Module 2 · suggested week 2
The universe operates through relationships, constraints, and change.
What you will be able to do
Read the core equations of physics as statements about constraint and change, know the assumptions behind them, and distinguish a physical model from physical reality.
Recognition
Rearrange to . Acceleration is the rate at which velocity changes, and velocity is the rate at which position changes. So a computer simulates motion by repeating two lines, thousands of times per simulated second. Physicists call this numerical integration: accumulating small changes, a Σ over timesteps.
let x = 0, v = 0;
const m = 2, dt = 0.01;
for (let step = 0; step < 1000; step++) {
const F = force(x, v); // whatever pushes on the object
const a = F / m; // Newton's second law
v += a * dt; // velocity accumulates acceleration
x += v * dt; // position accumulates velocity
}Look at the shape of that loop: state, plus a rate of change times a small step. Hold onto it. In Module 6 you will see . Same skeleton: parameters plus a step size times a direction. This is a genuine structural match (both are discretised update rules). It is not a claim that training is a physical process. The physics loop follows from laws of motion; the training loop is an algorithm someone chose.
Push a cart with a constant force. Double the mass and acceleration halves. Remove the force and the cart keeps moving: velocity only changes when a net force acts.
The net (total) force on an object equals its mass times its acceleration. Force causes a change in motion, not motion itself.
Does not say: Holds for everyday speeds and sizes in a non-accelerating reference frame. Fails near light speed (special relativity) and at atomic scales (quantum mechanics), where it survives only as an approximation.
Three things in that equation repay attention. The arrows mean vectors: force and acceleration have direction, and forces in different directions add like arrows, not like plain numbers. "Net" means the sum of all forces: a hiding in plain sight. And acceleration is a derivative: , the rate of change of velocity.
The pre-Newtonian intuition, that you need a force to keep something moving, is wrong, and seeing why it is wrong is the conceptual core. Friction is the force you forgot. That is a general research lesson: an apparently simple law often works only after you identify everything in the "net".
Newtonian mechanics predicts motion to extraordinary precision within its domain (spacecraft trajectories are planned with it, plus small relativistic corrections). It is also known to be an approximation to deeper theories. Both statements are true at once, and holding both is what "model versus reality" means.
Energy is a bookkeeping quantity that comes in forms (kinetic , gravitational potential , thermal, chemical, mass) and converts between them. A conservation law says some total stays constant in a closed system: energy changes form, but the sum does not change.
Conservation laws are powerful for reading because they let you skip the details. You don't need to simulate every instant of a pendulum's swing to know its speed at the bottom; energy at the top equals energy at the bottom.
Add up every form of energy in an isolated system; the total does not change over time.
Does not say: "Closed" is doing the work: energy flows in or out of open systems such as living cells and data centres.
Emmy Noether proved (1918) that every continuous symmetry of a physical system's laws corresponds to a conserved quantity. If the laws don't change over time, energy is conserved. If they don't change from place to place, momentum is conserved. If they don't change with orientation, angular momentum is conserved.
This is one of the deepest results in physics, and it is a theorem: mathematics applied to physical laws, not an analogy. When physicists-turned-ML-researchers talk about symmetry or invariance in networks (a convolutional net responds the same way to a shifted image), they are borrowing the concept. Whether a Noether-like conserved quantity exists in a given learning system is a specific, testable question, not a given.
Mass is a form of energy. The conversion factor is the speed of light squared, a huge number, so a little mass corresponds to an enormous amount of energy.
Does not say: This is the rest energy of an object at rest. For a moving object the full relation is E² = (pc)² + (mc²)². It says mass and energy are interconvertible, not that matter "is" energy in some mystical sense.
Dimensional analysis is the physicist's superpower and the habit most worth stealing. Every physical quantity has dimensions: length [L], mass [M], time [T]. An equation is only meaningful if both sides have the same dimensions. You cannot add metres to seconds.
Check : mass × (length/time)² = [M][L]²[T]⁻². Kinetic energy gives the same. ✓ The equations agree about what energy is dimensionally, which is a strong hint they belong to one framework.
