respect to time reversal transformation (according to which usual time direction or arrow is changed by inverse time arrow, and vice versa) . In other words there is no empirically strange phenomena for the observers by formal change of the usual time arrow (directed from past toward future) in opposite or inverse time arrow (directed from future toward past) by description of the classical mechanical motions. For example, as it has been pointed out by Feynman , we can record by a video camera (e.g. clockwise) rotation of a planet around Sun, as the consequence of the gravity interaction, and later we can move video material backward (formally simulating inverse time arrow) without any strange phenomena for observers. Namely, the planet rotates again around Sun (in opposite, anticlockwise direction) according to the same Newton's gravity law.
On the other hand it is well-known too that basic principles of the classical statistical mechanics and thermodynamics can include principally time irreversible phenomena (e.g. diffusion effects, second thermodynamic law, etc.) and in this sense it can be spoken about statistical or thermodynamic time arrow. By usual time arrow heat turns out from thermodynamic system with higher temperature onto thermodynamic system with smaller temperature and opposite directed heat transfer effectively never appears. For this reason heat transfer from the thermodynamic system with the smaller temperature onto thermodynamic system with the higher temperature that formally appears by inverse time arrow represents extremely strange phenomena for observers.
Between classical mechanics (of the rigid bodies) and classical thermodynamics and statistical physics there is domain of the classical mechanics (of the deformable bodies) that in different situations can tend in limit toward classical mechanics or classical thermodynamic. In the classical mechanics of the deformable bodies time reversal can be satisfied in some situations, while in some other situations time reversal cannot be satisfied. For example by an elastic spring, precisely a linear harmonic oscillator, time reversal is satisfied (backward motion of the video material does not imply any strange situation for observers). On the other hand by a phase transition from domain of the elastic deformation into domain of the plastic deformation and breaking phenomena of the spring time reversal becomes broken. Really, we can do a video record by usual time arrow of the event when a force can plastically deform spring coil in a linear wire without realistic inverse deformation corresponding to inverse time arrow. Also, by free fall of the glass and its breaking by strike on the floor time reversal is not satisfied (backward motion of the video material implies a extremely strange situation for observers in which extremely many parts of the glass on the floor tend mutually in the unbroken glass that later does a vertical shot).
In this work we shall consider a simple variation of the mathematical pendulum experiment dynamics and demonstrate that this dynamics is time irreversible. In the usual mathematical pendulum experimental set-up there are one vertical stative, string (practically massless, with constant length and changeable form) and (massive, but small) sphere whose dynamics is time reversible, as it is well-known. We add a new vertical stative with ring, through which mentioned string and sphere can be dragged. Sphere touches an inclined plane under the ring and it can move toward upper or down inclined plane end. Initially string is fixed in the ring in such way that part of the string between two statives has a non-stretched U form, while other part of the string under ring has a stretched | form, so that initial form of the string can be simply denoted by (U , | ). When fixation becomes cancelled sphere becomes to move toward down part of the inclined plane so that finally part of the string between two statives has stretched ― form, while other part of the string under ring has a stretched \ form, so that final form of the string can be simply denoted by (― , \ ). All this can be video recorded and by realization of this video record backward we can formally simulate time reversal of the system dynamics that implies the following transition from initial (― , \ ) into final {U , | ) string form. However, this simulated time reversal is principally different from real motion of the sphere with corresponding speed toward inclined plane upper end from (― , \ ) initial string form into (― , ι) final string form. In this way it is simply demonstrated that dynamics of the suggested simple variation of the mathematical pendulum is time irreversible. All this can be very interesting for a better understanding of the basic principles of the classical mechanical dynamics (for rigid and deformable bodies) as well as time reversal transformation.

