The Simple Harmonic Motion Mathematics

 The study of the second order Simple Harmonic motion(SHM) ODE is very interesting. It shows not only how oscillations generate from relatively simple physical laws but their mathematical analysis reveals also how deeply oscillations are related to complex exponentials and thereby the inseparable relation of complex numbers to the real world. Simple Harmonic Motion is the most basic physical oscillatory motion. The important thing is to get a physical understanding of the concepts and the equations involved. Mathematical understanding will help in solving the mathematical formulation of the physical phenomena. But another important thing that is required is the Physical understanding of the concept or phenomena. That alone will help in deriving a proper and ideal mathematical formulation. Computers and calculators merely do what their name suggests, they calculate only. But proper understanding is needed to tell the computer what to calculate. Using a few mathematical manipulations and substitutions, we can solve this differential equation which has important applications in Physics and Engineering. Complete and proper understanding of this article requires basic Calculus and Complex Numbers.

The statement of Hooke's law is :

The restoring force exerted by a stretched spring is directly proportional to the elongation or the change in length of the spring.

Mathematically,            

   where 'x' is the displacement of the Centre of Mass(CoM) from the mean position, and k is the spring constant, 'm' is the mass of the point object. The negative sign indicates that the force is restoring and thereby opposite to the direction of displacement. The left side follows from Newton's 2nd Law, the right side from Hooke's Law. Even though the above formulation is for undamped Spring-Mass systems, the mathematical treatment is general and can be applied from Simple pendulums to LC Circuits. This is one of the many advantages a mathematical formulation has, it tends to generalize and link apparently unrelated physical phenomena.

To solve this equation, we substitute

taking the second derivative

Using these substitutions in the SHM ODE yields

Using the initial substitution

Substituting in the differential equation,

                                                               

In this case, the constant of indefinite integration is not necessary-

But from the initial substitution, 

 

Therefore-

From this we can see that even for such basic differential equation, complex numbers have to be used.

Using Euler's formula we get

Adding both of them and dividing by 2 we get

Subtracting one from another and dividing by 2i we get

Thus we see that it is the linear combination of solutions of the differential equation that gives rise to sinusoidal oscillations. It also demonstrates how relevant complex numbers are to natural phenomena. It may seem that the article was a very complicated treatment of a relatively simple concept, after all, formula charts provide everything to calculate the physical properties of such systems. However, a proper understanding is very much needed while working on a real life problem. Formulas do provide shortcuts, but it fails to answer the question- Why does the system oscillate at all? 

Lastly, I would like to thank the reader for reading this post. Comments and criticism for improvement are welcome.