Dimensional analysis can even predict relationships before any detailed theory. If a pendulum's period depends only on its length and gravity , the only combination with units of time is . So the period must be proportional to , which is correct up to a constant ().
Dimensional analysis constrains the form of an equation, and stops there. It can tell you period but not the . It also assumes you've identified all relevant variables, the same trap as forgetting friction.
Scale changes which physics matters. At human scales gravity and friction dominate; at cellular scales viscosity dominates and inertia is nearly irrelevant (bacteria stop almost instantly when they stop swimming); at atomic scales quantum effects dominate. A model valid at one scale can be useless at another. That is a lesson ML scaling-law papers must face too: a trend measured from small to medium models need not continue.
Entropy is the most misunderstood concept in physics, and the most abused when borrowed by other fields. The careful statement: entropy measures how many microscopic arrangements (microstates) are consistent with what you observe at large scale (the macrostate).
A gas spread evenly through a box can be realised in vastly more ways than a gas bunched in one corner. Left alone, a system wanders among its microstates, and is overwhelmingly likely to be found in the macrostates with the most microstates. That is the second law of thermodynamics: the entropy of an isolated system does not decrease (overwhelmingly likely, for large systems).
Entropy is a constant times the logarithm of the number of microstates W. The log turns an astronomically large count into a manageable, additive number.
Does not say: Assumes all W microstates are equally likely. The general (Gibbs) form, S = −k_B Σ p ln p, drops that assumption, and has the same mathematical form as Shannon entropy.
Particles start in one corner and move randomly. The coarse-grained entropy (how evenly they spread across cells) rises and then fluctuates near its maximum. Nothing pushes them apart: spread-out arrangements simply outnumber bunched ones.
A wave’s wavelength λ (distance between crests) equals its speed c divided by its frequency f (crests passing per second). Faster oscillation means shorter waves, at fixed speed.
Does not say: Here c is the wave speed in the medium. For light in vacuum it is the speed of light, but for sound in air it is ~343 m/s. The relation is close to a definition for any regular wave; the physics lies in what sets c.
Change the frequency and watch the wavelength shrink so that λ·f stays equal to the speed. Change the speed (the medium) and both relationships shift.
Here means wavelength. In linear algebra it means an eigenvalue; in ML it often means a regularisation strength. Same letter, three unrelated jobs. Always look up the paper's definition.
Fields assign a value to every point in space: temperature in a room (a number per point), wind (an arrow per point), the electromagnetic field. A field is a function of position, . Waves are travelling disturbances in fields. In quantum mechanics, the wavefunction is a field-like object whose squared magnitude gives probabilities. That is the source of the ψ entry and of most quantum mysticism.
Symmetry, in physics, means a transformation that leaves something unchanged. Rotate a sphere: it looks the same. Shift an experiment to another lab: the laws are the same. Symmetries constrain what equations are even allowed, which is why theoretical physicists use them as a search strategy.
Models are not reality. Every equation in this module is a model with a domain of validity:
Status of the equations in this module
| Equation | Status | Breaks down when… | |
|---|---|---|---|
| $F = ma$ | Established within its domain | speeds near c; atomic scales; accelerating reference frames | |
| $E = mc^2$ | Established (special relativity) | used for moving objects without the momentum term; used as a metaphor | |
| $\lambda = c/f$ | Established; near-definitional | the medium is dispersive (c depends on f), so it holds per frequency | |
| $S = k_B \ln W$ | Established (statistical mechanics) | microstates are not equally likely; then use the Gibbs form | |
| Frictionless block | Deliberate simplified model | friction matters, which is almost always on Earth |
"Spherical cow" models (strip away everything except the variables you think matter) are a core physics habit and a core ML-research habit (toy models of superposition, linear networks, infinite-width limits). They are powerful because they are wrong in controlled ways. The honest question is always: which conclusions survive adding the complications back?
