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 the rotation of a single massive, small sphere (representing a rigid body) connected with practically massless, one-dimensional string (representing a deformable body with deformable form and practically constant length) for the center of a horizontal, tiny disc. We demonstrate that in this situation time reversal symmetry of the system dynamics is broken.

PACS numbers: 45.20.D-

Key words: Time reversal, time arrow

As it is well-known all Newton's basic classical mechanical dynamical laws and gravity law (for rigid bodies) are symmetric in respect to time reversal transformation [1]. In other words there is no empirically strange phenomena by formal change of the usual time arrow (directed from past toward future) in opposite time arrow (directed from future toward past) by description of the classical mechanical motions. For example, as it has been pointed out by Feynman [1], we can record by a video camera rotation of a planet around Sun as the consequence of the gravity interaction, and later we can move video material backward (formally simulating time reversal) without any strange situation for observers.

But, in the classical mechanics of the deformable bodies the time reversal symmetry of corresponding dynamics can be satisfied in some situations, while in some other situations this symmetry can be broken. For example by an elastic spring, precisely a linear harmonic oscillator, time reversal symmetry of the dynamics 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 of the spring the time reversal symmetry of the dynamics becomes broken. Really, we can do a video record by usual time arrow of the event when a force plastically deforms spring coil in a linear wire without realistic inverse deformation corresponding to time reversal. Also, by free fall of the glass and its breaking by strike on the floor time reversal symmetry 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 rotation of a single massive, small sphere (representing a rigid body) connected with practically massless, one-dimensional string (representing a deformable body with deformable form and practically constant length) for the center of a horizontal, tiny disc. We demonstrate that in this situation time reversal symmetry of the system dynamics of is broken.

Consider a simple classical mechanical system consisting of practically massless, one-dimensional string (representing a deformable body with deformable form and practically constant length L) and single massive, small sphere (representing a rigid body) fixed at one string end. Other string end is fixed for the center of a horizontal, tiny disc. Denote by D Euclidian distance between string ends, i.e. between the sphere and disc center. Initially whole system (disc, string and sphere) is in rest, sphere is placed in the disc center so that D equals zero while string has an irregular, non-stretched form. Later disc becomes to rotate (clockwise or anticlockwise) with some angular speed Ω (and under action of the centrifugal and Coriolis force) D increases toward L and string form becomes smaller and smaller irregular tending toward regular linear segment directed radially. If we do a video record of whole mentioned motion (in real, usual time direction) and later move this video record backward (simulating inverse time direction) we shall observe an unexpectedly strange fact that practically never occurs really (since realistic clockwise or anticlockwise disc rotation implies the same but not opposite effects of the string form ordering). Precisely, we can se how rotating sphere moves from disc periphery toward disc center. Or, we can see how regular, radially directed, linear segment form of the string turns out in more and more irregular form of the string whose ends tend to be closer and closer. In this way we demonstrate that dynamics of mentioned whole system (string and sphere) is not symmetric in respect to the time reversal transformation.

In conclusion we can shortly repeat and point out the following. In this work we consider the rotation of a single massive, small sphere (representing a rigid body) connected with practically massless, one-dimensional string (representing a deformable body with deformable form and practically constant length) for the center of a horizontal, tiny disc. We demonstrate that in this situation time reversal symmetry of the system dynamics is broken.

**Acknowledgements**

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

**References**

[1] R. Feynman, The Character of Physical Law (Cox and Wyman LTD, London, 1965.)