Wednesday, December 28, 2005

Particle kinematics

For the past while, I've been feeling stretched rather too thin, as If there are too many things to be working on and not enough time and focus on any one of them. So, I'm taking another cycle at a different pace.

Starting science, and within science, starting on physics, and within physics starting with mechanics, and within mechanics starting with classical mechanics, I've spent a couple of days working on particle mechanics.

The first subject is particle description, but this is fairly trivial. A particle is a geometric point, with certain fundamental and derived quantities belonging to or associated with it. As such, it is a mathematical abstraction, but a very useful one.

Kinematics, the description of the motion of particles, is a more interesting subject. I have this roughly divided into quantities of position, velocity, acceleration, higher quantities, and cases of motion. These underlie and do not depend heavily on particle kinetics (the study of momentum and force), particle energetics, or higher particle mechanics, since kinematics is the description of motion without regard to its causes. This overlaps somewhat with rigid-body mechanics, which can sometimes be treated as particles, but doesn't depend much on nonrigid bodies. other areas of classical mechanics are more useful in mechanics in general, other areas of physics, and other sciences are more useful in presenting applications of this study. Study of the particular individuals or scientific groups that have worked on this subject is a fairly low priority. Aristotle studied motion, and Galileo and Newton pioneered the modern and more effective study of it, but it has been otherwise developed and clarified by others.
I cannot give too much attention to the language and literature of particle kinematics before studying those subject themselves. Graphics and graphic methods are also used. However, the single most important area is mathematics. Many physics texts connect this to measurement. Until I have more informtion on philosophy, I can't really comment on this. Newtonian particle mechanics has been explored and tested thoroughly enough that it is no longer considered an experimental science, although instruments for study of the various quantities are continually being refined. By itself, without connection to other areas, particle kinematics is fairly simple, and the difficulty of learning or teaching it is tied to how much mathematics is involved. It is closely tied to mathematics in its extent among peoples and in history.

The hardest part is associated with the description of position. Verbal descriptions of "above", "behind", "to the right" and so forth, are insufficient for scientific purposes. It is necessary to refine exactly where the starting point is, and how far above, ahead, behind, or to the right some object is. Descriptions in two or three dimensions are more complicated and involve direction as well as distance. There are various possible ways (called coordinate systems) to describe position, and much of applied kinematics invovles translating descriptions of position from one coordinate system to another. Part of the reason is that it is easier to describe position in some systems, but easier to describe changes of position, or motion, in others. Another area of interest is particular paths of motion.
Once position is described, the next quantity of interest is velocity. This has no relation to location. Unless the motion is confined to some particular curve, only the changes are of interest. Velocity includes the rate of change of position, measured in some unit of distance per some unit of time (for example, miles per hour, or feet per second) and may involve conversion from one system of units to another. However, since position may take account of direction, a full description of velocity must also include direction. Changes in velocity may include changes in speed (the rate of motion) and changes in direction.
The next quantity of interest is acceleration; which is the rate of change of velocity. Again this has no inherent connection to either position or velocity unless the motion is confined to some particular curve. Acceleration is typically measured in velocity units per unit of time (as in, for sports car enthusiasts 0-60 mph in 10 seconds gives an average acceleration), which can be converted to others. Galileo was the first to study acceleration in connection with falling objects, and the accelleration due gravity is still a favorite subject of introductory mechanics, enough that while there is not e common unit of velocity, the "g" is a common unit of acceleration.
There are other nigher kinematic quantities, but these are of interest only to specialists.
There are important special cases of motion; including motion with constant acceleration, which includes sub-cases of constant velocity, or uniform motion, and constant position, or rest. Another case is simple harmonic motion, which is a mathematically simple type of confined, back-and-forth motion, and another is uniform circular motion, which is also confined to a plane, and is a case of constant speed, but changing velocity (because the direction of motion is always changing, towards the center). This counts as accelerated motion. These special cases are simple and important enough, that many students go no further in their study of kinematics. However, there is an infinite variety of curves, constant motion is rare in nature, and constant acceleration is nearly as rare, for those who might wish to stretch their skills in examining other cases.