I've been thinking about a possible physical representation of the Laplace Transform. While I was reading my advanced engineering mathematics book, I noticed that when the transformation is applied to electrical circuit elements such as resistances, inductive coils, capacitances,... they can be represented in terms of algebraic relations to the Laplace-transformed current and voltage. Before the transform, it is well-known in physics that a coil relates to differentials of voltage while a capacitance relates to anti-differentials of current. So the fact that we can simplify this differential relations into algebraic relations under the Laplace domain is really no surprise.

But what is physics behind it? What exactly is happening as the system gets transformed. The question becomes clear if I present another example. Suppose we have a ball free falling through viscous air. The speed of the ball is, after solving differential equation D[x,2]+p*D[x,1]+q*x=f, expressed as an exponential function. But when the transform is done, the speed of the ball is expressed simply as a rational function of Laplace variable s in the frequency domain (also known as Laplace s-domain).

If we can write every fundamental law of physics in this frequency domain, it will look completely different and if we can understand the new representations in a same way we do in the non-Laplace domain, it might cast us a new insight into something else, something different.

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