A “reversed viewpoint” of the momentum expression leads to a new energy expression and a new view of Mass and Energy.
The integration of the momentum expression, mv=p, mass times velocity equals momentum, with mass considered constant, leads to the expression for Kinetic Energy, KE=(mv^2)/2. This is well-known.
Apparently overlooked is the fact that since “mv=p” is an equation of the type, xy=K, that is two variables equal a constant, it can just as legitimately be integrated with velocity considered as constant and the mass allowed to vary, to obtain an analogous, “Alternative Energy” expression, AE=(vm^2)/2.
Since mass and velocity can both vary it seems reasonable to consider that a true energy expression would take into account both of these expressions. One way to do this is simply to write TE=KE+AE=(mv^2)/2+(vm^2)/2. We simply assume that a “True Energy Expression” would be the sum of the two alternative expressions.
Assuming that a quote attributed to Albert Einstein, “Mathematics is the Reality.,” has a degree of pertinence here, let us look at what conclusions may be reached by checking out this expression in various situations.
Three situations come immediately to mind. Where the energy expression will equal zero, where the energy expression will be at some maximum, and situations in between where whatever the expression applies to is a constant value, neither a minimum nor a maximum.
In the first case, we write, 0=(mv^2 + vm^2)2 . For the entire expression to equal Zero, both expressions would individually have to equal Zero, or the two would have to be such as to cancel. Let us consider this possibility.
We can remove the expression (mv/2) by multiplying both sides of the equation by “‘ 2/mv” , leaving 0 = m + v. Therefore, m = -v or v = -m. In either case, mass is a vector, directed quantity, opposite in direction from the velocity considered.
For an Energy Maximum situation, we can assume both Energy Expressions to be in the same direction and both maximized. That is we hit maximum velocity, the system could go no farther and additional energy on the same vector appears as mass. As the mass increase will change both of the expressions they will maximize at a point of maximum velocity in both expressions. As “c,” the Speed of Light” is generally considered to be a limiting velocity in our Universe, we assume that the limiting velocity is “c.” We can therefore write, Maximum Energy equals (mc^2)/2 + (cm^2)/2. We have noted that the two expressions must be equal, we also remember that “mass” can be considered a “velocity” vector. Hence, (mc^2)/2
(cm^2)/2 = (c^3)/2, and our maximum energy content appears to be c^3, for any system which we wish to analyze.
As this would be the maximum energy directed in any direction, it would be the maximum energy which can be directed outward from any point, or inward toward any point. In oscillatory motion, this would be the directed energy content at an extreme of vibration or pulsation.
Energy may perhaps be considered a "3-D Motion Package" of which mass is one dimension. Any three numbers multiplied together define a volume....
A common, and interesting, situation would be where the Energy Content is a constant, neither Zero nor Maximum. In this type of case, wherein mass and velocity can vary, but momentum and total energy content cannot, we have defined oscillatory or orbitory motion. As mass increases, velocity decreases and vice verso. we can rewrite our summary equation in several ways, considering that mv=p we can write, E (pm + pv)/2 or E/p=m+v, or E/mv=(m+v)/2. There will be some point at which m=v= p^0.5. At that point, when we put it all together we find that E=p^1.5. (E=(p)^3/2).
Another interesting situation would be wherein all motion was in the (vm^2)/2 package. The forward motion is a constant, but what does our motion package now mean? Forward motion is of a point along a line. If forward motion be static, then we can consider motions referenced to the point on the line. That is, as Kinetic Energy is known to reference to the motion of that point along a line, the AE expression would apparently apply to motions, i.e., vibration and rotation, referenced to the point, vibration/rotation/pulsation, etc. within the body/entity/system which we are considering.
It is said that Mass and Energy inter-convert. What appears to be the true case is that the two energy expressions inter-convert. Since Mass is a much more predominant factor in the second equation, when it decreases with permanent loss of energy from the system as some form of Kinetic Energy, we are “semi-correct in saying, “Mass is converted to Energy.”
It appears that neither Mass nor Energy has been well defined. There is extant, in some not well known work.* a definition of Mass as, ” The Pressure/Tension of the motions within a surface as against the remainder of the Substance of Existence.” This seems to fit, although it requires the acceptance of a “Substance of Existence.” This is, of course, would be an up-dating of the ancient idea of an “Aether” which was supposedly ruled out by the Michelson-Morley experiment…
Energy, by the same type of consideration, would be a more general term for a “Quantity of Motion.” (within the fundamental units of the substance.)
In summary, a mathematical look at the equation for momentum pointed out an alternative “energy expression” which seems to have been previously overlooked. Consideration of its possible significance raises some interesting questions about the nature of Mass and Energy and of their “interconversion.” Reference is made to a different theory of existence which may fit in with the ideas that appear here.
* See: http:/groups.google.com/group/oscillatorsubstance-theory
Post Script: After writing the above, several things were brought to my attention. One is that, while it has been assumed through out this paper that “p,”momentum, is a constant, this is not necessarily the case, In fact, momentum can be considered to be the ‘”instantaneous rate of change of energy.” If we do this we can integrate “p” as an entirety to obtain still another energy expression, E=(p^2)/2. Comparing this to the situation where in we decided that P^1.5 was a value for “E” gives the interesting situation that E=(p^2)/2 =p^1.5. This can only occur, if all units are dimensionless and “p” equals 4. This just might be the case as some very basic level with the number 4 indicating a dimensionality perhaps of rotation about three axes and pulsation about a ceentral dot…
The additional energy points out that there is a problem with units in the two other expressions, the interesting thing being that this expressiion has the units of m^2v^2. If either of the other expressions is multiplied by the “omitted factor.” om each they become equivalent to this expression. That is if Kinetic Energy be multiplied by the amount of mass involved, and Alternative Energy by multiplied by the amount of velocity involved, the two expressions become identical to teach other and to this “master expression.”