Introduction from lecture notes on introductory mechanics taught in 2019 and 2020
Why can we trust our memories? This has a lot more to do with physics than you might think. What do we mean by “trusting memories”? We mean that our previous experiences can be used to inform future situations. This means that what we learned in the past must be applicable to the future; that is, there is a continuity through time of our experiences. A hot stove you touched yesterday hurt, therefore you know that if you touch a hot stove tomorrow it will also hurt. We can make this more physically precise by stating that experiences exhibit a time-translation symmetry. This means that our learned experiences are always the same (a symmetry) throughout translating or moving through time. This is obviously extremely important for conscious beings like us, otherwise we could never learn.
Even more grand a statement that follows from this is that the laws of physics do not change in time. Now, I don’t mean that individual objects to not change in time; I mean that the way and rules for how objects interact with one another are always the same. For example, the rules of Monopoly are always the same, but any given game can have different outcomes. If the laws of physics do not change in time (they exhibit a time-translation symmetry), there ought be a concrete quantity whose value is unchanged, or conserved, in time. This is energy: that the laws of physics do not depend on time means that energy is conserved, and vice-versa. This relationship between a symmetry and a conservation law is called Noether’s theorem, after Emmy Noether, a German mathematician.
Noether’s theorem is perhaps the most important result in all of theoretical physics and provides extremely strong constraints on the interactions of objects. However, depending on the system you are studying, energy may or may not be conserved. We only believe that energy is conserved for the entire universe, the only truly closed system we can imagine. The energy of an object can change if work is done on that object. Work is necessarily a concept that is outside of the object or system that you are studying. Because you can’t go outside the universe, no work can be done on it and so energy is conserved.
However, not only do the laws of physics not depend on time, but they don’t depend on where you are or how you are oriented. That the laws of physics are independent of your position means that they exhibit a spatial translation symmetry. Just like with time translations, Noether’s theorem states that there is a conserved quantity: momentum. Momentum only changes if a force acts on your system or object. Further, the laws of physics don’t depend on your orientation: throwing a ball to the north or to the west exhibits the exact same phenomena. Thus we say that physics is rotationally-invariant: everything (i.e., the laws of physics) is the same if you rotated the system. For rotations, Noether’s theorem tells us that the corresponding conservation law is angular momentum. Angular momentum, a measure of an object’s rotation about a fixed axis, can only change if there is a torque on an object.
These three conservation laws, energy, momentum, and angular momentum, will be the central components of this course. We will describe systems under which they are conserved, and use that to our advantage when making predictions for future behavior given current data. We will also discuss how work, forces, and torques break conservation laws for open systems (systems that interact with an external environment). Fortunately and powerfully, this breaking of conservation is not arbitrary, and we will construct powerful relationships fitting it all together.
Though this class and topics are often referred to as “classical mechanics,” connoting “classical” in the Greco-Roman sense, the physics you learn this semester underlies all phenomena that we know. Conservation laws are the way that modern particle physics is formulated, and so these ideas are used throughout my own research. Though it may seem pedestrian or even pedantic at times, there is an amazingly rich structure lurking just beneath the surface. This semester, I’m thrilled to be your guide exploring Nature from this profound perspective.
A Physicist Abroad 