Symplectic
Geometry

Editions of this manuscript

A licenciatura thesis on the degeneracy of the symplectic form and on the intrinsic definition of second-order differential equations. The original is handwritten; what follows are four readings of it, from the most complete to the most faithful, and two companions to them: a commentary on the argument, and an outside assessment.

• English edition The complete edition. The manuscript translated and typeset, with an abstract, a plain-language account of the argument, an afterword on gauge theories and general relativity, a formal and an informal glossary, twenty-two diagrams redrawn after the author's own, references, and the seven manuscript pages in facsimile at the end.
• Transcription — Portuguese The text as it was written, in the language it was written in, typeset and with the diagrams redrawn after the originals.
• Transcription with the original diagrams The same Portuguese text, selectable, interleaved with the schemes and diagrams exactly as they appear in the manuscript.
• Facsimile A self-contained HTML facsimile of the original, its pages embedded as images, preserving the handwriting, the symbols and the marginal notes as closely as possible.
• The argument, explained A commentary written in 2026 for a reader who wants more than the narrative gives and less than the mathematics assumes. It follows the paper in nine steps: what a symplectic form actually measures, and why an instrument on a phase space must be handed two directions rather than one; how an observable comes to name a motion, worked through on the free particle; what the Poisson bracket buys, and why the work wants an algebra and not a list; where the construction breaks once non-degeneracy is dropped; what the extra freedom turns out to be, which is not new physics but the same state described twice; why the blindness is organised rather than arbitrary, with a counter-example showing what a jump in rank would destroy; how the double quotient repairs the damage by removing what cannot be seen instead of mending the form; what it takes for a vector field to deserve the name of an equation of motion; and what the two halves of the manuscript have to do with each other. It is a section of the English edition, and opens there.
• An outside reading An assessment of the manuscript written in 2026 by an AI model: what the work gets right, where its terminology is loose, and which of its steps are too compressed. Reproduced with its criticisms intact, and with notes where they have since been acted on.

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