4.1

 - Foreword

Parallelisms : Statics - Kinematics - Mechanics - Geometry

Statics - Kinematics - Kinetics - Mechanics

In physics this section would be the equivalent to kinematics that is concerned by the motions but not with the forces involved.

Say that we actually follow a progression that is very parallel to the development made by physics from "scratch" to "dynamics": after having looked at our world - the experiential learning from scratch, we started with learning how to define and design the space in a logical manner - a step equivalent to "geometry", next we defined and learned how to design and record equilibrium balances - a step equivalent to "statics", now in this section 4, we will define and learn how describe and record motions - a step equivalent to "kinematics", and in the next section 5 we will describe how motions can be generated - a step in equivalence with kinetics.

At this stage, like in mechanics, you will own tools that are self coherent and self verifiable and that allow to observe, describe and manage dynamic a system according to the knowledge you have of it.

Mechanics - Dynamics

For the readers who are aware, our choice of [q], [p] and [t] mimics the generalized coordinates that have been introduced in mechanics - Wikipedia : "Generalized coordinates are unspecified coordinates. By deriving equations of motion in terms of a general set of coordinates, the results found will be valid for any coordinate system that is ultimately specified".

We only infer here a kind of "generalization of the generalized coordinates" - say that our generalized coordinates embed the generalized coordinates of mechanics like a subset.

In example [q] will be any object - tangible or not - i.e. like a chair, the red color, a red chair, an idea, a concept and possibly also set of coordinates x, y, z or q_x, q_y and q_z.

Conversely, [p] can be any action - tangible or not - i.e. like buying, selling, supplying, painting, painting a chair, dreaming, thinking and possibly also any numerical values given at a mass time a velocity describing the momentum - say the "action" - of given body.

The time will not be expressed as continuously unfolding like usually in mechanics. [t] will be only expressed by time periods say cycles - i.e. 1 sec, 10 sec, 1 day, 3 days, 1 week, 1 month, 1 year.

This choice has been initially made because it corresponds to a very familiar users experience - out of the days that restart ... everyday, an experiment has a start and an end and can be restart as a cycle while the industry is also very familiar with production cycles and sub cycles.

On a mathematical viewpoint, Fourier series and transforms ensure that any sets of time cycles may be recomposed.

This choice of time periods instead of a continuously unfolding time appeared as a minor point at the time it has been done but it revealed later to infer a consequent and significant simplification with regards to the classical view point taken in mechanics.

It will be shown here below that the time in this fashion has first been incorporated in the phase space and next that the manifold that represents the phase space has received a real simplification because it can be constructed as the surface of a sphere.

This later point will be also suggested here below and next progressively demonstrated in the following pages.

Geometry - Mechanics

It is a noticeable difference with classical mechanics that the coordinates, the action (say the momentum) and the time does own here a common metric.

It is of course but nothing that the application of the page 3.5.

It can be also apprehended from a geometrical argument: remember that we have been so far playing with maps and figures that all are real 2-D planes and real maps like geographical maps.

In geographical maps as well as in 2-D planes, we only own the metrics of lengths and angles.

Say that whatever cluster type you may design or look at on a map, the calculus you may use for distances, perimeters, surface or whatever are all based only on one metric that is a length.

Our coordinates [q], actions [p] and time-period [t] have been defined nothing but being clusters only. By this fact they own all the three the same nature in our space and consequently the same metric - say the one that we previously recognized as being a bit.

Because of this unique metric but also because of this unique nature, one can also deduce that they all must play a similar role relatively to each other.

Easy is to imagine that when you design three clusters on an paper and when you call one [q] and the two other ones [p] and [t], the situation will not - and can not - be different if you permute the denominations [q], [p] and [t] in any fashions.

So we came in an unexpected and unusual situation - i.e. certainly unusual in day life but also to our knowledge in classical mechanics - that the coordinates - say "x, y, z" -, the action - say "mv"- and the time - say "t" - must play the exact same role because we must be able in our clusters drawings to permute them in any fashion.

Consequently, we could not define our phase space only with [q] and [p], but we had to add [t] so that our phase space is define by [q] x [p] x [t] and with only one metrics.

We remind that the phase space in mathematics, mechanics and physics is a space in which all possible states of a system are represented.

Because of this equivalence between the dimensions of the phase space and the activation of three "dimensions" - the dimension of the phase space are [q], [p] and [t] - , we can anticipate the phase space will have a "shape" that reflect all those three dimensions as equivalent hence that the phase space has a fair chance to be represented on the base of a sphere.