Symmetry

Symmetry is one of the most powerful ideas in physics. Emmy Noether, the most important woman in the history of mathematics, determined that every continuous symmetry in a physical system results in a conserved quantity. This is called Noether’s Theorem. Every beginning physics student learns that any system’s total energy, momentum, and angular momentum are conserved. Noether’s Theorem explains why. Each law comes from one of our universe’s symmetries. Let’s take a look.

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Conservation of Energy

If each conservation law stems from an underlying symmetry, what causes energy conservation? First of all, what does it mean for energy to be conserved? Suppose we take a box and count the total energy inside. If come back later and count again, we should get the same answer, provided nothing entered or left the box. So, for an isolated system, whether we turn the clock forward or backward, we should see the same energy. That’s energy conservation. If you pick any system and study its “equations of motions”, which are the equations which govern its behavior, you’ll find that the equations look the same at any time t0 and at time t0 + t1. Or in other words, the laws of physics are symmetric in time. Energy is conserved because the laws of physics (or the universe) are homogeneous in time, i.e., whether we turn the clock forward or backward the equations are the same.

Conservation of Momentum

Similar to the discussion for energy, if we compute the total momentum for the box now and the total momentum at a later time, then the total momentum should be unchanged. If we look at the laws of physics, like we did for energy, we’ll find that they don’t explicitly depend on position. The laws of physics are the same no matter where you are in the universe. If we are at position x0 and then shift to position x0 + x1, the laws of physics are unchanged. This symmetry is called the homogeneity of space and it causes momentum conservation.

Conservation of Angular Momentum

You’re probably catching on now. When the laws of physics don’t explicitly depend on a variable like time or space, we get a conservation law. So what about rotational symmetry? If we rotate a system through an angle, the equations of physics are the same. So, there is no preferred direction in space. This symmetry is called the isotropy of space and it causes the conservation of angular momentum.

The Underlying Mathematics

If you look at classical (or quantum) mechanics, you’ll encounter Noether’s theorem and see that there’s a conservation law between every pair of canonically conjugate quantities. Each member of a pair will be an “observable” quantity that we could measure. One member of the pair is what we usually think of as an independent variable, like time, and can produce transformations in the system (like translation in space or time). The other will be a physical quantity like energy. So, at this level of understanding, time and energy are canonically conjugate. The same is true for space and momentum and for angular position and angular momentum.

Summary

Every conservation law stems from some underlying continuous symmetry. Energy conservation is due to the homogeneity of time, momentum conservation to the homogeneity of space, and the isotropy of space produces angular momentum conservation.

To learn more about this, you can take a look at any of the following resources, listed in order of increasing sophistication:

Comments on the Apple Watch

My Apple Watch arrived yesterday. I chose the 42mm stainless steel model, with the black leather loop. It looks great and feels comfortable on my wrist. The only thing that I find a bit odd is that I can sometimes hear the magnets in the band moving against each other when I move my wrist. Not a deal-breaker, just a minor oddity.

I wanted the Apple watch primarily for the fitness tracking aspects. I was thrilled when I heard it had a gyroscope, accelerometer, and heartbeat sensor. I was less thrilled when I learned that Apple hasn’t opened up sensor access in WatchKit. I hope they’ll change their minds about that in the future. I have (or had) an app in mind that needs such data. Also, it would be interesting to look at gyroscope or accelerometer data from a session of Angamardana, which is the most intense workout I’ve ever encountered.

I do wish the watch’s Workout app would allow me to rename the “Other” exercise to something of my choosing or at least rename it in the Health app on my iPhone. The ironic thing is that if you manually add workout data in the workout section of the Health app, there are a myriad of workout types. The Health app does allow data exports though, so that’s nice.

Blog goals for 2015

I’ve been doing the 10 Days to a Better Blog online workshop as a way to breathe new life into my blog. I’ve never given it much attention until now. I’m on Day 8, where you focus on setting goals for your blog, using the SMART goals methodology:

  • Specific – target a specific area for improvement
  • Measurable – quantify or at least suggest an indicator of progress
  • Assignable – specify who will do it
  • Realistic – state what results can realistically be achieved, given available resources
  • Time-related – specify when the result(s) can be achieved

Some people aren’t a fan of this approach, but it suits me. I especially like the metrics. By the way, the fact that I’ve made it to Day 8 is nothing short of amazing.

