Although I am an engineer I love studying math and physics outside of work. I especially enjoy the history of math and how ideas that seem so obvious to us today were not always so obvious even to the most intelligent thinkers back 'in the day'.

I was reading about Emmy Noether, a German mathematician, and trying to understand her coming about of Noether's theorem. Although I am admittedly a dumb person, I try my best to understand things as well as possible.

Wikipedia has an informal formulation of the theorem as:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

Symmetry in this case I think just means that the property can be transformed about some dimension (spatial or time) and not change despite how much it is transformed or how long you transform it for as long as it is continuous. It makes sense with something like a rectangle on a paper where you can turn the rectangle 180 degrees clockwise and say that the shape property of the rectangle has not changed even if it took you 1 second to rotate the paper or 1 year. However, it becomes much more abstract when thinking about other properties such as momentum, energy, and mass. It might be a circular definition but my understanding is that if a system has a property that is symmetric in some way then that means that something is conserved.

In practical terms, if you translate a system through space, the momentum of the system does not change. This leads to the Conservation of Momentum. Since there are 3 spatial dimensions in which a system can move, momentum is a 3 element vector.

Likewise, if you rotate the system, it's angular momentum is conserved leading to Conservation of Angular Momentum. Angular momentum is also a 3 element vector since there are three spatial dimension to rotate the system about.

Finally, as a system moves through time its energy is conserved and thus gives Conservation of Energy. Energy is measured as a single scalar value since there is only one time dimension.

** ... or is there... **

What if, for fun, we assume there are 2 time dimensions? Now energy is actually a 2 element vector rather than a scalar. This also means that now energy can be rotated between the two time dimensions. This would then give raise to a Law of Conservation of Angular Energy, of which they would obviously credit its discovery to me ;)

I don't understand how one would go about rotating energy about various time dimensions. Nor do I understand if such a rotation could even be defined since one can only move forward in time.

Furthermore, if there are 2 time dimensions does what does that mean for Einstein's famous E=mc^2 equation? That would mean that mass must also be a 2 element vector. One could no longer say that they are 160lbs (if referring to their mass) if someone asks. It wouldn't make sense. They would have lie about it using 2 numbers when asked.

I think Newton's Second Law would be ok though. Isaac seemed like a pretty chill dude. He would have just replaced the m in F=ma with |m| (magnitude of mass) giving F=|m|a being published in Philosophiæ Naturalis Principia Mathematica and then gone on with watching apples fall from trees or whatever.

After reading more about conservation laws on Wikipedia I realized that it is a much deeper rabbit hole than I thought. Time for a beer.

Anyway, all of the above ramblings could also be total bullshit. I am, after all, just a simple hillbilly.

Bellow is the badass herself: Emmy Noether