Simulating motion (state += rate × dt) and training a network (θ −= η × gradient) share the skeleton of a discrete update rule. Gradient descent can even be analysed as a discretised "gradient flow" ODE.
Why this level: A real mathematical correspondence that researchers use productively. It does not make training a physical system; the "forces" are gradients of a chosen loss.
Gibbs entropy and Shannon entropy share the form −Σ p log p. Module 4 takes this apart carefully.
Why this level: Identical mathematics. Jaynes showed statistical mechanics can be derived as inference, a substantive link, but the physical (thermodynamic) and informational interpretations remain distinct.
Living things maintain low internal entropy by exporting entropy (heat, waste) to their surroundings. They do not violate the second law.
Why this level: Not an analogy: thermodynamics applies directly to organisms as physical systems, and the energy and entropy flows are measurable.
"Energy", "entropy", "force", "field", "quantum" and "resonance" are precise terms in physics and loose metaphors elsewhere. The metaphor is not wrong, but it borrows authority it hasn’t earned.
Free to read online. Volume I chapters on energy conservation and on "the relation of physics to other sciences" are perfect for this curriculum.
Watch for: Feynman’s habit of saying exactly which approximations are being made.
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Short concept videos on forces, energy and conservation, if you want to go deeper than Module 2.
Watch for: How every problem begins by choosing a system and listing what is conserved.
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Research-tested physics simulations (forces, energy skate park, waves) for intuition beyond Module 2.
Watch for: "Energy Skate Park" and "Wave on a String" pair directly with this curriculum.
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Thermodynamic entropy as a count of arrangements, not "disorder". The right setup for comparing it with Shannon entropy.
Watch for: Where energy spreading becomes probability. Hold that in mind when Module 4 puts Boltzmann and Shannon side by side.
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Frequencies as the hidden ingredients of a signal. The bridge from λ = c/f to spectra in physics and positional encodings in transformers.
Watch for: The integral as "average position of a wound-up graph". It is another sum in disguise.
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Integrals as accumulation and derivatives as rates, introduced through one geometric question rather than rules.
Watch for: The integral emerging as a limit of sums. That is the Σ→∫ bridge in the Rosetta Stone.
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The Feynman Lectures on Physics, Vol. I
Ch. 4 (conservation of energy) and ch. 3 (relation of physics to other sciences). Limitation: 1960s pedagogy; some framing is dated.
An ML paper that uses physics-style simplified models and phase diagrams. Limitation: toy models; transfer to large models is argued, not shown.
There are no right answers to check against. These questions are for thinking on the page, and your answers go to your notebook.
Write the update loop for something in your own work (a funnel, a content pipeline, an agent). What is the state, what is the rate, what is the timestep?
What is "conserved" in an agentic system you have built? Tokens? Budget? Attention? What breaks the conservation?
Find a non-physics article that uses "entropy". Rewrite the sentence using "number of possible arrangements". Does it survive?
Which of your favourite mental models is a spherical cow? What complication would you add back first?
F = ma · push a cart
Newton’s second law, a = Fnet/m: the same push gives a heavy cart less acceleration. With friction on, only the net force accelerates the cart; push below the friction threshold and nothing moves. This form assumes constant mass and speeds far below light (non-relativistic). Simulated, not measured.
Entropy in a box · coarse-grained
200 particles bounce in a box. We coarse-grain: forget exact positions and only count how many sit in each of 32 cells, then compute S = −Σ p log₂ p over those occupancy fractions. Remove the partition and S climbs toward its maximum (log₂ 32 = 5bits) because there are vastly more spread-out arrangements than bunched ones. Every particle’s motion is reversible, so the rise is overwhelmingly probable, not forced; with only 200 particles you will see small dips. This is a toy model, not a real gas.
λ = c / f · wavelength, speed, frequency
For a wave moving at speed c, one crest passes every period T = 1/f, so crests sit λ = c/f apart. Raise f at fixed speed and the wavelength shrinks. The same relation covers rope waves, sound and light; only the value of c differs (in a medium, c means that medium’s wave speed). Rope animation runs in real time. Light colours are approximate.