Excerpt from my 'Treatise of Classical Physics'

Classical physics is a paradigm and framework for quantitatively analyzing material/physical phenomena as perceived by man. The analysis it provides is meant to both predict and account for all the phenomena observed. We will first outline the basic axioms of the philosophy of classical physics:

The Universe operates in accordance with certain laws
All physical phenomena occur in accordance with a set of fundamental physical laws. Such laws include the conservation of energy, and Newton’s first law of motion. The laws are elucidated by identifying patterns of behaviour from empirical data. These physical laws can be expressed with the least ambiguity in mathematical language, and in this form serve to help the human mind analyse physical processes quantitatively and avoid the susceptibilities of qualitative speculation. From the most basic laws, a large number of corollaries may be derived. These will facilitate the analysis of physical phenomena.

Determinism
The principle of determinism asserts that all physical phenomena are completely determined by the laws of nature which govern them. It asserts that given a certain law, and some knowledge of the state of a system obtained through empirical methods, it is entirely possible to determine the state of that system at any arbitrary point in time. The accuracy of the calculated value is only limited by the limitations of the measuring instrument used to gather the data.

It should also be noted that the physical laws are generally assumed to be invariant with everything else; They are true forever. So what happens when laws transmutate to something else? Should there be a law of laws that governs the behaviour of laws? Sounds like a Russian doll problem. Of course, this is not a problem in the paradigm of classical physics; The absolute invariance of physical laws is assumed.


Mechanics
Much of the phenomena that surround us are about the motion of bodies. It is thus no surprise that a central theme of classical physics is mechanics, the study of motion. The fundamental axioms of motion are stated below:

Axioms of the Newtonian Mechanical Universe

1. The Universe is a collection of objects each described by a set of basic physical quantities, from which other parameters that characterize it may be derived. Derived parameters would be functions of those basic parameters. The basic physical quantities include mass, distribution of mass in space, electric charge, distribution of electric charge in space, position in 3 dimensions.

For purposes of mechanics, the quantities of position in each of 3 dimensions, and mass are most important.

The complete state of the Universe can be specified by knowledge of the above parameters of all the objects which populate the Universe.

Time is a quantity that only increases in magnitude. The state of the universe may be specified against the quantity of time. The state is single-valued and continuous with time.
[this is getting rather difficult to formalize…needs a better mathematical formulation than this crude and incomplete statement! Perhaps an analogy would be that each state of the universe is like a bead strung on the string called “time”.]

Orthogonality of Parameters
[some parameters are orthorgonal… e.g. motion in x-direction does not affect motion in y-direction…I’m lacking the math to formalize this.]

2. The fundamental object of the Newtonian Universe may be thought to be a point in space, which is enumerated by at least one of the fundamental parameters as specified above. Its behaviour will be subject to the axioms as stated in this treatise. Behaviour refers to the variation of the state of the object, the state being the set containing all the canonical coordinates which will completely specify the particle.

3. The Position Exclusion Principle
No two objects may have the same position coordinates.

4. Position coordinates of an object must be continuous and single-valued with respect to time.

5. Motion is the change of the parameters of position with respect to time.
To adequately describe motion, several parameters are derived from the fundamental parameters, and are defined as such:

Velocity is a vector quantity whose magnitude is the first derivative of position with time.

Acceleration is the first derivative of velocity with time, or equivalently, the second derivative of position with time.

Force
Force is a causative agent of acceleration. In other words, force causes a change in velocity. Forces are exerted by objects. Forces can have one of two natures: contact and non-contact.

Non-contact forces arise due to force fields (e.g. gravitational fields) and objects can experience such forces whether they are in contact with other objects or not.

Contact forces only arise when:
i) objects are in contact with each other
ii) the objects in contact are accelerating
We can always assume there is a contact force between objects in contact with each other, and if there is no such force in actuality, logical deduction will reveal their absence.

Energy
Energy refers to the capacity to enact a change in an object’s state of motion that is consistent with the enactor’s intention. Since force is the only causative agent for a change in motion state, the “enactor’s intention” refers to the direction of the exerted force.

6. Conservation of Energy




7. Newton’s first law of motion
In the absence of an external unbalanced force, an object at rest will remain at rest, and an object moving at constant velocity (not just speed) will remain moving at constant velocity.

8. Newton’s second law of motion
An object experiences an acceleration that is proportional to the net force it experiences and inversely proportional to its mass. By definition, the constant of proportionality is one, as it is arbitrarily defined that 1N of force will cause a mass of 1kg to accelerate at 1ms^2. After all, what 1 Newton of force or 1kg of matter means is entirely arbitrary and chosen for our convenience.

