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<h2 id="mp-hamilton-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Hamilton's Principle</h2> | <h2 id="mp-hamilton-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Hamilton's Principle</h2> | ||
<div id="mp-hamilton" style="padding: 10px;"> | <div id="mp-hamilton" style="padding: 10px;"> | ||
Of all the ways a ball could fly from your hand to a window, it takes one. Of all | |||
* [[Hamilton's principle/ | the paths light could follow from a lamp to your eye, it takes one. Physics has a | ||
* [[Hamilton's principle/ | single answer to why: along the true path, a quantity called the '''action''' is | ||
* [[Hamilton's principle/ | ''stationary''. | ||
The claim of this volume is that this is not one principle among many, but the | |||
deepest thing physics knows. Newton's laws, Maxwell's equations, general | |||
relativity and the Standard Model are each stated as an action, and 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 — [[Hamilton's principle/Mathematics/Noether's first theorem|Noether proved]] they follow from | |||
the action's symmetries. New theories are now built by writing the action a | |||
symmetry permits and letting the dynamics follow, which is how electromagnetism | |||
and gravity are derived here and how the nuclear forces were found. And the | |||
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. | |||
* [[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points|Stationary is not least]] — the long way round a great circle obeys the same rule at every point, and is plainly not the shortest. [[:File:OPL-HP-F-05.png|See the diagram]]. | |||
* [[Hamilton's principle/Applications/The Feynman path integral|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. | |||
* [[Hamilton's principle/Explainers/Comparing paths|Start without the mathematics]] — every page carrying equations has a plain-language counterpart, linked from it and linking back. | |||
'''([[Hamilton's principle|Full article...]])''' | '''([[Hamilton's principle|Full article...]])''' | ||
Revision as of 06:51, 9 September 2026
Introduction
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The Current Program (Round 3)
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Program History
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Hamilton's Principle
Of all the ways a ball could fly from your hand to a window, it takes one. Of all the paths light could follow from a lamp to your eye, it takes one. Physics has a single answer to why: along the true path, a quantity called the action is stationary.
The claim of this volume is that this is not one principle among many, but the deepest thing physics knows. Newton's laws, Maxwell's equations, general relativity and the Standard Model are each stated as an action, and 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 theories are now built by writing the action a symmetry permits and letting the dynamics follow, which is how electromagnetism and gravity are derived here and how the nuclear forces were found. And the 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.
- 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 1st Theorem
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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.
- Start reading →
- Introduction — the story, and why the paper matters
- The explainers — plain language, no mathematics
- The missing term — the one addition that made radio possible
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