Apart from having a qualitative description of quantities such as momentum, work and energy, why are these quantities considered so fundamental? What is the reason to define them in the first place? We could define other quantities but why these specifically? They help us solve the equations of motion, is that their only significance which gives us such an important reason to give a qualitative description to them? Is it because of Noether's theorem i.e. we define these quantities because they are invariant under certain conditions? Is the qualitative description justified in any sense?
closed as primarily opinion-based by Sebastian Riese, safesphere, Emilio Pisanty, Jon Custer, ZeroTheHero Aug 18 at 1:31
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Momentum and energy are fundamental because they are conserved quantities. The total momentum and the total energy of a closed system do not change with time. In a world where everything seems to change all all the time, it is nice to find that some things do not.
Noether’s Theorem is one way to understand why certain quantities like this are conserved. They are related to the symmetries of the system. Energy is conserved because of time-translation symmetry, and momentum is conserved because of spatial-translation symmetry.
We define any variable we please the ones you hear about, like force and momentum, are the ones which have historically proven worthwhile to pay attention to.
There are plenty of variables that have died off. For example, in the early days of thermodynamics, it was thought that heat and cold were two different fluids. A fire produced one, and ice produced another. It took a while before people realized that the two sets of equations they had for heat and cold fluids could actually be just one set of equations if you treated "producing" cold as "consuming" heat. Now, we only teach "heat."
The quantities you learn are those which offer the most bang for your buck. You learn about force and energy because we find they are most applicable at the most fundamental levels and in the most places. You probably don't learn about quantities like the electrostriction coefficient. For one thing, it's a fourth rank tensor, and you wont learn about tensors until later. For another, most people have absolutely no use for calculating the effects of electrostriction. And when you do have a use for it, guess what... you'll be taught about it!
The conserved values are, in my opinion, special. They are the closest to fundamentals as they come, because you can't just get them whenever you want them. You have to take them from somewhere. If I want to execute a Judo throw, the angular momentum (a conserved value) has to come from somewhere. Often this involves using our muscles, expending energy (another conserved value). These budgets for conserved values turn out to be quite applicable in real scenarios.
If you slip on ice. Why did you do that? Well, you are used to being able to control the angular momentum imparted on your body thanks to the forces of friction on your foot. Step on ice, those friction forces go away, and you suddenly must be much more careful with how you acquire and issue angular momentum, because you are given fewer chances to get it easily.
Going bigger, consider Delta-V. This is how space craft measure their maneuverability. In our daily life, delta-v is just "acceleration" and its easily acquired by pushing against the ground. In the case of rockets, however, this acceleration is hard earned, and budgeted dearly. You can't get Delta-V back in space. So people learning about space travel learn about the quantity known as Delta-V. Those of us which live on the ground (or fly in planes through the air) tend not to bother with it.