Датум креирања: среда, 04 јун 2014

Vladan Panković, Darko Kapor

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 present a simple classical mechanical system, simply called Newton's rotating mace (with two spheres connected with a tiny, deformable string) and demonstrate that dynamics of this system (that includes precession effects) is time irreversible. All this can be very interesting for a better understanding of the basic dynamical principles of the classical mechanics (of rigid and deformable bodies) as well as for time reversal.

PACS numbers: 45.20.D-

Key words: Newton's rotating mace, spheres, time reversal

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 (according to which usual time direction or arrow is changed by inverse time arrow, and vice versa) [1]. 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 [1], 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.

But, 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.

In this work we shall present a simple classical mechanical system, simply called Newton's rotating mace (with two spheres connected with a tiny, deformable string) and demonstrate that dynamics of this system (that includes precession effects) is time irreversible. All this can be very interesting for a better understanding of the basic dynamical principles of the classical mechanics (of rigid and deformable bodies) as well as for time reversal.

So, consider a simple classical mechanical system, simply called Newton's rotating mace (with two spheres). It holds a cylindrical hollow tube with height H and relatively small (much smaller than H) radius r of the circular basis area. Through this tube a practically massless, tiny, one-dimensional string with deformable form and constant length L=H+l is dragged where l represents some length much larger than r but much smaller than H. At both string ends two spheres with the same mass m and radius R are fixed where R is larger than r and smaller than l.

Suppose that initially tube is placed vertically so that one tube end is placed up while other is placed down at this vertical axis. Simultaneously sphere nearly the upper tube end is (manually or by some experimental arrangement) lifted vertically for l over this tube end and then upper string part (between upper tube end and upper sphere) is (without any other dislocation or deformation) switched for 180 degree in opposite direction of mentioned vertical axis. Then upper sphere and upper string part (without tube) do practically a resting mathematical pendulum with length l and mass m. Also, since string has a constant length L, down sphere at the down tube end is positioned strictly by this tube end. Then down sphere and string part inside tube, with length H, do practically a resting mathematical pendulum with length H and mass m.

Finally, since, according to supposition, string practically has no mass, upper and down sphere (with the same masses and at mutual height distance H-l) connected with string are in equilibrium in the Earth gravitational field. In other words here is a similar situation which occurs by a weighing machine in the equilibrium.

Suppose, further, that tube changes its previous vertical position in an inclined position so that down end of the tube become deflected for some length D in respect to the vertical axis at which upper end of the tube remains.

Finally, suppose that using an external mechanism (e.g. manual action or similar) tube does a precession around vertical axis with some sufficiently large angular speed Ω. It implies that there is corresponding centrifugal force acting at the down sphere (while, according to suppositions, there is no action of the centrifugal force at upper sphere). It, further, implies action of the corresponding component of centrifugal force at the string so that down part of the string (between down sphere and down tube part) becomes larger and larger tending finally toward l, while upper part of the string (between upper sphere and upper part of tube) becomes smaller and smaller tending finally toward zero.

It can be observed that the completely same effect appears if mentioned precession is clockwise or if it is anticlockwise oriented. It has an interesting consequence. Namely, according to remarkable Feynman suggestion [1], we can do a video record of the described dynamics of the Newton's mace rotating, for example clockwise, in usual time direction (from the past toward future). Later we can move this video record backward

that simulates inverse time direction (from the future toward past). In this case we can see how down sphere by anticlockwise rotation tends closer and closer toward down end of the inclined tube, while upper sphere by a vertical translation motion goes far away upper tube end till final distance l. However, it represents extremely strange phenomena for the observer, since as it has been previously pointed out, real precession of the Newton's mace, clockwise as well as anticlockwise, yield the completely same (but not opposite) final effect. It means in fact that by Newton's mace dynamics time reversal transformation is not satisfied or that here time irreversibility appears caused by deformable character of the string..

In conclusion we can shortly repeat and point out the following. In this work we present a simple classical mechanical system, simply called Newton's rotating mace (with two spheres connected with a tiny, deformable string) and demonstrate that dynamics of this system (that includes precession effects) is time irreversible. All this can be very interesting for a better understanding of the basic dynamical principles of the classical mechanics (of rigid and deformable bodies) as well as for time reversal.

**References**

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