A course in twelve sittings

Can the future be computed from the present?

One question runs the length of this volume. A perfect calculator, given every position and every velocity, claims it can predict everything forever. Twelve modules decide whether it can exist.

1. 01
   What a law is

   The demon is born. It looks unbeatable.
2. 02
   Coordinates and vectors

   The demon needs a language that survives a change of view.
3. 03
   The calculus you need

   The demon gets its instrument.
4. 04
   Motion

   The demon learns to run the tape forward.
5. 05
   Force and Newton's laws

   The demon gets its rule.
6. 06
   Many particles

   The demon gets its first shortcut.
7. 07
   Energy

   The demon can skip the middle of the calculation.
8. 08
   Least action

   The demon stops stepping and starts choosing whole paths.
9. 09
   Symmetry and conservation

   The demon gets a list of quantities it never recomputes.
10. 10
    Hamiltonian mechanics

    The demon gets a better map.
11. 11
    Liouville's theorem

    The demon is answered. Information cannot be destroyed.
12. 12
    Poisson brackets

    The demon is handed to quantum mechanics.

The ledger

Every idea in the volume, and the module that introduces it. Nothing is used before its own introduction; an automated check enforces that.

• state, s Everything you need to write down now so the rule can produce the next moment.
• law (update rule), f A rule that maps the current state to the next state.
• determinism One output per input: the current state fixes the next state completely.
• reversibility The rule has an inverse: from the current state you can recover the previous one.
• Laplace's demon An imagined intellect that knows the exact state of everything and computes the rest of history from it.
• coordinates, x, y The ruled directions of a chosen grid, used to turn geometry into numbers.
• frame The invisible graph paper an observer lays over the world before measuring anything.
• vector, v with an arrow A length together with a direction. Displacement is the archetype.
• components, v_x, v_y A vector's readings in one particular frame. Turn the frame and they change.
• magnitude, |v| A vector's length. The same in every frame.
• invariant A quantity that no change of frame can touch. The currency laws are written in.
• limit The value a shrinking-window measurement settles on. A destination, not a division.
• derivative, dx/dt The instantaneous rate of change: the slope a smooth curve is impersonating at a point.
• integral, ∫ The settled sum of slivers: the area under a rate curve, which rebuilds the total.
• velocity, v The derivative of position: how fast, and which way. A vector.
• acceleration, a The derivative of velocity: the dial one level deeper than the speedometer.
• force, F A push or a pull, with direction. The supplier of acceleration.
• mass, m A body's stubbornness: the exchange rate between force applied and acceleration bought.
• Newton's laws Left alone, velocity keeps; forces buy acceleration at the price of mass; pushes come in pairs.
• momentum, p Mass times velocity. The currency bodies trade when they push on each other.
• conservation A total that interactions inside the system cannot change, only relocate.
• system A chosen list of bodies. Forces are internal or external relative to the list.
• work, W Force times distance moved along it: the transfer that moves energy between accounts.
• kinetic energy, K Motion's holdings: one half m v squared.
• potential energy, U Holdings stored in position, paid back in full by an honest force like gravity.
• Lagrangian, L Kinetic energy minus potential energy: the per-instant grade of a path.
• action, S The Lagrangian summed along an entire path: one number per possible history.
• stationary Flat to first order: nudge the path and the action refuses to move. The real path's signature.
• symmetry A change to the whole setup that the Lagrangian does not register.
• Noether's theorem Each symmetry of the Lagrangian forces one quantity to be conserved: place pays momentum, time pays energy.
• phase space The map whose coordinates are position and momentum: one point is one complete state.
• Hamiltonian, H Total energy as a function of position and momentum: the altitude that steers the phase-space flow.
• Liouville's theorem Hamiltonian flow is incompressible: a blob of states shears freely but its area never changes.
• information The count of distinguishable worlds. Conserved by every closed classical flow; blurred only by finite readers.
• Poisson bracket, {A, B} The operation that reads one quantity's change along the flow another generates. All of mechanics, folded to fit one hand.