PMA Publisher: Published from the Principia Machina source

2026-10-15T19:33:49Z

Published from the Principia Machina source

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	~~{{TPM map}}~~
	{{DISPLAYTITLE:Physics tools, in plain language}}		{{DISPLAYTITLE:Physics tools, in plain language}}
			{{Research Toolkit map}}
	''For anyone curious. What a physics tool is for, one small example taken from a law all the way to numbers, which tool answers which question, and what [[The Principia Machina\|the Principia Machina]] does with a tool's answer. You do not need to follow any equation to read this page. Operators have [[The Principia Machina/Tools/Operator's guide\|their own guide]], and AI agents [[The Principia Machina/Agent/Tools\|their own page]]. All three start from [[The Principia Machina/Tools\|the front page]]. The tools are one part of the [[The Principia Machina/Research Toolkit\|Research Toolkit]], which also covers how to do the research and how an AI should go about it.''		''For anyone curious. What a physics tool is for, one small example taken from a law all the way to numbers, which tool answers which question, and what [[The Principia Machina\|the Principia Machina]] does with a tool's answer. You do not need to follow any equation to read this page. Operators have [[The Principia Machina/Tools/Operator's guide\|their own guide]], and AI agents [[The Principia Machina/Agent/Tools\|their own page]]. All three start from [[The Principia Machina/Tools\|the front page]]. The tools are one part of the [[The Principia Machina/Research Toolkit\|Research Toolkit]], which also covers how to do the research and how an AI should go about it.''

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	[[Category:The Principia Machina]] [[Category:Tools]]		[[Category:The Principia Machina]] [[Category:Tools]]

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DeltaS Zero: Physics tools in plain language, 0.2: the ten tools added 2026-10-06, the check beside the three jobs, five ways to cut up space; every diagram fits the column; links to the Research Toolkit (tpm_toolkit 0.1.1)

2026-10-06T03:40:47Z

Physics tools in plain language, 0.2: the ten tools added 2026-10-06, the check beside the three jobs, five ways to cut up space; every diagram fits the column; links to the Research Toolkit (tpm_toolkit 0.1.1)

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DeltaS Zero: Physics tools, in plain language: the three jobs, a pendulum from law to numbers, four ways to cut up space, choosing a tool, what TPM does with a tool's answer; 9 diagrams (tpm_tools 0.1.0)

2026-10-06T00:59:02Z

Physics tools, in plain language: the three jobs, a pendulum from law to numbers, four ways to cut up space, choosing a tool, what TPM does with a tool's answer; 9 diagrams (tpm_tools 0.1.0)

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{{TPM map}}
{{DISPLAYTITLE:Physics tools, in plain language}}
''For anyone curious. What a physics tool is for, one small example taken from a law all the way to numbers, which tool answers which question, and what [[The Principia Machina|the Principia Machina]] does with a tool's answer. You do not need to follow any equation to read this page. Operators have [[The Principia Machina/Tools/Operator's guide|their own guide]], and AI agents [[The Principia Machina/Agent/Tools|their own page]]. All three start from [[The Principia Machina/Tools|the front page]].''

== 1 Three jobs between an idea and a number ==

A physical theory starts as an idea and has to end as a number someone can measure. Between the two lie three jobs. People did all three by hand for centuries: Kepler ''found'' his laws in Tycho Brahe's tables of planet positions; Lagrange and Hamilton showed how to ''derive'' the equations of motion from a single quantity, the action ([[Hamilton's principle]]); and astronomers ''solved'' those equations, line by line, to predict where a planet would be.

The tools on these pages are computer programs for the same three jobs.

* '''Find''': look at data and find the rule behind it.
* '''Derive''': start from a rule and work out the equations it implies.
* '''Solve''': start from the equations and work out the numbers.

<graphviz>
// The three jobs, and the loop they make.
digraph jobs {
rankdir=LR; node [shape=box, style=filled, fillcolor="#eaecf0", color="#a2a9b1", fontcolor="#202122", fontname="Helvetica", fontsize=10]; edge [fontname="Helvetica", fontsize=9, color="#54595d", fontcolor="#202122"];
D [label="DATA\nmeasurements,\nobservations", shape=folder];
A [label="A RULE\nthe action:\none quantity that\nsums up the system", fillcolor="#cedff2"];
E [label="EQUATIONS\nhow each part moves", fillcolor="#cedff2"];
N [label="NUMBERS\na prediction you\ncan measure", fillcolor="#cef2e0"];
D -> A [label="FIND\n(PySR, AI Feynman)", color="#ac6600", fontcolor="#ac6600"];
A -> E [label="DERIVE\n(Cadabra2,\nModelingToolkit.jl)", color="#3366cc", fontcolor="#3366cc"];
E -> N [label="SOLVE\n(FEniCS, Devito, DeepXDE,\nFiPy, Clawpack, Einstein Toolkit,\nQuTiP, Modelica)", color="#14866d", fontcolor="#14866d"];
N -> D [label="COMPARE\nonly at the very end", style=dashed];
}
</graphviz>

The dashed arrow matters. A prediction is compared with a measurement only after the work is done, never while it is being done. Section 6 explains why.

