How do bicycles work?
Conservation of angular momentum plays a major role, but why is angular momentum conserved?
I grew up in Germany, where I received an excellent education in math and physics. In fact, I routinely observe that I learned mathematical concepts in high school that STEM PhD students in the US don’t necessarily know.1 I went on to pursue both an undergraduate degree and a PhD in theoretical physics, where I learned even more math and physics. I have since worked for many years as a scientist and professor in the US, and I have found that my German education has served me well. It’s rare that I encounter a topic in math or physics that I don’t have a basic understanding of that I can trace back to my high school and college education. I even learned about multi-layer perceptrons and backpropagation back in the late 1990s, topics that in 2025 I teach and use daily.2
And yet, there is one important concept that I was never taught, and that I only learned in my fifties. This concept is related to how bicycles work, so let’s talk about this briefly.3 The common explanation for bicycles is conservation of angular momentum: The wheels rotate, and the angular momentum associated with this rotation conveys stability to the wheels and in turn to the bicycle as a whole.
If you try to topple a rotating wheel, the wheel instead experiences a torque at a 90-degree angle to the direction of the toppling force. Therefore, instead of falling over, bicycles will turn into the direction of the fall. And so, balancing a bicycle becomes a simple task of keeping the bike pointed into a forward direction. As long as you keep the handle bars straight the bike won’t topple. I don’t want to go into more detail of how exactly all of this works, but suffice to say the root cause is conservation of angular momentum. If you’d like to know more, read up on the concept of precession.4 You can see a demonstration of this effect in this short video:
There’s more to bicycle stability than angular momentum. If you search the scientific literature, you find all sorts of research articles on this topic.5 But here, I’m interested in a deeper, more fundamental question: Why is angular momentum conserved? In my entire high school and college physics education, I was never provided with a reason. It was just a given. Angular momentum is conserved. Just like energy, linear momentum, mass, electric charge, etc. All sorts of things are conserved, and nobody can tell us why.
Well that’s not quite true. First of all, not all of these quantities are truly conserved. And second, physicists have known about the origin of these conservation laws for nearly a century. But for some reason, it was absent from my education. I never learned about it. That is, until a couple of months ago, when I watched a video by the YouTube science channel Veritasium.
The video is titled “The Biggest Misconception in Physics,” and the thumbnail states in large font that “Energy is NOT conserved.” Both the title and the hook are a bit click-baity, even if technically true. And yet, the video is very much worth watching. We learn that conservation laws are due to symmetries. Conservation of angular momentum follows directly from the assumption (or should I say observation?) that the laws of physics are the same in every direction. Similarly, conservation of energy follows from time invariance. If the laws of physics are unchanged as time passes, then energy has to be conserved. Curiously, these invariances do not necessarily hold under general relativity, which therefore implies that energy, momentum, etc. don’t have to be conserved at sufficiently large spatial scales and/or sufficiently long times.
We understand today that symmetries are related to conservation laws, but this was not known when Einstein was developing his theory of general relativity. He knew that in his theory energy was not conserved, and he considered this to be a problem. He thought he had to find a modification of the equations so that energy would be conserved. But he didn’t know how.
The person who sorted all this out was female mathematician Emmy Noether. She realized that conservation laws were due to symmetries, and therefore if the symmetries were broken conservation laws didn’t have to apply. Energy is conserved in an empty universe, but the moment you put stuff into the universe, and the stuff starts deforming space-time, symmetry is broken and energy is no longer fully conserved. The problem was not with Einstein’s equations, but instead with the expectation that energy should be conserved.
Noether’s influence on modern physics is difficult to overstate. Modern physics, in particular quantum field theory, is all about symmetries. The symmetries in the field equations determine what elementary particles are possible and how they interact. And yet, Noether is not that well known. I had heard of the name Emmy Noether before I watched the Veritasium video, but if you had asked me what her contribution was I wouldn’t have been able to tell you. And similarly, I knew about the importance of symmetries in field theories, but I didn’t know that similar concepts lead to basic conservation laws of energy and momentum, and that Noether had first pointed out this connection.
Of course, all of this reflects my own ignorance. Many theoretical physicists are fully aware of Noether’s contributions to mathematical physics and her explanation of the relationship between symmetries and conservation laws. But I think I should have learned this in undergraduate, and I didn’t. If somebody had ever said “energy is conserved because time is invariant” I am certain I would remember. After some reflection, I have a sense of what may have caused this gap in my education. Noether’s contributions are considered advanced topics in mathematical physics, and so they are often covered only in specialized graduate classes. And even when they are covered, they are frequently described in abstract mathematical terms that obscure their importance for basic physical concepts. I may even have encountered Noether’s theorem at some point in my education and not realized its importance, and thus not remembered it. As a case in point, read the Wikipedia page on Noether’s second theorem and tell me whether the page conveys how this theorem has shaped our understanding of modern physics. To be fair, the page on Noether’s first theorem is a bit better.
Since watching the Veritasium video, I have discovered a few other videos where Noether’s work has been mentioned. This is the typical pattern where once you’re aware of something you see it everywhere. As one example, here is a video by Angela Collier where she mentions Noether’s work. (Go to 1:04:17 in the video.) Clearly Angela Collier’s physics education was better than mine.
This post was inspired by a conversation on Substack Notes, though that conversation is only tangentially related to this post. I’d like to acknowledge Billt for pointing me to the research conducted by Kooijman et al. on what makes bicycles stable.
Among my US students, I see huge gaps in linear algebra, a topic we covered at great length in my high school math classes. I also rarely encounter students who know any complex analysis. (However, that’s a topic I learned mostly in my first year in undergraduate, if I recall correctly.) Maybe things would be different if I worked more with students with a physics or engineering background as opposed to a biology or biochemistry background.
If these terms don’t mean anything to you, suffice to say ChatGPT wouldn’t exist without multi-layer perceptrons and backpropagation.
Despite the title, this post is not actually about how bicycles work.
Things get even more interesting in motorcycles, which due to their higher speed and weight carry much more angular momentum than bicycles do. If you want to steer a motorcycle, you have to—somewhat unintuitively—push the handlebars into the opposite direction. This is called push steering or counter steering. The push initiates a lean in the motorcycle, which then causes the motorcycle to turn into the lean. It’s all an elaborate manipulation of the laws of conservation of angular momentum.
This is an interesting one: Kooijman et al., A bicycle can be self-stable without gyroscopic or caster effects, Science 332:339–342, 2011. You can read the pdf here.