Specific goals

My difficulty in the past has been maintaining momentum. When other aspects of life were looming large, the blog was the first casualty. So, I want to write at least one post per week. If I do more than that, great. I also want to make sure I cover the topics I plan to write about over the course of the year. That will keep the content somewhat varied and will also make sure I devote time to those topics in my life outside of the blog. So, for 2015 I want to:

  • Write one post per week.
  • Write posts covering all of the topics I want to write about.

Progress Metrics

This is easy enough.

  • Posting frequency
  • Post topic frequency

I’m also going to track the day of the week on which I tend to write. I’m going to aim for Wednesday, since those are generally less busy days for me, but I don’t want to commit to a specific day.

Assignable

Obviously, I’m going to do the writing.

Realistic

How realistic is all of this? I should be able to write one post per week. The only thing stopping me would be me. And I don’t plan on stopping myself this time around.

Time-related

I don’t want to try ramping up to one post per week. It only works if I start now. So, this is today’s post.

How I like to work

Since getting back from India and returning to work, I’ve been thinking about what I need in my work environment to work most effectively. I need three things,

  1. A relatively quiet area, with some random, unobtrusive background noise
  2. An uninterrupted block of time
  3. Plenty of scratch paper

And that’s all I need. But there are a few things I like to have as well: a Roost laptop stand, a bluetooth, backlit keyboard (don’t have this yet), a bluetooth mouse or trackpad, and a desk chair wide and deep enough for me to sit in Ardhasiddhasana.

You’ll notice that I mention a desk chair rather than a standing desk. I used a standing desk for about a year and realized that I’d rather just sit in Ardhasiddhasana instead. You might be wondering why the heck I want to sit like that. For those who don’t know, I teach classical hatha yoga and sit in that asana for the entirety of each class. It stabilizes the body and mind in such a way that you don’t need food, water, or bathroom breaks as much as everyone else does. It also increases mental alertness and focus. I’m sure you can see how that might be useful.

Why I’m blogging

Over the past decade, I’ve kept a blog at one time or another. The problem is that I never made it more than about seven posts in. Invariably I’d let other aspects of life overtake the writing. Of the posts I’ve written, my favorite was a group theoretic analysis of the Sim Sala Bim card trick. I learned that trick when I was in fifth grade and realized it was based on the symmetric group during my postdoc. I lost track of that post, otherwise I’d post it here.

So, all this has me thinking about why I wanted to blog back then and why I want to blog now (when I’m much more busy). I suppose there are few reasons:

  1. It gives me a record of things that I’ve thought about in enough depth  to write something coherent.
  2. It gives me an easy way to share something that I find interesting or worthwhile with others.
  3. When I write something technical, the writing helps me better understand the thing I’m writing about.
  4. The majority of my writing experience is with research papers, so blogging makes me a better writer.

This time, I’m determined to not let this fall by the wayside. This is due in part to my wife and I keeping a blog for our yoga studio, Adi Yoga of Nashville.

Pacing myself

As I sit in front of my laptop, I’m reminded of something one of my favorite authors, Steven Barnes said:

“You cannot run all the way to Disneyland. But you can walk there.”

I have always had a tendency to get interested in something and do it to the exclusion of all else. Sometimes it served me well, sometimes it didn’t. Whenever it happened, my mother would say that I got on a “kick”. I wish I could remember some of them. But anyway, I did the same thing throughout graduate school. My field was theoretical condensed matter physics. Instead of spending all my time learning more condensed matter theory, I spent time diving into some of the “sexier” subfields, like string theory, general relativity, and quantum gravity. It was fun, but it didn’t directly, or even tangentially, relate to my research. It was a diversion of sorts. (At least it was fun!)

As I’ve gotten older this issue has gone away. I still want to learn new and interesting things, but I have responsibilities now, at home and at work. So I mostly pick topics that relate to things that I need to focus on. For example, I develop machine learning software. So when I want to learn something new, I’m likely to pick some algorithm or numerical technique.

Looking back, part of my frenetic desire to learn as much as I can was a desire to squeeze as much as I could out of life. No telling when it might end. Now, for the same reasons, I try to spend my time more wisely. I’m not going to live forever and I can’t do everything. So I pace myself.