The Pendulum an Open Dynamic System?

Edgar Paternina

Abstract. In this paper the pendulum and its approximation factor that can be validated with what is observed in the reality “out there”, is presented by using the complex basic unit system concept based on Euler relation. This paper is a result and a promise made in Physics Forum, in its sub forum Theory Development, where I have been participating under the pseudonym of Epsilon Pi. Here I want to show that it is possible to cope the fundamental equations of physics from a point of view or framework that includes the third, which means mathematically speaking, by using complex numbers.

Key words: pendulum, synergy, time, decoupled, Newton’s laws generalized

1. Introduction

This is the first of a series of articles where a new methodology to cope the fundamental equations of physics is presented; new in the sense that it deviates definitively from that prevailing methodology where Aristotelian logic is so central, expressed as: if we have A=B, B=C, then C=A, which means that the third is always excluded. It was Hermann Weyl the so-called philosopher of relativity the one that in his beautiful book “Simetría” wrote: “La fe en la equivalencia derecha-izquierda se ha mantenido en la ciencia incluso ante hechos biológicos como los que citaremos a continuación, que parecen sugerir su no equivalencia aún más fuerte que la desviación de la aguja magnética que tanto impactó al joven Mach.”(10, 19). Since then this faith in that equivalence between right and left has prevailed in western science even in spite, as pointed out the translator of the book:”El tiempo se encargó, como tantas veces ha ocurrido con las “verdades” físicas, de echar por tierra esta afirmación. En 1956, pocos años después de publicarse este libro. T.D. Lee y C.N. Yang postularon, en un trabajo publicado en Physical Review, la violación de la paridad en las interacciones débiles.”(10, 16)

What we have found is then that there is a different way to cope reality in which the third is included; a way that is fundamental when coping with the radical duality of the universe such as in those cases where:

- momentum cannot be related to position

- wave cannot be reduced to a particle

- time cannot be coupled or reduced to space

In this case we have always, as I said before, a radical duality which means that we cannot reduced the one to the other: we cannot couple time to space, wave to particle, momentum to position as they are decoupled, but decoupled in this case does not mean that both terms cannot be put in a common ground or in a framework that includes them both; this is where the “third included” enter into the scene, but the important thing is that there is a mathematical framework, I mean, complex numbers based on Euler relation, that have at the background the complex plane, sort of canvas to represent the dynamic nature of reality, that permits us to represent in a most rigorous way that framework with the third included. The great mistake of philosophers at all times had been to suppose that reality is simple, and as so of that science whose mathematical representation was based just on an equal sign or as Hermann Weyl put it in bilateral symmetry, with no chance to have a radical duality included, I mean, not just opposition, the two signs(+/-), the Aristotelian binary logic or symmetry, but also complementarity.

This first paper has to do with the pendulum, the second one with the Schrödinger wave equation, the third one with the Lorentz transformation group, and the fourth with gravitational fields. It is worth to point out that in this way we will put under a same roof and methodology the fundamental equations of physics so it certainly is not a TOE, but a unified methodology to include all those phenomena of physics, even those that had resisted a good documentation.

2. The Pendulum , an open dynamic system?

We will start our quest with that entity, that practically motivated Galileo to dedicate his all entire life to that enterprise we now call western science, I mean, the pendulum; another reason for starting with it has to do with what T.S.K. wrote:

"How else are we account for Galileo's discovery that the bob's period is entirely independent of amplitude, a discovery that the normal science stemming from Galileo had to eradicate and that are quite unable to document today?"(8,124)

As a matter of fact and keeping the due proportions, the pendulum behaves as a non classical system as it is a "particle" with mass "m", and with a wave behavior.

The pendulum will be represented then as a basic unit system concept or else:

i(q)

DS = |DS| e

where we have Euler relation as:

i(q)

e = Cosine (q) + i Sine (q)

At this point it is important to recall that in this most elegant equation we have a radical duality included or else the third:

on the first term we have the cosine, with an even parity, as it remains the same when the sign of the angle is changed
on the second term we have the sine, separated by the symbol i, with an odd parity as it changes its sign when the sign of the angle is changed.

Additionally we must point out the most remarkable property of this relation: it has to do with the fact it remains with the same form with change, or with those mathematical processes that represent change, I mean, integration and differentiation, that makes it the ideal tool to represent the covariant laws of nature.

Normally is said “The cosine is perfectly general, since the addition of a phase angle of -90° converts the cosine function into the sine.”(9, 512), but this addition of -90° means all difference, as it means the complex plane, but additionally this difference has to do with another radical duality, I mean, the two kinds of parities, very important in modern physics as: “…..if you look at the original state and by making a little computation on the side discover that an operation which is a symmetry operation of the system produces only a multiplication by a certain phase, then you know that the same property will be true of the final state- the same operation multiplies the final state by the same factor. This is always true even though we may not know anything else about the inner mechanism of the universe which changes a system from the initial to the final state.” (6, 17-7)

So far so good with Euler relation.

