Aleksandr Mylläri

Fractals, Entropy and the Arrow of Time

Why can we not move backward in time? According to Relativity, time is just one more dimension like length, width and height where we can go as easily backward as forward. What makes time special? Newton’s laws are symmetric in time and so is even Schrodinger’s equation which controls the happenings at the quantum level. In Schrodinger’s equation one can also switch from a positive sign of the time coordinate to a negative sign without us noticing any difference in the system behavior. There have been suggestions that the initial point and the final point in the evolution of the system may not be symmetric because of the nature of the measuring process while others note that the processes related to measurement could also be symmetrically reversed. Therefore we have to look elsewhere for the cause of asymmetry: we have a strong sense that the past and the future are quite different, and it is not up to us or to any physical system to decide whether we want to advance into the past or into the future.

Three Body Problem in Perspective

The three-body problem is one of the simplest puzzles in nature. It may be stated very simply but the solution can be extremely hard. The formulation of the problem is as follows: Let us introduce three point-like bodies with given mass values, and say that they attract each other by the universal law of gravitation. We assume that the initial positions and velocities are known. Our task is to predict their coordinates and velocities in an arbitrary time in the future or in the past. The problem is simple, but as we have seen in previous chapters, the solution is not. Many famous scientists beginning from Isaac Newton tried to solve this puzzle. For a long time, all attempts failed except in special cases.

From Newton to Einstein: The Discovery of Laws of Motion and Gravity

The three-body problem is Sir Isaac Newton’s problem. It is not only an academic problem, but a serious problem, affecting the whole existence of mankind. Nobody before Newton had thought of it, but as soon as it dawned on Newton, he started to work on it feverishly, even though it gave him a head ache (Fig. 2.1).

Black Holes and Quasars

The standard three-body problem uses Newton’s law of gravity which weakens as the square of distance from the mass point. Einstein’s law of gravity weakens faster than the inverse square law. The difference between the two laws is normally not very much. But if we need to be accurate, as for example when a position on the Earth’s surface is determined by the GPS system, then it is necessary to use General Relativity rather than Newton’s law of gravity. Also, it is necessary to use General Relativity to determine the motions of planets in the Solar System; Newton’s law is not accurate enough for many purposes.

The Solar System

Our Solar System has one star, the Sun, in the center. The Sun is the only significant source of light which is generated deep in the interior of the Sun by nuclear reactions. The other major bodies in the system are the planets, too small to have the high temperature and pressure to initiate nuclear reactions inside them. Therefore they shine mostly by reflected sunlight. It would be difficult to detect the planets from a great distance.

From Comets to Chaos

Now we go back to the beginning of the eighteenth century, to the time when nobody had heard the words Lagrangian, Quantum Mechanics or General Relativity. The three-body problems arising in Quantum Mechanics will not be discussed in this book. General Relativity will enter our discussion mostly in the final chapter. The Lagrangian methods of solving the problems are central today, but more from the technical point of view. Here we do not need to go to this level of technical detail, and it isn’t even possible without extensive use of mathematics. Thus in the following we will go at a descriptive level to some of the problems arising after Newton to see what has been achieved.

Padovan-like sequences and Bell polynomials

We study a class of Padovan-like sequences that can be generated using special matrices of the third order. We show that terms of any sequence of this class can be expressed via Bell polynomials and their derivatives using as arguments terms of another such sequence with smaller indices. Computer algebra system (CAS) Mathematica was used for cumbersome calculations and hypothesis-testing.