Using “Newtonian” space and time, and dividing space by a standard co-ordinate system (the origin being of a totally relative nature).
Quoting from J. C. Maxwell, article 18 of his treatise on matter and motion; “so there is nothing to distinguish one part of space from another except its relation to the place of material bodies… We cannot describe the time of an event except by reference to some other event… or the place of a body except by reference to some other body… All our knowledge, both of time and place is essentially relative”.
Well, yes it is, but this is not unusual or bad. If space where finite, indeed you should be able to compartmentalize it due to its edges. But with infinite space, yes what Maxwell says is true. Except you should not preclude that this means there are no types of absolutes. For real observations and experiments there is no way to include absolutes, except perhaps by first knowing the absolutes going on in the un-observable primary sphere of events. Here one must resort to theoretical considerations first.
Consider any particle A in space at an instant of time, and with a constant velocity in a straight line. Let the particles position at this instant define the origin of the co-ordinate grid. Let the direction of motion of the particle define the X-axis, the other axes are then perpendicular to it (and passing through the point of origin).
For a subsequent instant (after some interval of time) the velocity of the particle and its position can be figured from its previous position, so therefore its theoretical absolute velocity and relative position [Classically this couldn’t be applied to any real (observed) system because a point is an area of space which may also have superimposed motion ad infinitum. But not so here as all motion is relative to a theoretical position in space and space is motionless. But from a theoretical sense it can be considered because you state particle A as having no other motion than its linear velocity. Or it is at absolute rest if it has no motion].
As seen in general in this hypothesis the use of absolute motion is not so removed from calculations as it is in classical mechanics because when PP collide their absolute motion (all motion) must be involved to conserve motion, and in many cases is completely stopped for an instant in time.
Because we assume a particle has a constant velocity, and because no forces are considered and (therefore) all motion is transferred through collisions, and when two rods are in contact the motion is transferred instantly (see appendix B), therefore all velocity is always of a constant nature and no true acceleration exists on a primary level.
So, no calculations involving acceleration are used here.
Absence of motion is absolute rest, that is, over any interval of time (duration) a particle does not change its position in space. The except to this is 180 degree opposite motion (see Contrary Motion section below).
Motion is a particle moving with some velocity, that is, over any interval its position in space changes (same exception as in first paragraph).
If all points of a rod move in a constant linear direction, the rod has a constant linear motion. Because the rod is assumed immutable, all points of the rod always move in unison.
Rotational motion can be defined for the rod as any uniform circular motion around a point on the rod that has a continuously changing point of rotation.
Also there is orbital/circular motion as described in chapter 3. And faux orbital/circular motion as described in chapter 8. Also orbital motion caused by torque release and following a rotational motion as described in chapter 3.
Plus rods can have compounded motion as explained in chapter 8 under vectors.
CHART A-1 All Possible Motions
A. Simple Motions
B. Combined Motions
Any combination of the simple motions. Each to other of the same or different motions. Except torque does not combine with torque. Theoretically rotation and rotation could combine if in complimentary directions, but in practice is probably always re-rotation off one or the other point of contact.
For a rod (A) moving through space with some velocity (S) and rotating with some velocity (R) - if the midpoint of the rod could be as a piece of chalk and the plane it travels in space be as a blackboard, it would, over some time interval, scribe a line as S1. However, its real velocity is as S+R which can be figured with standard geometry (given the initial and final configurations) as in figure A-1.
So the apparent value S1 < S+R, but by definition motion must displace space so where is it to be made up? It can be accounted as space covered simultaneously for the allowable combined motions as above. So the “observed” value for displacement is not equal to the actual total displacement.
Section 4 Contrary Motion Vectors
From chapter 2 – Direct Hits. If the motion of the particles is a single vector of motion then it is as stated in chapter 8, but if either or both particles are moving through space with compounded motions (overlapping) then treat each vector component separately in the calculation of the impact results. So they may stop all of each other’s motion, or they may not, depending on the angles of the motion vectors. For example if B has a single vector and A has two vectors as in figure A-2 then at impact B’s motion is stopped but for A only A1 vector is stopped (both A and B undergo torque & rotation). But the A2 vector component continues, and in pulling away frees up some of B’s original motion to continue, and A will slide relative to B as it torques and rotates. End results therefore will vary depending on the speeds involved.