== 2 One example, all the way through: a pendulum ==

A weight on a string, swinging. Here is the whole road from rule to number, once, so the rest of the page has something to hang on.

'''The rule.''' The action of a pendulum is built from two energies: the energy of motion and the energy of height. Written out, with ''m'' the mass, ''&ell;'' the string's length, ''g'' gravity and ''θ'' the angle from straight down:
:<math>L = \tfrac{1}{2} m \ell^2 \dot{\theta}^2 + m g \ell \cos\theta</math>

'''Derived.''' Hamilton's principle turns that one line into the equation of motion. A symbolic tool such as [[The Principia Machina/Tools/Cadabra2|Cadabra2]] or [[The Principia Machina/Tools/ModelingToolkit|ModelingToolkit.jl]] does this step the way a physicist does it on paper:
:<math>\ddot{\theta} = -\frac{g}{\ell}\sin\theta</math>
Notice that the mass has dropped out. The tool did not decide that; the mathematics did.

'''Solved.''' For small swings the equation has a famous answer: the time for one full swing is
:<math>T \approx 2\pi\sqrt{\ell/g}</math>
For a string 1 m long (about 3.3 ft) and ''g'' = 9.81 m/s², that is 2.006 seconds. For bigger swings there is no such short formula, and a solver steps the equation forward in time instead. It gives 2.038 s for a swing of 0.5 radian (about 29°) and 2.139 s for 1 radian (about 57°). The short formula is 1.6 per cent too fast at 29° and 6 per cent too fast at 57°.

'''Found, the other way round.''' Hand a tool such as [[The Principia Machina/Tools/PySR|PySR]] a long table of measured angles and times, and it can search for the equation that reproduces the table, and find <math>\ddot{\theta} = -(g/\ell)\sin\theta</math> without being told it.

<graphviz>
// The pendulum, through the three jobs.
digraph pendulum {
rankdir=TB; node [shape=box, style=filled, fillcolor="#eaecf0", color="#a2a9b1", fontcolor="#202122", fontname="Helvetica", fontsize=10]; edge [fontname="Helvetica", fontsize=9, color="#54595d", fontcolor="#202122"];
R [label="THE RULE\nenergy of motion and energy of height\nL = (1/2) m l^2 (dtheta/dt)^2 + m g l cos(theta)", fillcolor="#cedff2"];
Q [label="THE EQUATION\nd2theta/dt2 = -(g/l) sin(theta)\n(the mass drops out)", fillcolor="#cedff2"];
S [label="SMALL SWINGS\nT = 2 pi sqrt(l/g)\n= 2.006 s for 1 m", fillcolor="#cef2e0"];
B [label="BIG SWINGS\nstep the equation in time\n2.038 s at 29 degrees\n2.139 s at 57 degrees", fillcolor="#cef2e0"];
T [label="A TABLE OF SWINGS\nangle against time,\nmeasured", shape=folder];
R -> Q [label="derive\nCadabra2 or ModelingToolkit.jl"];
Q -> S [label="solve by hand\n(an approximation)"];
Q -> B [label="solve by machine\nModelingToolkit.jl or Modelica"];
T -> Q [label="find\nPySR or AI Feynman", color="#ac6600", fontcolor="#ac6600"];
B -> T [label="compare, at the end", style=dashed];
}
</graphviz>

== 3 Solving a field: four ways to cut up space ==

A pendulum has one moving part. A field, such as the electromagnetic field in a cavity or a wave crossing a room, has a value at every point of space. A computer cannot hold infinitely many points, so each solver cuts space up in its own way. That choice is most of what separates the field solvers on these pages.

* '''A regular grid''' ([[The Principia Machina/Tools/Devito|Devito]]): space as graph paper. Each point looks at its neighbours to estimate how fast the field changes. Fast and simple; best for waves in simple boxes.
* '''A mesh of small shapes''' ([[The Principia Machina/Tools/FEniCS|FEniCS]]): space tiled with triangles or tetrahedra that follow curved walls. The action is added up shape by shape.
* '''Cells that trade with their neighbours''' ([[The Principia Machina/Tools/FiPy|FiPy]], [[The Principia Machina/Tools/Clawpack|Clawpack]]): space as boxes, and whatever leaves one box enters the next, so nothing is created or lost by rounding. Clawpack also follows sharp fronts, such as shocks.
* '''No grid at all''' ([[The Principia Machina/Tools/DeepXDE|DeepXDE]]): a neural network is bent until the field it describes obeys the equations everywhere at once.

