It is fair to say that Force is the linchpin of mechanics. Sir Isaac Newton made it so when he penned his Second Law,
F=dP/dt. (1)
Newton called P the “quantity of motion.” In that great man’s own words,
“The quantity of motion is the measure of the same, arise from the velocity and quantity of matter conjointly.”
In modern parlance, we define P to be the product of an entity’s mass times its velocity:
P=mv. (2)
Hence Newton’s Second Law can be written as
F=m dv/dt+v dm/dt. (3)
It is one of the most important formulas in physics. There are many different kinds of forces in Nature … gravity, electromagnetism, friction, gas pressures, nuclear, … The list seems practically endless. Yet they all agree with Eq. 3.
Now a matter which grew in the interest of physicists, after it became clear that the propagation speed of light is the same in all inertial frames, had to do with measurements in different inertial frames of reference. More explicitly the question “If I measure things in inertial frame K, then what will the same measurements, made in frame K’ (using meter sticks at rest in K and K’ respectively), be? For thousands of years the answer seemed to be so trivial as not to require any thought. But the revolutionary discovery, that all inertial observers will measure the speed of light to be the same, stimulated much thought in some of the most celebrated minds of all time. With regard to space and time measurements, the “self-evident” and trivial Galilean transformations evolved into the practically dumbfounding Lorentz transformations.
Once the Lorentz transformations of space and time intervals had become known, curious minds began to wonder how other key variables like mass, the electric and magnetic fields, etc. transform. Before long the venerable Force of Newton became a matter of interest. Involving space, time and mass … all different in different frames … it is not a trivial derivation. But in due course it was figured out. Given a force F (and its rectangular components) in frame K, the components using the measuring rods etc. in K’, are related to the components measured in K by the force transformations:
Fx’=[Fx-(v/c2)(F dot u)]/(1-vux/c2), (4a)
Fy’=(Fy/γ)/(1-vux/c2), (4b)
Fz’=(Fz/γ)/(1-vux/c2). (4b)
(In these equations, u is the velocity of the entity being acted upon by F.) Clearly that is more difficult to remember than the Galilean transformation, F’=F. But the heuristic value is enormous. For if one knows Eqs. 4a thru 4c, then one can make immediate conclusions about gravity, electromagnetism, friction, gas pressures, nuclear, … the list is practically limitless. Do you wish to know what body temperatures onboard a (future) space ship will be from an earthbound point of view? A good starting point might be the Lorentz force transformation. Do you wish to calculate the pressure of light reflected by mirrors at rest in K and K’ respectively? Start with the force transformations. Gravity, electromagnetism, friction, etc… they’re all expressed on the right side of equations with Force on the left side.
The author’s advice to all minds … both those starting their journeys in the many facets of physics, and to older minds who have perhaps been dissuaded over the central importance of Newton’s Force and the transformation thereof … Learn this transformation. It is the mother of all transformations for those wondering about pushes and pulls in different frames of reference.