Feynman left us many interesting facts/tools and one of the most powerful I have ever come across was his observation that “The same equations have the same solutions”. By this he means that equations of the same form have similar solutions. Now this may sound like it is stating the obvious, which I guess it is, but it is an incredibly powerful tool in the right hands.
Maybe around 30 years ago I found two areas where this statement had some very interesting outcomes, but I have not been able to push the results to a final conclusion. As I am now getting too old for this sort of young man’s thinking it is time for me to put this out for general consumption so that someone with a more agile mind/imagination can finish off this work.
The first subject area is mechanics and the derivation of a set of “Mechanical Maxwell Equations”. I shall give you the outline of the ideas and it’s up to you to finish it off The starting point for equations of the same form are the Newtonian gravitational force equation and the equation for the force between two charges. Using the similarity in form you can “equate” 1/4 Pi Epsilon nought with G – I call this the equivalence of the constants. Charge then equates to mass and distance is the same in both equations. We have two more quantities in electromagnetism that we need an equivalent mechanical quantity for, namely B and E. By looking for similar equations in electromagnetism and mechanics it isn’t too difficult to find that E’s equivalent is a, the linear acceleration, and that B’s equivalent is omega or angular velocity (very interesting that B should equate to omega!!). You now have all the necessary information to create your own set of four “Mechanical Maxwell” equations. I have done this, and luckily they turn out to be dimensionally correct – but I have no idea what they mean or what they are saying. I will leave it up to you to tell me
The second area where “The same equations have the same solution” shows us an interesting route for new thoughts and ideas is in the area of Quantum Mechanics, specifically Hidden-Variable theory. The form of the de-Broglie/ Bohm equation for a hidden-variable form of wave equation has the same form as the condensate wave function in superfluid physics. So the Quantum Potential which is the source of all the trouble in de-Broglie/Bohm Hidden-Variable theory is related to a term in superfluid physics which is only important when the superfluid density varies rapidly with position, for instance near a wall or inside a vortex. When we are dealing with the bulk fluid we can omit the term – which is extremely interesting.
So there you have it. Two interesting areas of research. The first might lead to a nice paper, and the second might lead to a Nobel Prize. Over to you.