Need for a new formalism of Mechanics

 After Newton made his mathematical explanation of the entire natural phenomenon with his fundamental three laws, all the scientists started to work out many physical problems using that.

The three laws of Newton were conceived to be an essential thing for every motion and all the problems were tried to solve based on these.

But, solving of the problems headed into a great many difficulties.

There is no any single doubt about the applicability or pertinence of these three laws.

Newton’s three laws are always applicable but it is not always a useful one.

The reason may be,

Newton’s definitions starts with the definition of a concept called particles. It says that, the motion of each particle follows from the net external force acting on that particle.

 But all the entities in nature are not a single particle. They all are some system of particles with definite shape, size, etc. Besides, all the forces will not act at a single particle, but rather at a certain position of the system of particles.  

So, we need a new way or mechanism to deal with these kinds of problems. The reason is because, if you try to apply Newton’s laws for each of these molecules, then note that, there are some order of 10^23 molecules in a mole of substance.

Though Newton’s laws are applicable here,

though it is certain that it will give a correct answer,

it cannot be used here since we know for sure that it will never be achieved writing equations of motion for all these particles.

It may look, this need somewhat satisfied by the introduction of center of mass concept but not completely.

Centre of mass concept is just a simplified version of Newton’s law, where the third law is utilized completely, where all the internal forces are taken as action, reaction pairs and canceled out from the equations of motion.

But this doesn’t alter anything about the nature of the force. The force is again somewhat regarded as acting on a single particle, which is too much different from the real situation.

All these problems need a general way to solve the system of particles without considering the internal properties of the system.

Let us say the system is just a shadow. We don’t know whether there are particles inside it and we know just the forces acting on that system.

Thus, there is a great need for the sake of simple explanation of mechanical system.

Looking for a new idea with this condition, it is found that the idea is there already in our Physics, i.e. Thermodynamics.

In Thermodynamics, though Volume, Pressure and Temperature are the manifestation of microscopic phenomena, we characterize the thermodynamic system just only with the macroscopic variables.

It is just the final external effect i.e. Effective motion in mechanics is significant.

There is no any concern about “the way how the final external effect is achieved”.

As a result, one thing is to be sure that, the new mechanism should involve a more geometrical approach rather than the physical approach. And so, Mathematics plays a more fundamental role in this mechanism. 

Explaining the system with this new approach started by introducing a new concept namely, Constraints.

Correspondingly, the concept of constraints led to some new other essential concepts such as, degrees of freedom; generalized coordinates; configuration space, etc.  All these things exhibit the greatness of classical mechanics in other fields of Physics.