I apologize in advance for the length of this post, if you wish to reduce reading skip to paragraph 5. If you are really lazy, skip to the final paragraph

I've had long running difficulties with these classical concepts. Not to say I can't apply or use them. I've taken three different mechanics courses (due to bureaucratic reasons) and gave excellent performances each time. However, I still maintain a feeling of unease as to what I'm actually dealing with.

All the instruction I've ever had gave defininitonal based explanations: (net) F = ma, momentum = mv, and energy is work and work is energy (this is also a point of confusion). 'These things are what they are becausethat's how they are defined.'I don't want definitions, I want derivations. Not in a strict mathematical sense but conceptually. I understand these are rather nebulous demands, so I will demonstrate my current train of thought.

Noticing that Newton defined Force in terms momentum, I like to use momentum as the base for all the other concepts. I'll start with the assumption:Within a closed system, the amount of mass and the amount of velocity associated with that mass never change. Closed system in this case defined as a selection of particular masses that are imagined as the only masses in existence (no 'outside masses act on the 'inside' masses in any way). I love this starting point because it's simple and intuitive. Perhaps you could argue against the phrase "amount of velocity" but I feel it is sufficient.

Because of the use of "associated" with our assumption, velocity is added only as many times as there is mass that holds that velocity and vice-versa. So a quantity of velocity is addedm times, mathematically stated as m*v. We have now arrived at the textbook definition oftotalmomentum: m_{total}*v_{associated total}= constant. Good job guys.

If our closed system were to be breached and momentum altered without adding mass, (classically speaking) velocity is the only quantity subject to change. The derivative of velocity is acceleration, therefore the change in momentum would be m*a and is given the name "Force". This brings into question statements of 'forces' being applied but canceling out.

If force is thechangein momentum, can a force exist if there is no change? Momentum can be separated into known components because that is our starting point; our initial condition's are the masses and their velocities. It is from there wederivenotions of momentum. Our reasoning doesn't allow for reverse engineering, starting from momentum and then deriving masses and their velocities. Now apply that idea to force. Given simply just force (or momentum) allows for infinite possibilities. So the statement "a force is applied" is false, we can only say "a force is noted". Force is a derived term. In a closed system of a ball and the earth, if the ball were to be at rest on the earth (ignoring internal changes) textbooks would say there are two forces: gravity down and normal force upwards, canceling out. Why can we say that? Under that logic, couldn't there be an infinite combination of forces? Maybe one leftwards and one identical in quantity rightwards. My current reasoning allows for forces to exist only when there is a change in momentum. Text books, too, only definenetforce, anything other doesn't seem to technically exist. What I want from you, Internet, is a logical thought process that satisfies current conventions.

Furthermore, energy mathematically seems to be the integral of momentum with respect to time m*v to (m*v^{2}/2). But energy is also constant in a closed system which doesn't hold when momentum's constant is integrated with respect to time. Maybe I should have started with energy as the base concept and momentum and force as the derived concepts, I don't know. Also work seems to be the change of energy, via force (W = integral of f*dx). Force represents a type of change in energy and momentum but I'm not to sure on how to explicitly state this.

These are my problems with classical mechanics. I suppose the heart of my difficulties lies in what are fundamentally derived concepts and what are initial statements, assumptions, or definitions. I struggle with momentum, energy, and force because they all seem so tauntingly related but I can't quite glue them together myself.

I am grateful for any enlightenment