Stoat,
I found what seems like two different methods to try to incorporate classical physics with more recent physics:
One is Lorentz's transformation and Einstein's interpretation:
(sorry about the equations, some symbols did not transfer correctly_ square root & greek letters)
This one relates to relative motion, and I think "radiation from accelerated bodies" is not addressed.
The second one is another attempt which includes flow of energy.
It starts with "F=ma does not hold in modern physics", then at end concludes that it still does!!
1)
"Selected Topics In Modern Physics: Special Relativity
There are two different 'relativity' theories (both due to Einstein). One is called General Relativity and is a theory of gravitational phenomena. The other, Special Relativity is the subject of this section.
Principle of Relativity: The equations of physics must have the same form for all observers in relative motion. Equivalently this states that there is no way (via physics) to detect absolute motion. All motion is relative, that is, compared with someone else.
As we saw in mechanics, the quantity that is easily detectable is acceleration. You know if you are accelerated relative to an inertial frame. (You can feel it when an elevator speeds up or slows down)
Relative motion means that one observer is moving with constant velocity relative to another. Between any two observers who are using different coordinate systems, we must have a transformation rule which takes you from one system to the other. For example, if O' is moving with a velocity u in the x direction relative to O, we have two possible choices for the transformation rules:
Galilean Transformations:
x' = x - u t;
y' = y; z'=z
Lorentz Transformations:
x' = (x - u t); y' = y; z'= z
(assumed: t'= t) t' = [t - (u/c)x]
where = 1/(1 - u2/c2).
Einstein's 'Dilemna': Newton's Laws (Theory of Mechanics) takes the same form (i.e. satisfies the Principle of Relativity) if the Galilean Transformations are used. Their form changes if the Lorentz Transformations are used.
Maxwell's equations (Theory of E&M) take the same form if the Lorentz Transformations are used, but change their form if the Galilean Transformations are used.
There can be only one rule. Hence, either Newton's Laws are correct (and the rule is the Galilean T.), or Maxwell's Equations are correct (and the rule is the Lorentz T.)
The theory of Special Relativity states that Maxwell's Equations are correct, and hence Newton's Laws need modification. Einstein created 'relativistic' mechanics by the following:
Keep: F = dp/dt as 2nd Law and p means m v
but change m = m(v) = m0 .
m0 is the mass of the object in its rest frame. The definition of KE must also be changed. If we define the kinetic energy as the work done by Fnet accelerating the object from rest, then we find:
KE = m c2 - m0 c2 .
Hence, Einstein called m0 c2 = E0 the 'rest energy' and moved it to the left side of the equation. He then called KE + E0 the 'total energy E'. Hence his famous equation: E = m c2. It is also possible to express the total energy in terms of momentum. This is:
E2 = p2 c2 + m02 c4.
Thus 'mass' can be considered as a form of energy (this is called the principle of equivalence) and one can look for ways to transform 'mass energy' into other forms.
Effects of the Lorentz Transformations:
In the Lorentz Transformations c is the speed of light in a vacuum and it is treated as a universal constant. (The speed of propagation of a light wave is measured to be the same by all observers independent of their motion with respect to the source. Doppler shifts occur with light but only the wavelength and frequency are changed.) All experiments over the past 90 years have confirmed this fact. Because of the Lorentz Transformations two observers in relative motion will never agree on the length of an object or the time for some event to occur! The reason is that for each observer, a 'length' is a distance between two points (say on the x-axis). Thus if O measures a length of an object at rest with respect to him L = x2 - x1, then the length of the object as measured by O' will be
L' = x2' - x2' = (x2 - x1) = L
Hence observer O thinks that O' sees a longer object. If we turn this around then O' thinks he is at rest and the object is moving so he sees an object with a shorter length than its rest frame length. That is:
L = (1 - u2/c2) Lrest similarly the time for an event is:
t = (1 - u2/c2) trest
For instance, let observer O' be in a space ship moving with respect to you at a speed of u = .98 c . Then u2/c2 = .96 and the square root factor is: (1 - .96) = .04 = .2 = 1/5. Hence a measurement of the length of his spaceship by you will yield a value 1/5th as long as he measures and, a one day long event (as measured by you) will appear to occur for him in 4.8 hours."
2)
"Relativity
Main article: Special relativity#Force
According to special relativity, an object's mass is not constant and instead depends on the object's velocity. While the relation
F = d(mv)/ dt
is still valid, the operation
d(mv)/ dt = m (dv/dt)
is not, and so the equation F = ma no longer holds.
Open systems:
So-called variable mass systems that are not closed systems, like a rocket burning fuel and ejecting spent gases, can not be directly treated by making mass a function of time in the second law.[14][15]
The reasoning, given in An Introduction to Mechanics by Kleppner and Kolenkow and other modern texts, is that Newton's second law applies fundamentally to particles.
In classical mechanics, particles by definition have constant mass. In case of well-defined systems of particles, Newton's law can be extended by summing over all the particles in the system.
In this case, we have to refer all vectors to the center of mass. Applying the second law to extended objects implicitly assumes the object to be a well-defined collection of particles.
However, 'variable mass' systems ... are not well-defined systems. Therefore Newton's second law can not be applied to them directly.
The general equation of motion for a body whose mass m varies with time by either ejecting or accreting mass is obtained by rearranging the second law and adding a term to account for the momentum carried by mass entering or leaving the system,[13]
F + u (dm/dt) = m (dv/dt)
where u is the relative velocity of the escaping or incoming mass with respect to the center of mass of the body. Under some conventions, the quantity u dm/dt on the left-hand side is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F.
Then, by substituting the definition of acceleration, the equation becomes, once again,
F = ma"
Stoat,
There must some equation (maybe in thermodynamics) which relates a force to motion caused by energy spent from a mass. For example what changes are required to the equaton F = ma if you consider the second law of thermodynamics, the universe being a dissipative sytem, matter accelerating as it tries to reach equilibrium towards a main attractor?
Note: Einstein's approach is to quantify energy into particles (which probably makes it easier for calculations).
The term "rest mass" is also used in today's physics. I think there is no such thing. If energy flows within a mass, then it is not at rest.
