Path Integrals

I’m sitting in an electromechanics class listening to our professor lecture. He emphasizes that we should keenly pay attention to the lecture, not taking notes, but clinging to every word and anticipating the next. “It is imperative you focus on the concepts. Think about what is being taught and try to guess what is going to happen next.” His favorite word is imPERative. He says it just like too.

Today’s lecture is on finding energy and coenergy from electromechanical systems; basically the energy output of moving an iron object near some copper coils with current flowing through them.

  • Coenergy as described by current i and flux linkage λ
  • Most often the system in question, a motor or an electric generator at a power plant, are multiport— multiple energies and coenergies than can be described by the relationship as such
  • In order to perform these calculations path integrals are computed going along lines OB -> BA or OC -> CA

Coenergy calculations are actually not too complicated. A couple of integrals and derivatives and you’re there.

“Now we will just compute the coenergy as it has a much simpler integration.” My whirring thoughts came grinding to a halt as I tried to rationalize his statement. It had been two years but path integrals were about the only thing I remembered from multi-variable calculus. My hand shot up abnormally quick, wobbling my coffee thermos on the desk. He gave a slight nod in my general direction.

“Are we able assume the path is conservative because the system is electrically linear? Or do we have to check it?”

He smiled. “We can’t assume the system is conservative. You can never assume any two paths are the same.” He wasn’t referencing electrical engineering anymore.

Picture a mountain. There are easier and harder ways up the mountain that ultimately lead to the exact same point. Life is very similar, it is full of these destinations and points with a limitless number of possible paths. The main distinction, and by consequence the reason life is so much harder than electromechanics, is that we can’t calculate anything.

You want to climb higher up this abstract mountain of life, up to the heights of your dreams and goals. You see your peers there and you know their stories — their hardships, and sacrifices. But you cannot know what your journey will be along the path they took because in real life, no path is conservative. Life isn’t something you can integrate with respect to time and space and reduce to a function that describes when and where to walk. No matter what steps you take, guessing and theorizing, you will never know if you are taking the best path forward. Even worse, when it is all said and done, and you have either failed or succeeded, you won’t be able to know what other paths would have resulted to.

Knowing the unknown path is futile, looking forward and even in retrospect. I sat through the rest of the class trying to decide how I should feel about this. Am I mad that I’ll never know what lies ahead and what could have been? Or would knowing torment me more than anything else ever could?

“It is imPERative you check that the flux linkage and current partial derivatives to make sure they match up. If they do not, your path is not conservative and you can’t solve for the energy values.”

“What can we do then?”

“Write down that the path is not conservative. But that’s it, nothing else.”