Mechanics to Relativity
A self-study course from Newton's second law to Einstein's field
equations, published whole: 41 units, each a slide deck with speaker notes, clickable glossary terms (press G
in a deck), a reading mode (R) and M to come back here, and a problem sheet with a hint ladder
and a worked solution folded under every problem. Commands such as ./mr problems 07 in the decks
refer to the course repository. See also the curriculum, and before unit 00
the readiness sheet.
Part A the equation of motion
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00
Newton's second law is an ODE — and dimensions tell you the answer's shape before you solve it
Three forces can be integrated in closed form, and one page of linear algebra fixes the form of the answer to almost anything else.
slides · reading mode · problemsspine · needs nothing · ~2 h
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01
Determinism, and the one hypothesis it rests on
Newton's laws determine the future — as a theorem, with a hypothesis, and a frictionless dome on which that hypothesis fails and the future does not.
slides · reading mode · problemsspine · needs 00 · ~2 h
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02
Phase portraits: what you know when there is no formula
Almost no equation of motion can be solved in closed form. This unit is the set of questions you can still answer exactly: how many equilibria, of what type, with which basins, and whether anything periodic exists.
slides · reading mode · problemsspine · needs 01 · ~2 h
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03
Where things are: curves, frames, and the frame that lies to you
Differentiation in several variables, the frame a curve carries with it, and the exact price of doing mechanics in a room that is turning.
slides · reading mode · problemsspine · needs 00 · ~2½ h
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04
Work, and the arrival of vector calculus
Energy in three dimensions is an integral along a path, and asking when that integral forgets the path is what builds the gradient, the curl and the first theorem of this course that is really about topology.
slides · reading mode · problemsspine · needs 03 · ~2½ h
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05
Flux, Gauss, and gravity stops being a force between pairs
Integration over regions and over surfaces, the divergence theorem, and the moment gravity becomes a field with a local law rather than a sum over pairs.
slides · reading mode · problemsspine · needs 04 · ~2½ h
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06
The damped driven oscillator, which is most of linear physics
One equation, solved completely — and with it the habit that runs to the end of the course: find the response to a single kick, and you have the response to everything.
slides · reading mode · problemsspine · needs 01, 02 · ~2.5 h
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07
Many oscillators, and the spectral theorem earning its keep
A coupled system is an uncoupled one in the wrong basis — and the right basis exists because two quadratic forms can be diagonalised at once when one of them is definite.
slides · reading mode · problemsspine · needs 06, 04 · ~2 h
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08
The central-force problem, and the two conservation laws that solve it
Six equations, two conserved quantities, and what is left is a one-dimensional problem you already know how to read off a graph — with the Kepler orbit falling out in closed form at the end.
slides · reading mode · problemsspine · needs 04, 06 · ~2.5 h
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09
Where is the planet at time t? Kepler's equation and the birth of Bessel functions
The orbit's shape came in closed form; its timetable is a transcendental equation with no closed-form inverse. Solving it by series is how Bessel functions entered mathematics — and where the series stop working is a number you can state exactly.
slides · reading mode · problemsspine · needs 08 · ~2.5 h
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10
Real planets are not points: multipoles, Legendre, and secular perturbation
One family of polynomials, one expansion of 1/|r − r′|, and the first perturbation calculation in the course — which predicts the inclination of a real satellite.
slides · reading mode · problemsspine · needs 05, 08 · ~2½ h
Part B the action
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11
The brachistochrone, and how to differentiate with respect to a curve
A number that depends on a whole curve has a derivative too, and setting it to zero turns "nature extremises this integral" into a differential equation.
slides · reading mode · problemsspine · needs 04 · ~2½ h
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12
Lagrangian mechanics: coordinates stop mattering
Write one scalar, T − V, in whatever coordinates the constraints suggest, and the equations of motion follow — correct in every coordinate system at once, with the constraint forces never named.
slides · reading mode · problemsspine · needs 11, 08 · ~2 h
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13
Noether: every conservation law you have met is one theorem
A one-parameter family of transformations that leaves the Lagrangian alone, or changes it only by a total time derivative, hands you a conserved quantity — and the proof is three lines.
