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A basic class of
processes, especially for the material objects is represented by
those specific processes in which the variable attribute is the
spatial position of an object (in case of an individual
process) or of a set of objects (subject which shall be minutely
analyzed in the next chapter focused on fluxes), which are called
motion processes. We have seen in the previous chapter that
the position of an object is in fact the position of the object’s
inner reference against an external reference. The object’s
motion is therefore a variation of the position of this internal
reference against the external reference. This motion may be
decomposed into motions of the internal reference system’s (RS)
components against the same components of an external RS. The spatial
RS which were briefly mentioned in the previous chapters are the T
reference (common component of the reference axes), and R
reference (the specific directions of the three axis X, Y, Z).
The spatial position of the T inner reference of the object is
given by the position vector
which
unites the two references - external T reference (application
point which is invariant by definition), and the internal T
reference of the object. Since the position vector has an invariant
application point, it belongs to the class of bound vectors, its
other attributes besides the application point being the size
(module, magnitude) and the direction (defined in relation to the
external R reference).
The overall (global) motion of an object shall be made-up, according to the above-mentioned issues, from the variations of the two elements which define the object’s spatial position, the two components belonging to the spatial RS: T and R references. Because we are dealing with two qualitatively different attributes, we shall have two types of specific processes.
Definition 4.8.1: A specific process of external variation of the inner T reference position of an object is named translation.
If the direction of the translation process remains invariant, we are dealing with a pure translation (motion on a rectilinear pathway).
Definition 4.8.2: A specific variation process of a vector’s direction is named rotation.
If the spatial position of the application point of the rotating vector is invariant, we are dealing with a pure rotation. One may notice that pure rotations are mostly assigned to the position vectors, either against the internal R reference (for the object’s elements) or against the external reference, for the object as a whole.
Definition
4.8.3: A translation motion of an object during which its
position vector performs a pure completed rotation (2
radians)
is named revolution motion (around the application point of
the position vector).
Comment 4.8.1: The reader is invited to observe the approaching differences on the objects motion, between the objectual philosophy and the classic approaches. The objectual-processual approach specific to this paper takes strictly into consideration the variable attribute, because a single qualitative attribute must be variable in case of a specific process. According to this approach, the motion of an object means the motion of its inner reference system against an outer reference. T reference belongs to the inner RS and it can be considered as a null rotation point, therefore, it is able to generate only translation motions, and R reference is made-up only from directions, and as a result of this fact, it is able to operate only rotation motions. Thus, it becomes very clear that the rotation motions can be attributed only to the vectors (either the position ones or the ones which make-up R reference). The rotation of an object means also the rotation of its inner R reference (against the outer R reference), that is a rotation which generates however a lot of revolution motions of all the inner elements of the objects which have non-null inner position vectors.
Depending on the above-mentioned definitions and on the variable attribute, we could point out few specific motion processes, which shall be presented only because that the reader to understand the objectual-processual approach of the motions:
invariant
both as a module and as a direction, but the angular position of the
inner R reference is time variable; in this case, we are dealing
with a rotation of the object, decomposable along its three
possible simultaneous rotation axes and determined against the outer
R reference. As we have also mentioned in the comment 4.8.1, it is
worth noticing that a rotation of a compound object is decomposable
in the same number of inner revolution motions of the object’s
elements.
time variable
but only as a direction, the variations being coplanar, a circular
revolution motion of the object occurring in this case (it is
decomposable as well and determined against the outer R reference).
Comment 4.8.2: If there is an invariant relation between the direction of the position vector and the inner R reference of the object which operates the revolution motion, then a rotation motion of the position vector determines also a self-rotation of the object (around an axis which is parallel with the rotation axis of the position vector). This is the case of S-type media (which shall be defined in the following chapters, but which are momentarily assimilated with the solids), in which there are invariant relations both between T references and between R references of the constitutive objects (the free translation and the free rotation of the elements are forbidden). In this case, a rotation motion of the S-type compound object determines, besides the revolutions of the inner elements, also a simultaneous rotation of each constitutive element (a curl vector field with an even distribution is generated).
variable
,
but the successive variations are collinear, a pure translation
(obviously, an external one) being produced in this case. As also
regards the pure translation (except the case when its direction
coincides with the direction of the position vector), we are
simultaneously dealing with a rotation of
(finite but non-null), and if the condition mentioned in the comment
above exist, we shall simultaneously have a self-rotation.
Comment 4.8.3: It is well-known that in order to maintain an electro-magnetic communication with a space probe, directional (paraboloidal) aerials are attached both on the probe and on the ground station, whose axes must be kept as collinear as possible (an invariant relation must exist between their directions). The space probe is a S-type object with an inner RS against which the aerial has a steady direction. The probe trajectory is usually a closed curve, its position vector (against the terrestrial RS) has a variable direction, therefore, either the ground control centre, or a control system within the probe must permanently initiate the forced rotation of the probe in order to align the axes of the two aerials, simultaneously with the rotation of the axis of the aerial located on the ground. In this case, the direction of the ground aerial axis corresponds with the direction of the position vector of the probe against the terrestrial RS, direction against which the control system of the probe’s rotating position must maintain an invariant relation. It is clear that the probe’s rotation depends only on the direction of the position vector (regardless of the distance towards the probe), and the intensity of electro-magnetic flux received from the probe (which depends only on the distance, in case of the alignment of the aerials axes) is independent from its direction. This example implies an artificial invariant relation between RS of two objects (that is a informational relation maintained by some automatic control systems), but there are also numerous situations in which this relation is natural. That is the above-depicted case of S objects, but also of the natural satellites with isochronous orbital and axial movements (such as the Moon, the great Jupiter’s satellites, etc.). As also regards these satellites, there is an invariant relation between the inner R reference of the satellite and the direction of its position vector against the central planet.
At the paragraph 4.5, we have seen that a certain specific and individual process may be decomposed into concatenated SEP, and these SEP are uniform variations with the invariant direction (vectors). According to the above-mentioned motion types, we may notice that the translation process belongs to this SEP class. Therefore, any type of translation motion is decomposable in series of elementary translations. We might also add that a pure translation with a constant velocity is a P1-type of process, while its temporal density (uniform velocity) and its direction, a S1-type of state.
Copyright © 2006-2011 Aurel Rusu. All rights reserved.
