Time and Dynamical Systems

What is Time? Cause and effect are described in laws of evolution we think of physics as in a classical framework. The evolution paths in phase space are determined by mathematical relations.

To follow the discussion and reasoning it is not necessary to know the exact meaning of all the words and concepts. One just needs to obtain intuition and some understanding of basic ideas.

A trajectory is a curve, a graph or path through a phase space. It tells you by given laws how a place in the total phase state flows through space from an initial point. Time is then a parameter for the curve one might say an evolution is an operator-valued function of time.

The operator E thus gives a vector E( t)x (or then P( t)x ) giving the state or position on the trajectory at time t starting from x and beginning with t= 0.

In some very simple examples the phase space is only one- or two-dimensional or discrete and even finite. It is usually much larger and in principle can be infinite-dimensional. The graphs give an idea of evolution operators' behavior and some kinds of trajectories. There are many other considerations of all kinds for instance in properties of continuity and smoothness.

Determinism

Is the state of our Universe pre-decided, is the future fixed and unchangeable? The age old question that we could not answer and probably will not ever has very serious implications. Apparently attempts to answer always leads to contradiction. Does a person have a free will? It has to be a matter of faith. However we can work with mathematical models.

An evolution operator P determines a curve in phase space from a given initial point by a parameter t or "time".

Defining properties of causality are deterministic and accommodated by concepts from dynamical systems. For a semigroup you had in effect the two crucial axioms, basic properties of the essential nature of time was given in E( 0) = I and E ( t +s) = E( t) E( s) = E( s) E( t)

For understanding the concepts freehand sketches can be handy.

The defining property

A dynamical system in turn has as a third axiom a further restriction, a rule of existence of inverses on a trajectory as a function of time, that is if for each time t the inverse P( -t) exists and,

P( -t) P( t) = P( t) P( -t) = = P( t - t) = P( 0) = I

It means that P followed by its inverse brings us back to the original position, or the other way around, the inverse first, which means we return to here and now but from the past. In both cases we end up again with the starting operator.

The inverse returns you back to exactly where you began.

As an example I throw a stone up into the air and catch it again coming down. Let us assume the stone turned at height 15 meters. The way down is also 15 meters and the stone ends where it started, in my hand. The resulting displacement is zero.

Or suppose we take a container with 750 liters of water and then pour out 150 liters leaving 600. Subsequently we add 150 liters. The container now has 750 liters of water, the same as it first had, the same as in the original situation.

In effect in our current considerations you can see "back into time" and even to minus infinity and provided that the evolution operators are invertible in this sense. The trajectory is invertible (can be back-tracked) but only as a bystander and has to be deterministic in the true sense. So that for our current considerations you may say we can then see back into time.

Recurrence

In dynamical systems there are many recurrence results under differing conditions. The most famous and easiest to prove would be Poincaré's Theorem. The idea is that under certain conditions any point in phase space is revisited in a finite time. The result in effect says that the trajectory always returns to its initial point. There are all kinds of extensions and generalisations.

All this is very much related to my speculations and that time is infinite. Time must be of a theological scale.

The process can only be observed as a bystander, an outsider, it is quite boring frankly and the trajectory has to be deterministic in the true sense. It is like a flash the total, the whole exists the path exists as an instant, as once, a lightning bolt, and cannot be influenced neither in the past nor future.

History of an evolution

At time zero the trajectory is itself and proceeds from there and in principle the entire history is known and it can be calculated.

As I see it a divine being as an external independent agent from the "outside" is the only thing that could stop or change the course of history should it want to. Divine intervention, in the true sense of the word.

The start of time

For a dynamical system time cannot have an origin, "a big bang" an "act of creation".

When the operators are defined for time negative too the trajectory goes back as far as you like so that it has no beginning it all has been forever and will ever be, but for semigroups yes indeed there can be a beginning, a start or point in phase-space where time is zero.

At this point we run into other difficulties. There are considerations of perturbations as a significant practical problem as well as entropy, measures of disorder.

These matters are studied amongst others in ergodic theory, measure conserving transformations.

A dynamical system is thus a special kind of semigroup of operators on a vector space

For these things to work out we need to have enormous time intervals however time can be conceived itself to be much longer, infinite, eternity. These are very fundamental issues in philosophy involving the existence of a beginning of time as well as the question of determinism.

Research and the public

Popular imagination finds the ideas very appealing and while the proofs can be hard and often require specialised knowledge, many of the actual results in themselves are not hard to understand.

More relatively recent theoretical investigations are of chaos, fractals, perturbations and noise, there are also others including stochastic processes in general.

The research involves a broad background, measure theory, topology and abstract analysis as well as some familiarity of partial differential equations in exact sciences and many applications including in probability theory and mathematical statistics.

The ideas and findings contribute to almost every branch of science.

There is still much research to be done it doesn't look like we are to run out of questions. This new exploration seeming in differing studies contribute many exciting discoveries to 20th and 21st century mathematics. Studies are very much driven by applications and by necessity.