slides · reading mode · problemsspine · needs 12 · ~2 h
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14
Hamilton: phase space, and flows that preserve volume
Trade n second-order equations for 2n first-order ones, and the new space turns out to carry a bracket, a class of allowed coordinate changes and a preserved volume — none of which configuration space ever showed you.
slides · reading mode · problemsspine · needs 13, 02 · ~2½ h
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15
Hamilton–Jacobi: mechanics was optics all along, and free motion is already geodesic motion
One nonlinear PDE replaces the equations of motion; its solutions are wavefronts, its trajectories are rays, and a particle in a potential is a free particle in a rescaled geometry.
slides · reading mode · problemsbranch · needs 14, 11 · ~2½ h
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16
Rigid bodies, and what a tensor actually is
Angular momentum is not parallel to angular velocity, and the object that relates them cannot be described by nine numbers — only by nine numbers per basis, related to one another by a law that is a theorem, not a definition.
slides · reading mode · problemsspine · needs 12, 05, 03 · ~3 h
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17
Integrability and chaos: determinism without predictability
Solutions exist, are unique, and depend continuously on the data — and are still useless for prediction after a few dozen orbits. Both halves are theorems, and they do not conflict.
slides · reading mode · problemsbranch · needs 14, 02, 01 · ~2½ h
Part C fields and the crisis
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18
From N masses to a field: the wave equation
Let the number of oscillators go to infinity and the system of ordinary differential equations becomes one partial differential equation — the first field in this course, and the first equation whose type decides what you are allowed to ask of it.
slides · reading mode · problemsspine · needs 07, 12 · ~2½ h
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19
Fourier and Sturm–Liouville: the modes are a basis, and that is a theorem
Separation of variables produces modes and then asks you to believe that every initial shape is a sum of them. It is true, it is not obvious, and the reason is that one differential operator is symmetric with respect to one inner product.
slides · reading mode · problemsspine · needs 18, 09, 10, 11 · ~3½ h
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20
The Fourier transform, complex analysis, and why causality is a statement about poles
A linear system that cannot answer before it is asked has a frequency response with no poles in one half of the complex plane — and that one fact ties refraction to absorption, turns unit 06's step function into a theorem, and evaluates integrals nothing else can.
slides · reading mode · problemsspine · needs 06 · ~3½ h
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21
Perturbation theory: the series that diverges and is still right
Almost nothing has an exact answer. What it has instead is an expansion in a small parameter — often divergent, usually non-uniform, and, handled correctly, accurate to ten digits.
slides · reading mode · problemsspine · needs 06, 02, 08 · ~3 h
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22
Continuum mechanics: the stress tensor, and conservation laws in local form
A deformable body has no single "force on it", only a force on every surface inside it — and proving that this force is bilinear in the surface is the course's first tensor that is derived rather than defined.
slides · reading mode · problemsspine · needs 16, 05, 18 · ~2½ h
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23
Static fields: electrostatics and magnetostatics, the same vector calculus twice
Coulomb's law is Newton's with a sign changed, so unit 05 transfers whole; what is new is the boundary-value problem, the vector potential, and a theorem that says how many field equations there must be before you have seen any of them.
slides · reading mode · problemsspine · needs 05, 10, 19 · ~2½ h
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24
Maxwell: the displacement current, and light
One term, added because Ampère's law contradicted charge conservation, turns four field equations into a wave equation whose speed is computed from two bench-top constants — and it is the speed of light.
slides · reading mode · problemsspine · needs 23, 20, 18 · ~3 h
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25
The crisis: Maxwell's equations do not survive a Galilean boost
The chain rule alone shows the wave operator breaks under x ↦ x − vt; asking which substitution does not break it produces the Lorentz transformation, with c supplied by the equation rather than by a postulate.
slides · reading mode · problemsspine · needs 24, 03 · ~2 h
Part D Minkowski
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26
Minkowski space: relativity as the geometry of a quadratic form
Homogeneity, isotropy and the group axioms leave one free constant, a speed; fix it, and every paradox of special relativity becomes a statement about one indefinite bilinear form.
slides · reading mode · problemsspine · needs 03, 14 · ~2½ h
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27
Relativistic dynamics: four-vectors, and why colliders exist
Energy and momentum are the components of one vector whose Minkowski length is the mass; square the conservation law and a reaction threshold falls out in three lines, in any frame you like.
