Survey of classical mechanics: Generalized coordinates and momenta. Lagrangian equations. Derivation of Hamilton’s equations from Lagrangian. Poisson brackets. Hamilton as representing invariant E under time for conservative systems.
“Pull quote”: Something simple and seemingly obvious but actually deep and foundational
Some notes (not checked for typos!)
However, influence and importance are two things, and unfortunately the two aren't always correlated. As Sydney Brenner recounts in detail here, the founding script for molecular biology should really have been John von Neumann's 1948 talk at Caltech as part of the Hixon Symposium, titled "The General and Logical Theory of Automata". In retrospect this talk was seminal and far-reaching. Brenner is one of the very few scientists who seems to have appreciated that von Neumann's influence on biology was greater than Schrödinger's and that von Neumann was right and Schrödinger wrong. Part of the reason was that while many biologists like Crick and Watson had read Schrödinger's "What is Life?", almost nobody had read von Neumann's "General and Logical Theory of Automata".
Later I looked up the species on the Internet and found that it was a male Ceanothus silk moth, very prevalent in the Western United States. I found out that the reason it’s not seen very often is because the males live only for about a week or two after they take flight. During that time they don’t eat; their only purpose is to mate and die. When I read about it I realized that I had held in my hand a thing of indescribable beauty, indescribable precisely because of the briefness of its life. Then I realized that our lives are perhaps not all that long compared to the Ceanothus moth’s. Assuming that an average human lives for about 80 years, the moth’s lifespan is about 2000 times shorter than ours. But our lifespans are much shorter than those of redwood trees. Might not we appear the same way to redwood trees the way Ceanoth moths or ants appear to us, brief specks of life fluttering for an instant and then disappearing? The difference, as far as we know, is that unlike redwood trees we can consciously understand this impermanence. Our lives are no less beautiful because on a relative scale of events they are no less brief. They are brief instants between the lives of redwood trees just like redwood trees’ lives are brief instants in the intervals between the lives of stars.
I have been thinking about change recently, perhaps because it’s the standard thing to do for someone in their forties. But as a chemist I have thought about change a great deal in my career. The gist of a chemist’s work deals with the structure of molecules and their transformations into each other. The molecules can be natural or synthetic. They can be as varied as DNA, nylon, chlorophyll, rocket fuel, cement and aspirin. But what connects all of them is change. At some point in time they did not exist and came about through the union of atoms of carbon, oxygen, hydrogen, phosphorus and other elements. At some point they will cease to be and those atoms will become part of some other molecule or some other life form.
Sometimes popular culture can capture the essence of science and philosophy well. In this case, chemistry as change was captured eloquently by the character of Walter White in the TV show “Breaking Bad”. In his first lecture as a high school chemistry teacher White says,
“Chemistry is the study of matter. But I prefer to think of it as the study of change. Now, just think about this. Electrons change their energy levels. Elements, they change and combine into compounds. Well, that’s…that’s all of life, right? It’s the constant, it’s the cycle, it’s solution, dissolution, just over and over and over. It’s growth, then decay, then transformation. It is fascinating, really.”
Changes in the structure of atoms and molecules are ultimately dictated by the laws of atomic physics and the laws of thermodynamics. The second law of thermodynamics which loosely states that disorder is more likely than order guarantees that change will occur. At its root the second law is an argument from probability: there are simply many more ways for a system to be disordered than to be ordered. The miracle of life and the universe at large is that complex systems like biological systems can briefly defy the second law, assembling order from disorder, letting it persist for a few short decades during which that order can do astonishing things like make music and art and solve mathematical equations enabling it to understand where it came from. The biologist Carl Woese once gave an enduringly beautiful metaphor for life, comparing it to a child playing in a stream.
“If not machines, what are organisms? A metaphor far more to my liking is this. Imagine a child playing in a woodland stream, poking a stick into an eddy in the flowing current, thereby disrupting it. But the eddy quickly reforms. The child disperses it again. Again it reforms, and the fascinating game goes on. There you have it! Organisms are resilient patterns in a turbulent flow—patterns in an energy flow.”
Woese’s metaphor perfectly captures both the permanence and impermanence of life. The structure is interrupted, but over time its essence persists. It changes and yet stays the same.
Although thermodynamics and Darwin’s theory of evolution help us understand how ordered structures can perform these complex actions, ultimately we don’t really understand it at the deepest level. The best illustration of our ignorance is the most complex structure in the universe – the human brain. The brain is composed of exactly the same elements as my table, my cup of coffee and the fern plant growing outside my window. Yet the same elements, when assembled together to create a fern, somehow when assembled in another, very specific way, create a 3-pound, jellylike structure that can seemingly perform miracles like writing ‘Hamlet’, finding the equations of spacetime curvature and composing the Choral Symphony. We have loose terminology like ’emergence’ to describe the unique property of consciousness that arises when human brains are assembled together from inanimate elements, but if we were to be honest as scientists, we must admit that we don’t understand how exactly that happens. The ultimate example of change that makes the essence of us as humans possible is still an enduring mystery. Will we ever solve that mystery? Even some of the smartest scientists on the planet, like the theoretical physicist Edward Witten, think we may not. As Witten puts it,
“I think consciousness will remain a mystery. Yes, that’s what I tend to believe. I tend to think that the workings of the conscious brain will be elucidated to a large extent. Biologists and perhaps physicists will understand much better how the brain works. But why something that we call consciousness goes with those workings, I think that will remain mysterious. I have a much easier time imagining how we understand the Big Bang than I have imagining how we can understand consciousness…”
In other words, what Witten is saying is that even if someday we may understand the how and the what of consciousness, we may never understand the why. One of the biggest examples of change in the history of the universe may well remain hidden behind a veil.