So only impact
within a 90° range gets compounded. When < 90° the force in the two or more vectors can force movement of the
rod without “pulling it apart”, and still cover the distance and direction
involved. >90° it does not, and would “pull apart” the rod. But this
situation never arises from direct impacts >90° as it cause the linear motions in the rods to stop, as described
with figure A-2. However in many situations after rotation this can occur. Figuring for this is below.
Vectors after Torque and Rotation
For vectors after rotation causing midpoint to midpoint contact and other cases as per chapter 2, consider this:
Principle 19.2 For cases where divergence of two linear vectors is > 90o the two motions are not compounded but drawn together as one
vector. However not as standard vector analysis either, but the drawn vector is in the spot as triangulated, but with the length of each vector added.
Principle 19.4 If two linear motions within a rod (A) are 180o opposite the motion within the rod, the rod is stationary in space until impact from another rod (B). Then, in principle for a midpoint to midpoint collision, all motion in the direction of the (direct) impact is transferred to the impacting rod (B), and the rod hit (A) goes off with the its other linear motion.
Of course the real case of impact in 19.4 is more involved. B torques and rotates, then causes A to re-rotates on it. Then when A reaches its midpoint, at the instant of transfer of motion it will have 3 linear motions, the two 180 degree opposites and the new one at discharge from B. My take on this is as follows.
Principle 19.5 For combining three linear motions, two of which are in 180 degree opposite directions, each old motion "separately" compounds with 1/2 of the new (non-opposite) motion as it where to two new resultants. Those two resultants compound to a further resultant, and in that process the first 3 linear motions are erased and the resultant consists to the two secondary motions combined to single motion.
Principle 19.6 following principle 18, in cases of a rod torqueing in contrary directions, or torqueing and rotating in contrary directions, the rebound is always at a midpoint from the two opposing motions, and expressed as a resultant linear motion.
For rotational and circular motion the situation is more puzzling as to a possible explanation in that indeed the linear and rotational motion cause 90° divergence ( that is points on the rod that are going forward and backward simultaneously). But indeed spatially in space the motion is expressed again as a type of overlapping motion. That is in rod A’s rotation on B, A’s linear motion is complete and the rotational motion is added on. So a Principle :
Principle 20: For motions to be compounded they must be expressed in movement “displacing space”, (which can be overlapping) completely and separately as to the distance and direction of each of the compounded motions, relative to the initial position of the rod.
Discussion on Rotational and Linear Motion in General
Further thoughts on combined motions.
Further thoughts on combined motions, particularly rotational and linear. I am inclined to stay with rod tearing apart if in two opposite directions. Somehow with rotation it is possible to combine without tearing apart. ONE point there is moving same as linear, perhaps somehow that makes all the difference. This theoretically could make a rod traveling linearly also rotate off a point and at the same time another point. There is always a common point that goes linearly without change while rotation occurs, even if it is not one of the points rotation is occurring around, and even if that point is outside the rod. This is true to for orbital motion also, rather in contact with the pusher rod or free from contact. Note this common point can change as long as moment to moment it is having common linear motion to both. With two opposite linear motions no such point of common motion occurs. See also the new section on torque and follow mechanics in chapter 3.
I going to take a leap and lay out a principle I think must then be true:
Principle 19.7 Opposite motions in a rod can occur in compliment/combined fashion if there is at least one common point of motion. Otherwise the resolve to a singular new motion, except in the case of perfectly opposite (180 degree) motion which causes vibration with in the rod.