In figure 1 we have on the one hand a maximum angle qmax, for a corresponding displacement Smax. That point, where these values are reached, is a point of static equilibrium within a dynamic system . It is from this point that we have a returning point.

In this same figure we have two axis:

a dual or symmetry axis, X and
a non dual axis Y

These two axis define the complex plane in this case, as it is this plane the one that serves as referential frame or background to represent dynamic entities such as the pendulum.

It is important to note at this point that in that figure 1, the inertial force is identified with the tangential force and the principle of equivalence between weighting mass and inertial mass hold, so it is not strange to find that the tangential force as a component of the weight be equal to the inertial force.

Figure1. The pendulum and the complex plane

3. A non classical solution for the pendulum?

The solution of this pendulum system will be done in two complementary phases, but in a certain sense, independent from each other:

· one phase related with the spatial representation, in which we have those special points of equilibrium

· one phase related with the dynamic behavior properly speaking that will permit to introduce time.

In regards to the former, if we integrate the trajectory DS between S = 0, and S = Smax, then we have:

Smax Smax Smax

ò DS = Cosine (qmax) ò DS + i Sine (qmax) ò DS

0 0 0

so:

i(qmax)

S = Smax e

We have thus obtained a static expression, as it were, a potential expression, associated with the path in space. Its dynamic counterpart will be introduced by multiplying this expression by the basic unit of time given by:

i(wt)

e where, q = wt

Time is then an independent sphere of reality introduced in this case by means of a non linear process, but immersed too in the complex plane. Time is not reduced in this case, to a time-made-quantity, or to a time assimilated to a space dimension as that one used by contemporary physicists, but in this case we include both aspects of time. Time is not described just by a process taken from extension as Bergson would say, but by a rotation, or a process that can be qualified as qualitative, as a counterpart process derived from extension.

It is precisely here where we have a clear deviation with that conception of time as conceived by contemporary physicists and specially as is presented by Stephen W. Hawking in his Historia del Tiempo(7, 178), which is as a matter of fact a consequence of that conception of time introduced by Minkowski in his well-known paper Space and Time(5,75); paper in which the bases of the geometrization of space was introduced for the first time and that was then the starting point of general relativity.

Hawking wrote:

“... se supone... que con cada una de esas historias está asociada una pareja de números, uno que representa el tamaño de una onda y el otro que representa su posición en el ciclo(su fase)... hay que sumar las ondas correspondientes a historia de la partícula que no está en el tiempo <<real>> que Ud. y yo experimentamos, sino que tiene lugar en lo que se llama tiempo imaginario... Para evitar las dificultades técnicas en la suma de Feynman sobre historias, hay que usar tiempo imaginario. Es decir, para los propósitos del cálculo hay que medir el tiempo utilizando números imaginarios en vez de reales. Esto tiene un efecto interesante sobre el espacio-tiempo: la distinción entre tiempo y espacio desaparece completamente. “(7, 178)

Even though we recognize as a good point the utilization of complex numbers, as a indispensable tool to represent physical reality, it is evident though the reductionistic tendency in that conception of time of contemporary physicists in which time and space are a same thing, or in other words, they are always conceived as been coupled, which certainly does not agree with the arrow of time where time is an irreversible entity, not an entity that like space can run in both directions. This is but a mathematical abstraction used for the sake of applications.

On the other hand we do not agree anymore with the term “imaginary” used by Descartes, for the first time, for complex numbers, as the starting point of this deduction of the pendulum formula is precisely the complex plane that coincides with its dynamic reality. Instead of the imaginary character what seems clear is that dual character, or bilateral symmetry associated with its behavior in space and with the term i. Hermann Weyl was then right to dedicate a whole book to this important concept that today is the base of those processes that have to do with the chemistry of nuclear interactions.

The complex reality of the pendulum is then represented by

i(wt + qmax)

S(t) = Smax e

This expression is then a rotational system in the complex plane and not just an instantaneous picture of privileged moments; so our problem is reduced to find the state of this complex expression, which means we must find a relation between its two state variables Smax and qmax.

The natural state of the pendulum is related with a cyclical phenomenon, kind of spiral movement, in which each step is followed by another, going back to sort of new origin, but, is not this an external manifestation of the same principle of synergy as we have sort of “spin” property that can give us a whole greater than the sum of its parts, where that greater is explained by its permanent interchange with its environment?