I think about change a lot not just because I am a chemist but because I am a parent. Sometimes it feels like our daughter who is now two and a half years old has changed more in that short time than a caterpillar changes into a butterfly. Her language, reasoning, social and motor skills have undergone an amazing change since she was born. And this is, of course, a change that is observed by every parent: children change an incredible amount during their first few years. Some of that change can be guided by parents, but other change is genetic as well as idiosyncratic and unpredictable. Just like you can coax simple arrangements of atoms into certain compounds but not others, as a parent you have to make peace with the fact that you will be able to mold your child’s temperament, personality and trajectory in life to a certain extent but not beyond that. As the old alchemists figured out, you cannot change mercury into gold or gold into mercury no matter how hard you try. And that’s ultimately for the better because, just like the diversity of elements, we then get a diversity of novel and surprising life trajectories for our children.
Children undergo change but they are are also often the best instruments for causing it. Recently I finished reading Octavia Butler’s remarkable “Parable of the Sower” which is set in a 2024 California that is racked by violence and arson by desperate, homeless people who break into gated communities and burn, murder and rape. The protagonist of the story is a clear-eyed, determined 18-year-old named Lauren Olamina who, after her family is murdered, starts out by herself with the goal of starting a new religion called Earthseed amidst the madness surrounding her. Earthseed sees God as a changeable being and embraces change as the essence of living. Lauren thinks that in a world where people have to deal with unpredictable, seismic, sometimes violent change, a religion that makes the very nature of change a blueprint for God’s work can not just survive but thrive. For an atheist like myself, Earthseed seems as good a religion as any for us to believe in if we want to thrive in an uncertain world. Butler’s story tells us that just like they always have, our children exist to fix the problems our generation has created.
Change permeates the largest scales of the universe as much as it does ourselves, our children and our bodies and brains. One of the most philosophically shattering experiences in the development of science was the realization by Galileo, Brahe, Newton and others that the perfect, crystalline, quiet universe of Aristotle and other ancients was in fact a dynamic, violent universe. In the mid 20th century, astrophysicists worked out that stars go through a life sequence much like we do. When they are born they furiously burn hydrogen into helium and form the lighter elements. As they age they can go in one of several directions. Stars the size of the sun will first blow up into red giants and then quietly settle into the life of a white dwarf. But stars much more massive than the sun can turn into supernovae and black holes, ending their lives in a cosmic show of spectacular explosion or fiery gravitational contraction.
When our sun turns into a red giant, about 6 billion years from now, its outer shell will expand and embrace the orbits of Mercury, Venus and Earth. There is no reason to believe that those planets will survive that encounter. By that time the human race would either be extinct or would have migrated to other star systems; the worst thing that it could do would be to stay put. Even after that we will not escape change. The science of eschatology, the study of the ultimate fate of the universe, has mapped out many changes that will be unstoppable in the far future. At some point the Andromeda galaxy will collide with our Milky Way galaxy. Eventually the stars in the universe will run out of fuel and cease to shine; the universe will become a quieter and darker place. Soon it will only contain black holes and at a further point even black holes will evaporate through the process of Hawking radiation. And way beyond that, the laws of quantum mechanics will ensure that the proton, usually considered a stable particle, will decay. Matter as we know it will dissolve into nothingness. The accelerated expansion of our universe will ensure that most of these processes will inevitably take place. The exact fate of the universe is too uncertain to predict beyond these unimaginable gulfs of time, but there is little doubt that the universe will be profoundly different from what it is now and what it has been before.
The elements from which my body and brain are composed will one day be given back to the universe (I like to think that they will perhaps become part of a redwood tree). That fact does not fill me with a feeling of dread or sadness but instead feels me with peace, joy and gratitude. The ultimate death of the universe described above causes similar feelings to arise. Sometimes I like to sit back, close my eyes and imagine a peaceful, lifeless universe, the galaxies receding past the cosmic horizons, the occasional supernova going off. The carbon, oxygen, nitrogen and other heavier elements in my body came from such supernova explosions a long time ago; the hydrogen came from the Big Bang. Those are astounding facts that science has discovered in the last few decades. Of all the things that could have happened to those elements forged in the furnace of a far off supernova, what were the chances that they would assemble into the exact specific arrangements that would be me? While we understand now how that happens, it could well have gone countless other ways. I feel privileged to exist as part of that brief interval between supernova explosions, to be able to understand, in my own modest way, the workings of our universe. To be a tiny part of the change that makes the universe what it is.