slides · reading mode · problemsspine · needs 26, 14, 16 · ~2½ h
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28
Electromagnetism was relativistic all along: the field tensor
E and B are not two fields but two blocks of one antisymmetric tensor, and a boost stirs them into each other — which is why a neutral wire pulls on a charge moving beside it.
slides · reading mode · problemsspine · needs 27, 25 · ~2½ h
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29
Classical field theory: Noether again, and where Tμν actually comes from
A field is a mechanical system whose coordinates are labelled by points of space. Everything follows from one functional — the equations of motion, the conserved currents, and the object Einstein's equation puts on its right.
slides · reading mode · problemsspine · needs 28, 13, 11 · ~2½ h
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30
The equivalence principle, and why flat spacetime cannot hold gravity
One experimental coincidence — inertial mass equals gravitational mass — predicts falling light and slow clocks before any geometry exists, and then proves that no field on Minkowski space can be gravity.
slides · reading mode · problemsspine · needs 27, 05, 16 · ~2½ h
Part E curved space
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31
Manifolds: doing all of it without coordinates
Coordinates stop being data. What survives is a space, the vectors tangent to it, the tensor fields on it, and one derivative — the Lie derivative — that needs nothing but a flow.
slides · reading mode · problemsspine · needs 16, 30 · ~3 h
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32
Metric, connection, geodesics — free fall is a straight line
A metric measures. A connection differentiates. Together they decide what straight means — and the straight lines turn out to be the worldlines of bodies nobody is pushing.
slides · reading mode · problemsspine · needs 31, 11 · ~3 h
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33
Curvature: tidal forces, with the mask off
Two covariant derivatives fail to commute, and the failure is a tensor. That tensor is what no change of coordinates can remove — and, component for component, it is unit 05's tidal tensor.
slides · reading mode · problemsspine · needs 32, 05 · ~3½ h
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34
Differential forms and Cartan: Stokes' theorem was always one theorem
One algebra with a sign in it, one derivative, one theorem. Maxwell's four equations become two, the four integral theorems become one, and unit 04's hole acquires a number.
slides · reading mode · problemsbranch · needs 31, 24, 33 · ~3 h
Part F Einstein
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35
The Einstein field equations, twice
Write down what a law of gravity must satisfy, and discover that in four dimensions exactly one equation satisfies it. Then derive the same equation again from an action, and watch the same constant appear.
slides · reading mode · problemsspine · needs 33, 30 · ~3 h
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36
Schwarzschild: the first exact solution, and what its two singularities are
Solve the vacuum field equations for a round, static star, once and slowly. The answer breaks down in two places: one is a badly chosen clock, and the other is the end of time.
slides · reading mode · problemsspine · needs 35, 13 · ~3½ h
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37
The classical tests: three numbers, and one of them is 43
Unit 08's orbit equation acquires exactly one term. From it come the 43 arcseconds of Mercury, the 1.75 arcseconds of starlight at the Sun's edge, a radar echo that returns late, and an orbit that no longer has to be stable.
slides · reading mode · problemsspine · needs 36, 21, 08, 10 · ~3½ h
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38
Gravitational waves: ten components, two polarisations, one chirp
Linearise the field equations about flat spacetime and they become unit 24's wave equation. Count what the gauge leaves and two polarisations remain; solve with the retarded Green's function and a binary star's orbital period shrinks by 2.40×10⁻¹² seconds every second, which is what the sky does.
slides · reading mode · problemsbranch · needs 35, 21, 24, 20 · ~3 h
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39
Cosmology
Assume the universe looks the same everywhere and in every direction, and Einstein's equation collapses to one ordinary differential equation for one function of time — which, solved with measured numbers, says the universe is 13.8 billion years old.
slides · reading mode · problemsbranch · needs 35, 22 · ~3 h
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40
Rotating and charged holes, and where this course stops
Every hole that has ever formed rotates. Kerr's solution has an ergosphere you can mine, a hidden symmetry nobody expected, and — at the end — a temperature this course cannot derive.
slides · reading mode · problemsbranch · needs 36, 15 · ~3 h