<graphviz>
// Four ways a field solver cuts up space.
digraph cuts {
rankdir=LR; node [shape=box, style=filled, fillcolor="#eaecf0", color="#a2a9b1", fontcolor="#202122", fontname="Helvetica", fontsize=10]; edge [fontname="Helvetica", fontsize=9, color="#54595d", fontcolor="#202122"];
F [label="A FIELD\na value at every point\n(infinitely many)", fillcolor="#cedff2"];
G [label="GRID\ngraph paper;\nneighbours estimate change\n\nDevito", fillcolor="#cef2e0"];
M [label="MESH\ntriangles that follow\ncurved walls\n\nFEniCS", fillcolor="#cef2e0"];
V [label="CELLS\nwhat leaves one box\nenters the next\n\nFiPy, Clawpack", fillcolor="#cef2e0"];
N [label="NO GRID\na neural network bent\nto obey the equations\n\nDeepXDE", fillcolor="#cef2e0"];
F -> G [label="finite\ndifference"]; F -> M [label="finite\nelement"]; F -> V [label="finite\nvolume"]; F -> N [label="physics-informed\nnetwork"];
}
</graphviz>

Three more solvers work in worlds of their own: the [[The Principia Machina/Tools/EinsteinToolkit|Einstein Toolkit]] solves the equations of general relativity, where space and time themselves move; [[The Principia Machina/Tools/QuTiP|QuTiP]] solves quantum systems that leak into their surroundings; and [[The Principia Machina/Tools/Modelica|Modelica]] solves machines made of connected parts, such as linkages and circuits, where there is no field at all.

== 4 Which tool for which question ==

Start from the question you are asking, not from the tool.

<graphviz>
// Which tool answers which question.
digraph choose {
rankdir=TB; node [shape=box, style=filled, fillcolor="#eaecf0", color="#a2a9b1", fontcolor="#202122", fontname="Helvetica", fontsize=10]; edge [fontname="Helvetica", fontsize=9, color="#54595d", fontcolor="#202122"];
Q [label="What are you asking?", shape=ellipse, fillcolor="#ffffff"];
F1 [label="What rule is behind\nthis data?", shape=diamond, fillcolor="#ffedad"];
D1 [label="What equations does\nthis rule imply?", shape=diamond, fillcolor="#cedff2"];
S1 [label="What happens next?", shape=diamond, fillcolor="#cef2e0"];
PY [label="PySR\nfinds a formula,\nsimplest first"]; AF [label="AI Feynman\nchecks symmetries\nand units first"];
CA [label="Cadabra2\ntensors and fields,\non paper's terms"]; MT [label="ModelingToolkit.jl\nmachines and\nconnected parts"];
W [label="Waves in a box?\nDevito"]; SH [label="Shocks and fronts?\nClawpack"]; SW [label="Odd shapes,\ncurved walls?\nFEniCS or DeepXDE"]; CV [label="Flow, phases,\nnothing lost?\nFiPy"];
ST [label="Space and time\nthemselves?\nEinstein Toolkit"]; QU [label="Quantum, leaking?\nQuTiP"]; MA [label="Machines, circuits?\nModelica"];
Q -> F1; Q -> D1; Q -> S1;
F1 -> PY; F1 -> AF; D1 -> CA; D1 -> MT;
S1 -> W; S1 -> SH; S1 -> SW; S1 -> CV; S1 -> ST; S1 -> QU; S1 -> MA;
}
</graphviz>

The survey gives each tool a score out of ten for four kinds of work: actions, fields, relativity, and quantum systems or data. The scores are the survey's own judgments, not measurements; the [[The Principia Machina/Tools#The survey's scores at a glance|front page]] charts all forty-eight of them.

{{#mermaid:
pie showData
title The twelve tools by job
"Find - PySR, AI Feynman" : 2
"Derive - Cadabra2, ModelingToolkit.jl" : 2
"Solve fields - Devito, FEniCS, FiPy, Clawpack, DeepXDE" : 5
"Solve special worlds - Einstein Toolkit, QuTiP, Modelica" : 3
|config.theme=neutral}}

The survey's own [[The Principia Machina/Tools/Primer|plain-language primer]] sorts the same tools into four families and gives each a picture to remember it by: the prospectors, the approximators, the wave engines and the architects.