In that figure 1 we have that

Smax = L qmax

where L is the radius or the length of the cord, so replacing we have:

i(wt + qmax)

S(t) = L qmax e

4. Equality and dynamic equilibrium and the pendulum

We have then two kinds of forces:

· inertial or tangential forces and

· gravitational forces, due to the earth gravitational field which is in this case the external environment in which the pendulum is immersed.

When we talk about the acquisition of the steady state and a dynamic equilibrium, we mean that along the trajectory there is:

· kind of non equilibrium between inertial and gravitational forces, that makes it possible precisely that pendulum movement,

· but we have too privileged static points that make it possible to determine the steady state of the pendulum.

On the other hand there is a third type of force, due to friction and air that make its entropic contribution to the whole system, but this type of force must be compensated as sort of secondary regulation if we want to preserve the steady state of the whole system. It does not have any sense to think that will have sort of perpetual motion in this case. It is correct to say that if we do not compensate these losses, entropy is the negative of synergy.

If we take into account the inertial forces and a resulting force along the trajectory, we must then apply Newton’s second law, but this time in a generalized way, I mean, expressed in complex numbers. So the first derivate of

i(wt + qmax)

S(t) = L qmax e

is:

i(wt + qmax)

dS/dt = L qmax i w e

and its second:

i(wt + qmax)

d²S/dt² = -( L qmax w² ) e

being, amax = ( L qmax w² ), the maximum value of acceleration.

So according to that second law, we have that at the maximum point we have the inertial vector as:

Fmax = m (L qmax w² )

If we apply the composition of forces as shown in figure 1, in that point of a maximum arc, Smax, the total forces on the bob become minimum in that moment inertial force is equal to weighting force. Both, this equality of forces, as that tendency to preserve the form, make us think in the generalization of the first and third Newton’s laws not just applied to linear movements anymore.

We have then a change in the direction of movement in that Smax point, having in this way the pendulum movement.

The weighting vector gives us in that Smax point:

Fmax = m g Sine(qmax)

And

L qmax w² = g Sine(qmax), so

w = Ö (g Sine(qmax) /( qmax L))

being this the expression for the angular velocity, but if w = 2 p f and T = 1/f, we have the period as:

T = 2 p Ö ((L/g) ( qmax /Sine(qmax))

Which is precisely the pendulum formula, with correction factor included that can be validated not only with what is observed but with that solution obtained classically by means of an elliptic integral(1,744), as is shown in table 1.

Angulo	0°	10°	20°	30°	60°	90°	120°	150°	180°
K	1.571	1.574	1.583	1.598	1.686	1.854	2.157	2.768	¥
2k/p	1.000	1.002	1.008	1.017	1.073	1.180	1.373	1.762	¥
q/Sine(q)	1.000	1.005	1.0206	1.047	1.209	1.570	2.418	5.235	¥
Error-%	0.000	-.304	-1.267	-2.853	-11.23	-24.8	-43.22	-66.34	N/A

Table 1. The pendulum and its correction factor by two methods

This factor, Ö(qmax /Sine(qmax)), tends to one when qmax is small, having then sort of linear behavior, but this is not the case in general what seems to suggest that time and space are not always coupled or time cannot be always assimilated to space dimension.

On the other hand it is from this pendulum formula where all those observed laws found by Galileo, can be derived such as:

· the oscillation period does not depend of the amplitude within the limits, as can be seen in the table, of 10°

· the period of the pendulum does not depend of the substance or material

· the period of the pendulum is directly proportional to the square root of its length

· the period of the pendulum is inversely proportional to the square root of the acceleration or local gravity.

ã, 1991, 1999, 2001. Edgar Paternina. Electrical Engineer.

email: [email protected]

August 27, 2004

References

1. Beer/Johnston. Vector Mechanics for Engineers. McGraw Hill. 1962

2.1 Bergson Henry. Creative Evolution. UPA.1983

2.2. Bergson Henry. Memoria y Vida. Altaya.1994

3. Drake Stillman. GALILEO. Alianza Editorial. 1980

4. Dibner* Drake. A Letter from Galileo Galilei. Burndy Library. 1967

5. Einstein Albert et all. The Principle of Relativity. Dover P.Inc.1952

6. Feyman Richard. Física. Mecánica Cuántica. Volumen III. Fondo Editorial Interamericano. S.A.

7. Hawking Stephen W. Historia del Tiempo. Editorial Crítica. 1988

8. Kuhn T.S. The Structure of Scientific Revolutions. The University of Chicago Press . 1996

9. Scott R.E .LINEAR CIRCUITS I & II. Addison-Wesley PC. 1967

10. Weyl Hermann. SIMETRÍA. McGraw Hill/Interamericana de España. 1991