General Discussion on Motion
So equal volumes travelling at equal speeds sweep equal areas. But a full rotation of a rod with a fixed point of rotation on the rod is not the same as a orbited rotation around a fixed point off the rod. So how to relate the area swept from the two. In linear motion all subsections travel same vector and area swept. Rotating rod not so, but the area swept per time is same for same velocity of linear motion. So, a philosophical question of why can subsections of a rod rotating have larger area swept without rod breaking etc.?
Rotation around one point, but it can also be around two points at once, however they seem to become reciprocal to each other such that a midpoint between the two becomes the point of rotation. Likewise, for three or more points of rotation, mathematical points on a rod, they resolve to a single point and single rotation. So, then rotation becomes a singular point. With real rods in contact rotation is not free wheel (multiple rotation points) but constrained as per principles already given. But even here without realizing it these principles resolve rotation, when occurring, to a single point. But is not the momentum in all the rod all the time, that is upper half sweeps more area, but any point has all the momentum of the rod in it as motion in rod is instantaneous and everywhere.
Linear and rotational motion can combine. But considering linear only and rotational only, is there another motion other than that? A curve is it seems just their combination. So, every curve should be resolvable into a rotation and a linear component. If the linear motion is one dimensional then the combo becomes a sinusoidal curve? If two dimensional linear (two compounded linear motion) then what? Since two linear motions are resolved into one, then with the rotational its just a sinusoidal in a different direction.
With two points, getting from one to the other is a straight line, or series of straight lines with angles. Or circular curve, irregular curve, or elliptical curve, etc. In real world of rods, all curves are brought back to linear by doublet formation. Then all curves are generated by rotational and orbiting as perfect circles. So even if random of all sorts in start of the universe it would resolve to those things only. So only macro motions by of fits and starts create other types of motion on the macro level. On the primary level no other motion than rotation or linear or their combined motion of a sine.
Okay I think I got it, its this simple, a curve is which any two consecutive points are continuously deviant from a straight line. Consecutive points within the context of inf #’s that is. Continuously changing in that from one set of points to the next set is continuous deviation from a straight line between those points. Straight line has no deviant points. Curve is always deviant. Only two types of motion then, curve or straight. Curve can have any rate of deviation though. Rate of change is a measure of any chord, and a parallel chord and the angle they engender between points, which will be the same for the same curve, different for different curves. I do NOT believe a curve can be continuously changing its angle. Only segment to segment can this be done.
It’s weird how a rod rotates, seems as if ½ being held so can't displace like linear. And is faster at top, so what’s going on? Even moving in different directions on opposite sides of the rotation point. Like if icicle and throw it and it moves in two directions it would be because it broke. But twirl it around your finger and it does two different directions so to speak, but does not break. With linear all moves in unison, so if rotation does not break must all be unison too. But unison spin. So, rod is one piece but spinning around a point. So faster and slower section is still a unison motion, in fact necessary for unison motion in spin, if all motion of rotating rod was equal , it would not be rotating after all, that would have to be linear.
Indeed, opposite ends of the rod in linear motion travel in the same direction and speed. Opposite ends of rotating rod travel opposite direction and different speed. But this is the nature of spin, Its around, not from, a point. But all the rod is involved, even at end point is still spin just from point around itself. Fundamental I guess. But interesting to note, spin is around a mathematical point, not a mass area. If a mass area it would violate principle of immutability and break the rod. So mathematical point is not area, but still involved in real stuff.
So out of this rambling several thoughts to keep. Rod has only two primary types of motion through space. Linear, or Spin. The rest of the rod motions through space are a combo of these two. But can only be expressed in a rod as loopy motion, turning into a sinusoidal motion at lower rotation to higher linear differentials of speed.
This is true even for multiple compounded motion of a rod, as the rotations and linear motion are each resolved into one expressed motion, though the reality may be different.
Orbital motion in particular is derived from rotational motion so in pure form is still a perfect circle. The only exception to all this is the in the 3P type of case while the orbited rod is orbiting down it will not have a perfect circular motion as its primary motion.
All other motions then in the universe are from macro scale effects of fits and starts of N from these primary motions.