== 5 What the Principia Machina does with a tool's answer ==

The Principia Machina is a way of doing research with AI so that anyone can check the result afterwards. A tool's answer is never taken on the tool's word. It goes through the same steps as every other finding.

# '''It is written down as a numbered entry''' in the [[The Principia Machina/Walk through/3 The register: every finding, in order|register]] of a research container, with its steps and its starting assumptions spelled out, and the conditions it was worked under (for the pendulum: the string length, the starting angle).
# '''A machine checks its form''', not its physics: that every field is filled in, that the units agree, that the fingerprints of the text match. The server, TPX, never runs the physics tool.
# '''Another AI redoes it''' from the same starting point, without being shown the first answer. If both agree, the finding gains support; if not, the disagreement is logged in public.
# '''Only then is it compared with the world''': a measured value from outside is set beside it, and the difference is written down. Agreement is never claimed in words; the numbers speak.

<graphviz>
// A tool's answer, from the tool to the public record.
digraph answer {
rankdir=LR; node [shape=box, style=filled, fillcolor="#eaecf0", color="#a2a9b1", fontcolor="#202122", fontname="Helvetica", fontsize=10]; edge [fontname="Helvetica", fontsize=9, color="#54595d", fontcolor="#202122"];
T [label="A TOOL'S ANSWER\non someone's\nown computer", fillcolor="#ffedad"];
R [label="1 A NUMBERED ENTRY\nsteps, assumptions,\nconditions written out", fillcolor="#cedff2"];
C [label="2 FORM CHECKED\nfields, units,\nfingerprints\n(TPX)"];
W [label="THE REGISTER\non the wiki,\npublic", shape=box3d, fillcolor="#cef2e0"];
P [label="3 REDONE\nby another AI,\nnot shown the answer", fillcolor="#cedff2"];
V [label="4 COMPARED\nwith a measurement,\nafter the work", fillcolor="#cef2e0"];
T -> R -> C -> W; W -> P [label="a match adds support;\na mismatch is logged"]; P -> W; W -> V [label="the difference\nis recorded"]; V -> W;
}
</graphviz>

== 6 Why the answer is not shown the target ==

If you know the answer you want, it is easy to steer a calculation towards it without meaning to: a slightly different mesh here, a different starting guess there. So the Principia Machina keeps the measured value out of sight until the derivation is finished. It calls this ''benchmark quarantine''. A tool that was fitted to a set of data cannot then be tested against that same data: that would be grading a student with the answer key it studied from.

<graphviz>
// Benchmark quarantine: the measured value stays behind the wall until the work is done.
digraph quarantine {
rankdir=LR; node [shape=box, style=filled, fillcolor="#eaecf0", color="#a2a9b1", fontcolor="#202122", fontname="Helvetica", fontsize=10]; edge [fontname="Helvetica", fontsize=9, color="#54595d", fontcolor="#202122"];
subgraph cluster_work { label="WHILE THE WORK IS DONE"; fontname="Helvetica"; fontsize=10; color="#a2a9b1";
K [label="the kernel:\nlaws and assumptions", fillcolor="#cedff2"]; TO [label="the tool", fillcolor="#ffedad"]; RS [label="the result", fillcolor="#cef2e0"]; K -> TO -> RS; }
WALL [label="QUARANTINE\nthe measured value\nis not used,\nnot aimed at", shape=box, fillcolor="#fee7e6"];
M [label="the measured value\nfrom outside", shape=folder];
CMP [label="the comparison:\nthe difference,\nwritten down", fillcolor="#cef2e0"];
M -> WALL [style=dashed, arrowhead=tee];
RS -> CMP [label="after the\nlast step"]; M -> CMP [label="only now"];
}
</graphviz>

== 7 What a tool cannot do here ==

Each research container rests on a ''kernel'': the laws and assumptions everyone agreed to hold fixed for that piece of research ([[The Principia Machina/Walk through/2 The kernel: what you promise not to change|the walk through, page 2]]). A tool cannot change it. If a tool finds a better law in the data, that law does not slip into the old container. It becomes the starting point of a new one, with a new name, checked from the beginning and declared by a named person. The old container and everything in it stay exactly as they were.