2. Time irreversibility by a simple variation of the mathematical pendulum experiment

Consider the following simple classical mechanical system consisting of a small (with practically neglectable radius), massive sphere (representing practically a rigid body) and practically massless, one-dimensional string (representing practically a deformable body with changeable form, with constant length L, and, without any internal tension if the Euclidian distance between string ends is smaller than L). One string end is permanently fixed for the sphere.
Consider also the following experimental arrangement in which mentioned system string+sphere is settled. It includes two, left and right, vertical statives with the same height H (somewhat larger than L) at mutual horizontal distance D (significantly smaller than H and L). At the top of the first, left stative the free end of the string is permanently fixed so that in absence of the other connections of the string+sphere system with the second, right stative this system can behave as a mathematical pendulum. As it is well-known motion, i.e. dynamics of this mathematical pendulum is definitely completely time reversible. In other words, if we do a video record of this mathematical pendulum motion in usual time direction, and if we later emit this video record backward (formally simulating inverse time direction) nothing strange will happened for the observers.
At the top of the second, right stative a ring (or similar technical element) with the small radius (comparable with sphere radius) is settled. Also, the sphere is dragged through this ring and string is temporarily (manually) fixed in this ring in such way that practically non-stretched first string part between two statives does an U form. Simultaneously, sphere under and right of the ring, using gravitation, stretches second part of the string in form of a vertical, linear segment | whose length is chosen to be practically one half of the first string part length. In this way initially left string part between statives has length 2/3L, while right string part, under and right from ring, has length L/3. Right string part touches top of an inclined plane slant whose down end is placed right in respect to top. Also, initial form of the string can be simply denoted by
{U , | ).
Suppose now that in the initial time moment (manual) fixation of the string in the ring is cancelled so that the sphere becomes to move with constant acceleration down inclined plane. By this motion right string part below and right ring becomes stretched line segment closer and closer toward inclined plane slant with larger and larger length. Simultaneously, left ring part, between two statives, stands practically non-stretched with smaller and smaller length in U form with smaller and smaller height..
In the final time moment left string part, between two statives, becomes stretched line segment parallel to horizontal plane ― with length D, while other, right string part, below and right to ring, becomes stretched line segment \ with length L-D so that final form of the string can be simply denoted by {― , \ ).
If we do a video record of the discussed motion of the string+sphere system, in the usual time direction, and if we later emit this video record backward (formally simulating inverse time direction) observers can see the following. Sphere climbs along inclined plane slant with corresponding initial speed and constant deceleration and second part of the string, between the ring and sphere, stretched in form of a linear segment tending to a vertical direction becomes shorter and shorter, from initial value L-D till final value L/3. Simultaneously, left ring part, between two statives, turns out from initial stretched linear segment in more and more non-stretched U form with larger and larger height. Length of this string part tends finally to 2/3L. Then, final form of the string can be simply denoted by {U , | ).
However, by realistic experimental procedure, we obtain completely different situation. Namely, we start from initial state in which sphere climbs along inclined plane slant with corresponding initial speed (done manually or by corresponding experimental apparatus by us) and constant deceleration. By this motion right string part, between ring and sphere is non-stretched in ι form or form of the left party of U. Simultaneously, left string part, between two statives, stands practically stretched line segment parallel to horizontal plane ― with length D.
In this way it is demonstrated that by described experiment representing a simple variation of the mathematical pendulum experiment dynamics of the system is not time reversal.
In discussion we can observe that described phenomena is somewhat similar to the Landau's continuous phase transition phenomena . Namely, we can consider left string part, i.e. its length, l , as an "ordering parameter" with "critical value" D and when this "ordering parameter" arrives its "critical value" an irreversible "phase transition" from non-streched in the stretched form of the whole string appears.

3. Conclusion

In conclusion we can shortly repeat and point out the following. In this work we consider a simple variation of the mathematical pendulum experiment dynamics and demonstrate that this dynamics is time irreversible. In the usual mathematical pendulum experimental set-up there are one vertical stative, string and sphere whose dynamics is time reversible. We add a new vertical stative with ring, through which mentioned string and sphere can be dragged. Sphere touches an inclined plane under the ring and it can move toward upper or down inclined plane end. Initially string is fixed in the ring in such way that part of the string between two statives has a non-stretched U form, while other part of the string under ring has a stretched | form, so that initial form of the string can be simply denoted by {U , | ). When fixation becomes cancelled sphere becomes to move toward down part of the inclined plane so that finally part of the string between two statives has stretched ― form, while other part of the string under ring has a stretched \ form, so that final form of the string can be simply denoted by {― , \ ). All this can be video recorded and by realization of this video record backward we can formally simulate time reversal of the system dynamics that implies the following transition from initial
{― , \ ) into final {U , | ) string form. However, this simulated time reversal is principally different from real motion of the sphere (with corresponding speed toward inclined plane upper end) from {― , \ ) initial string form into {― , ι) final string form. In this way it is simply demonstrated that dynamics of the suggested simple variation of the mathematical pendulum is time irreversible. All this can be very interesting for a better understanding of the basic principles of the classical mechanical dynamics (for rigid and deformable bodies) as well as time reversal transformation.

Acknowledgements

Authors are very grateful to Prof. Dr. Tristan Hübsch for illuminating discussions.

References

 R. Feynman, The Character of Physical Law (Cox and Wyman LTD, London, 1965.)
 L. D. Landau, E. M. Lifshitz, The Statistical Physics (Pergamon Press, Oxford, 1960.)

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