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Introduction
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The impetus of Ipsum Invarians might be easiest to see if you look at these reductions as fuzzy images, not proofs:
Music is merely a variety of tones. Chemistry is little more than the behavior of electrons. Biology happens when molecules copy themselves. Digital logic is nothing but one gate, many times over. Computation is simply addition, ad infinitum.
Then what does physics come down to? Less poetically: what is the most austere basis we can use to derive physics? This research program postulates Hamilton's Principle as the kernel of physics. It has embraced AI to expose the difficulties with this postulate.
The Current Program (Round 3)
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- ... that nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit?
- ... that ut enim ad minima veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam?
- ... that nisi ut aliquid ex ea commodi consequatur quis autem vel eum iure reprehenderit?
- ... that qui in ea voluptate velit esse quam nihil molestiae consequatur, vel illum qui dolorem eum fugiat?
- ... that quo voluptas nulla pariatur at vero eos et accusamus et iusto odio dignissimos?
Program History
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- Round 1 – nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit.
- Round 2 – quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est.
- Round 2 review – omnis dolor repellendus ut aut reiciendis voluptatibus maiores alias consequatur.
- Round 3 opening – aut perferendis doloribus asperiores repellat, sed quia consequuntur magni dolores.
Hamilton's Principle
Newton's laws, Maxwell's equations, general relativity and the Standard Model are each formulated as an action via Hamilton's Principle. In every case the equations of motion are consequences rather than starting points. Conservation of energy, momentum and charge are not separate laws at all — Noether proved they follow from the action's symmetries. New models are now built by writing the action contrained by symmetry and letting the dynamics follow. This is how electromagnetism and gravity were derived and how the nuclear forces were found. Hamilton's Principle alone survives the passage to quantum mechanics intact: the action becomes the phase carried by every possible history, and the classical path is simply where those phases fail to cancel. The action is the deepest thing physics knows.
- Stationary is not least — the long way round a great circle obeys the same rule at every point, and is plainly not the shortest. See the diagram.
- Why the rule holds at all — nature selects nothing. It sums over everything, and what looks like a choice is the residue left by interference.
- Start without the mathematics — every page carrying equations has a plain-language counterpart, linked from it and linking back.
Noether's theorems
Klein read the paper to the Göttingen Scientific Society on 26 July 1918. Its last section answers a claim of Hilbert's: that energy conservation in general relativity fails to take the form it takes everywhere else in physics. There are two theorems in the paper, and they are not versions of one another.
The first takes a symmetry with a fixed number of parameters and returns that many conservation laws, which hold for real motions. The second takes a symmetry that is allowed to differ from place to place and returns identities between the field equations — identities that hold whether or not those equations are satisfied. The second theorem is why a gauge theory has fewer independent field equations than it has fields, and why gravity has no local energy.
A 90-page reading edition: both theorems proved in full and kept apart, the mathematics they need, the physics they explain, and a plain-language counterpart for every page that carries equations.
- Göttingen and the energy problem — the question the paper was written to settle
- The primer — plain language, no mathematics
- The two theorems side by side — the distinction most accounts lose
The Work of James Clerk Maxwell
On Physical Lines of Force
In four instalments between March 1861 and February 1862, James Clerk Maxwell
filled space with spinning cells and little rolling particles, to explain the mechanics behind magnetic fields. His machine was wrong. On the way to building it,
he wrote down the displacement current, derived the speed of light from two
electrical measurements that had nothing to do with optics, and founded field
theory.
A 65-page reading edition: Maxwell's argument in his own order and register, the modern equations derived from his results, and a plain-language explainer for every major idea.
- Introduction — the story, and why the paper matters
- The primer — plain language, no mathematics
- The missing term — the one addition that made radio possible
Featured plate
| |
| The shift from Newtonian vectors to scalar fields and variational principles defined modern theoretical physics. | |
| Core Frameworks | |
|---|---|
| Classical Mechanics | Lagrangian mechanics |
| Dynamical Optimization | Principle of least action |
| Electromagnetism | Maxwell's equations |
| Vacuum Dynamics | Spacetime topology |
| Key Equations | |
|
| |
The narrative of 18th- to 20th-century physics lies in a profound mathematical restructuring. This era marked the transition from treating nature as a collection of forces acting on point masses to understanding it through continuous fields, spacetime manifolds, and unified variational principles.
The foundation was laid when physical systems began to be analyzed not through Newtonian vectors, but through scalar quantities—kinetic and potential energy—operating within generalized coordinates. By positing that the true dynamical trajectory of any system is the one that leaves the action integral stationary, theorists established a universal mathematical language.
This analytical approach seamlessly extended beyond mechanical gears and pulleys into the continuum. It provided the structural mathematics required to unify electricity and magnetism into a single vacuum field theory, and later, to model gravity not as a pulling force, but as the dynamic geometric curvature of spacetime itself. Whether applied to subatomic particle behavior or cosmic expansion, these foundational principles continue to govern the leading edge of modern physics.
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