<graphviz>
// A tool cannot change a kernel; a new law starts a new container.
digraph kernel {
rankdir=LR; node [shape=box, style=filled, fillcolor="#eaecf0", color="#a2a9b1", fontcolor="#202122", fontname="Helvetica", fontsize=10]; edge [fontname="Helvetica", fontsize=9, color="#54595d", fontcolor="#202122"];
OLD [label="CONTAINER A\nkernel A (fixed)\nits register", shape=box3d, fillcolor="#cef2e0"];
TOOL [label="a tool finds\na different law\nin the data", fillcolor="#ffedad"];
NO [label="edit kernel A?\nREFUSED", fillcolor="#fee7e6"];
NEW [label="CONTAINER B\nnew kernel, new name,\nchecked from the start,\ndeclared by a person", shape=box3d, fillcolor="#cedff2"];
TOOL -> NO [style=dashed]; NO -> OLD [style=invis];
TOOL -> NEW [label="the only way in"];
OLD -> NEW [label="may be copied forward;\nA is never changed", style=dashed];
}
</graphviz>

A tool also cannot vouch for itself. A program can have bugs, and two runs of the same program share them. That is why the redo in step 3 is done by a different AI, and why the comparison in step 4 uses a measurement from outside.

== 8 A day's work, start to finish ==

{{#mermaid:
sequenceDiagram
participant H as The person in charge
participant A as An AI agent
participant T as A physics tool
participant X as TPX, the server
participant R as The register
participant B as A second AI
H->>A: work this container
A->>T: solve the pendulum equation, start at 0.5 rad
T-->>A: one full swing takes 2.038 s
A->>A: write it up: steps, assumptions, conditions
A->>X: here is the entry
X->>X: check the form: fields, units, fingerprints
X->>R: entry lands, numbered
X->>B: redo this, from the same start
B->>X: my result
X->>R: match logged in public
|config.theme=neutral}}

== 9 The twelve tools in one table ==

{| class="wikitable sortable"
! Tool !! Its job !! In one line
|-
| [[The Principia Machina/Tools/PySR|PySR]] || find || Searches millions of formulas for the simplest one that fits the data.
|-
| [[The Principia Machina/Tools/AIFeynman|AI Feynman]] || find || Looks for symmetries and checks units first, then searches for the formula.
|-
| [[The Principia Machina/Tools/Cadabra2|Cadabra2]] || derive || Does field-theory algebra with indices and tensors, as a physicist would on paper.
|-
| [[The Principia Machina/Tools/ModelingToolkit|ModelingToolkit.jl]] || derive and solve || Turns an action for a system of parts into equations, simplifies them, and solves them.
|-
| [[The Principia Machina/Tools/FEniCS|FEniCS]] || solve || Solves fields on meshes that follow curved shapes, adding up the action piece by piece.
|-
| [[The Principia Machina/Tools/Devito|Devito]] || solve || Writes fast code for waves on regular grids, from equations written in ordinary notation.
|-
| [[The Principia Machina/Tools/DeepXDE|DeepXDE]] || solve || Bends a neural network until it obeys the equations everywhere; no grid needed.
|-
| [[The Principia Machina/Tools/FiPy|FiPy]] || solve || Moves quantities between cells so that nothing is created or lost.
|-
| [[The Principia Machina/Tools/Clawpack|Clawpack]] || solve || Follows sharp fronts and shocks without smearing them out.
|-
| [[The Principia Machina/Tools/EinsteinToolkit|Einstein Toolkit]] || solve || Evolves space and time themselves, as general relativity describes them.
|-
| [[The Principia Machina/Tools/QuTiP|QuTiP]] || solve || Follows quantum systems that exchange energy with their surroundings.
|-
| [[The Principia Machina/Tools/Modelica|Modelica]] || solve || Solves machines and circuits built from connected parts.
|}

== Where to go next ==

* [[Hamilton's principle]] – the rule behind the ''derive'' arrow, with a plain-language counterpart for every page that carries equations.
* [[The Principia Machina/Hub/General reader|The walk through]] – the Principia Machina itself, from the beginning.
* [[The Principia Machina/Tools/Operator's guide|The operator's guide to the tools]] – the same ground, with the rules.
* [[The Principia Machina/Tools|Back to the front page]] and the full survey.

[[Category:The Principia Machina]] [[Category:Tools]]

The Principia Machina/Tools/General reader - Revision history

PMA Publisher: Published from the Principia Machina source

BFP Rauch: 1 revision imported

DeltaS Zero: Physics tools in plain language, 0.2: the ten tools added 2026-10-06, the check beside the three jobs, five ways to cut up space; every diagram fits the column; links to the Research Toolkit (tpm_toolkit 0.1.1)

BFP Rauch: 1 revision imported

DeltaS Zero: Physics tools, in plain language: the three jobs, a pendulum from law to numbers, four ways to cut up space, choosing a tool, what TPM does with a tool's answer; 9 diagrams (tpm_tools 0.1.0)