This law establishes the relation between mass and motion. That is, mass is a fundamental property of an object which resists change to its motion. The resistance to a change in motion is known as inertia. Hence mass the cause and the measure of inertia.

10. Newton’s third law of motion
When an object experiences a force, it exerts an equal and opposite force on the object that exerted that force on itself.

[post note: the following is one of the couple of examples in the treatise which used the axioms alone to make deductions about some mechanics problems]

Example 3
Consider two boxes connected to each other by a rigid rod. We have the following diagram with all the forces labeled:


F is a pulling force that acts directly on M1.
A2 and R1 are an action reaction pair between M1 and M2.
A3 and R2 are an action reaction pair between M2 and M3.

Since M1 is stuck to M2 which is stuck to M3, and we claim that all the masses cannot be deformed, and that they’re perfectly stuck together, this can only mean that they must all experience the same acceleration. Since A = F/M, we write the following expression to equate the accelerations of each mass:

A3/M3 = (A2-R2)/M2 = (F-R1)/M1

Newton’s third law tells us that the magnitude of forces in an action-reaction pair is the same. Hence,

R1 = A1 and R2 = A3

So to make the algebra look nicer, we write this:
R2/M3 = (R1-R2)/M2 = (F-R1)/M1

Suppose the two boxes have a mass of 1kg i.e. M3 = 1 and M1 = 1. We then have:
R2 = (R1-R2)/M2 = (F-R1)

Suppose that M2 was a massless string. A string cannot extend beyond its length, so M3 and M1 cannot move any further from each other than the string’s length. But unlike the case when M2 was a rigid rod, they now can move closer to each other. Consequently, we cannot claim that the accelerations of all the masses are equal. Instead, we can only say that M3 can have an acceleration that is equal or greater than that of M1. By considering the property of the string alone, this is a correct deduction. However, further reasoning taking into account this particular situation of F pulling on the system eliminates the possibility of M3 accelerating more than M1. Why? This is because the instant M3 accelerates faster than M1, the string M2 becomes less taut. When this happens, the string no longer passes the force to M3. Consequently, M3 has zero acceleration. Since at any instant when M3 accelerates more than M1, the acceleration disappears, this means that at any point in time, M3 can never have a greater acceleration than M1. Thus the only possibility is that the acceleration of M3 and M1 is the same.

During the discussion, we have conveniently left out any consideration of the mass of the string, M2. Since the string is flexible, the motion along its length is complicated and does not easily render itself to a simply discussion. Moreover, our force diagram remains the same, since our conclusion above that acceleration of all parts of the system is constant implies that the string must be taut at all times.

Given the special case that the string M2, has zero mass, we can multiply M2 across to give the following:
R1 – R2 = 0
This implies that
R1 = R2

Hence we have proven a frequently used assumption: “that the tension in the string is constant.” This has been placed in quotes because it is not very technically correct. I explain:

We should be careful when using the notion of a massless string. We must note that by A=F/M, anything massless that experiences a force will have infinite acceleration! This clearly cannot happen because it implies that the object can have infinite kinetic energy when no work was done by the force at all, since at the very instant the force is applied, the object takes off at infinite speed. This is a violation of the law of conservation of energy. Also, the string does not experience any net force, since R1 = R2. How can this make sense when it is clearly supposed to be a rigid body (it is taut in this situation) undergoing the same finite acceleration those two masses are experiencing too? This is a contradiction…if there is finite A, there should be finite F. But M cannot simultaneously be zero. Therefore, when using the idealistic notion of a massless string, the string should not be regarded as a mass (its massless anyway). Instead, it is simply some ideal device that :
i) constrains the maximum distance between the two objects it connects
ii) exerts an equal force upon the two objects it connects

Hence we prove that the “massless string” is physically impossible and its construct in the Newtonian mechanical world is contradictory and paradoxical. It can at most be an approximation to make computation simpler.

Therefore, the statement “that the tension in the string is constant” should be amended to “the string exerts an equal force upon the two objects it connects”. When we speak of tension in the string, we really are referring to the forces which are pulling the particles of the string apart. When the forces exceed those forces which keep the string intact, the string breaks.

Technical Conclusion
We have done all our reasoning in the above examples purely with the 10 axioms we have listed in the previous section. An axiomatic approach is tedious, and not practically efficient. But it exposes the many assumptions we make when we solve these problems; assumptions which we take for granted, which we sometimes fail to justify, and which we never worry about when we unknowingly make them.

We have proved the assumption that the tension felt by two objects connected by a taut, massless string is constant.

A solution to a mechanics problem is found by listing all the forces and parameters of the system in consideration. All the variables must behave according to the 10 axioms. This can be expressed mathematically. We can then proceed to solve the equations simultaneously.