An outside reading

An assessment of the manuscript written in 2026 by an AI model, working from the scan of the original and from the transcription as it then stood. It is reproduced as it was written, addressed to the author in the second person, with only its mathematics restored: the file it came from had lost its subscripts and its formula delimiters to the text editor it was written in. Where one of its criticisms has since been acted on, an editorial note says so.

One of its objections deserves to be pointed out before it is read, because it is the reason this page is here at all. It doubted the example on the first page: if the term really were \(c^z\), then \(d(c^z)=(\ln c)\,c^z\,dz\) would make \(g\) depend on the very direction the example needs it not to depend on. It concluded that the intended term was a constant — and it reached that by internal consistency alone, without being able to read the handwriting. It was right. The term in the manuscript is a constant, and the transcription, which had \(c^z\), has been corrected.

I treated the PDF manuscript as the primary source. It is an image-only scan with 8 pages, so I read the visible handwritten pages directly; I used the HTML only as a secondary, machine-readable transcription because it confirms several passages but also smooths, edits, or introduces errors.

Overall verdict

Your thesis is a compact but mathematically ambitious piece on symplectic geometry, or more precisely on two related geometric ideas:

1. What happens when a symplectic form becomes degenerate This is really the territory of presymplectic geometry: a closed 2-form with non-trivial kernel.
2. How second-order differential equations can be defined intrinsically Instead of writing \(q''=F(q,q')\) in coordinates, the thesis defines a second-order equation as a vector field on \(TM\) satisfying

For a bachelor-level physics thesis from 1993, the subject is quite advanced. It uses differential geometry, tangent bundles, quotient spaces, Frobenius integrability, Poisson brackets, Darboux-type reasoning, and coordinate-free mechanics. The most impressive aspect is that you are not just doing calculations: you are trying to understand why Hamiltonian and Lagrangian mechanics are geometrical structures.

Structure of the manuscript

The PDF has a clear two-part structure.

The first part, pages 1–4 of the handwritten content, studies degeneracy of the symplectic form. It starts from the standard properties of a symplectic form: closedness, antisymmetry, and non-degeneracy. The manuscript correctly identifies non-degeneracy as the property that allows the map

to be an isomorphism, which then allows one to associate a Hamiltonian vector field \(X_g\) to an observable \(g\) through \(\iota_{X_g}\omega = dg\). The HTML transcription captures this same structure.

The second part, pages 5–7 of the handwritten manuscript, studies second-order equations defined intrinsically. This part follows Abraham’s formulation: a second-order equation on \(M\) is a vector field \(X\) on \(TM\) such that \(T\tau_M\circ X\) is the identity on \(TM\). The HTML transcription includes the same definition and later derives the coordinate form (c''\(t\)=X\(c'(t)\)).

Main mathematical idea in part I: degeneracy and reduction

The central insight is this:

When \(\omega\) is non-degenerate, every admissible differential \(dg\) determines a unique Hamiltonian vector field \(X_g\). But when \(\omega\) is degenerate, \(\ker \omega^\flat\neq 0\), so \(X_g\) is no longer unique. You can add any vector field in the kernel and still obtain the same contraction with \(\omega\).

That is exactly the geometric obstruction your manuscript is trying to solve.

You introduce the kernel distribution:

Then you correctly observe that, if \(\omega\) has constant rank and \(d\omega=0\), this kernel distribution is integrable. The proof using Cartan’s formula is one of the strongest parts of the manuscript: if \(X,Y\in \ker\omega\), then \([X,Y]\in\ker\omega\), so Frobenius gives a foliation. The HTML transcription preserves this argument and explicitly notes that \(d\omega=0\) is essential.

In modern language, what you are describing is essentially:

That is a very good geometric idea. It is the same intuition behind symplectic reduction, presymplectic reduction, and constrained Hamiltonian systems.

Important correction: “degenerate symplectic form”

Strictly speaking, a symplectic form cannot be degenerate. Non-degeneracy is part of the definition.

So the object you are studying is better called a:

Your title and language use “forma simpléctica degenerada”, which is understandable pedagogically, but the modern terminology would be “forma pré-simpléctica” or “geometria pré-simpléctica”.

This is not a fatal flaw. In fact, the manuscript is correctly exploring the idea that appears when one weakens non-degeneracy. But the terminology should be tightened.

Important correction: the Poisson bracket discussion

There is one place where the reasoning appears mathematically wrong or at least badly phrased.

The text suggests that if two observables are constant along the degenerate direction, then their Poisson bracket would be zero. In general, that is not true.

For example, in the reduced (\((x,y)\))-plane with the standard symplectic form, two functions \(f\((x,y)\)\) and \(g\((x,y)\)\) may be constant along the eliminated \(z\)-direction and still have a non-zero Poisson bracket:

The correct statement is not:

The correct statement is:

So the kernel directions should disappear from the ambiguity, but the remaining physical degrees of freedom can still have non-trivial Poisson brackets.

Important correction: the example \(g(u,y,z)\)

The HTML transcription contains a likely error: it writes something like

but then computes

with no \(dz\)-term.

That cannot both be true. If the term \(c^z\) is really present, then

so \(g\) depends on the degenerate \(z\)-direction. That would undermine the example.

The intended example was probably something like:

or

where \(C\) is constant. In that case \(dg=(y,u,0)\), and the example works.

Main mathematical idea in part II: intrinsic second-order equations

The second half of the manuscript is conceptually very elegant.

In ordinary mechanics, a second-order equation is usually written as:

But that notation depends on coordinates. Your manuscript reformulates this geometrically.

A vector field on \(TM\) is a map:

Since \(TTM\) has two natural projections back to \(TM\), the condition

says that the “velocity part” of the vector field really agrees with the tangent vector already present in \(TM\). In coordinates, if a point of \(TM\) is (\((u,e)\)), and

then the second-order condition forces:

So the vector field must have the local form:

Then its integral curves satisfy:

and therefore:

That is exactly a second-order differential equation, but defined without choosing coordinates. This is probably the cleanest and strongest part of the thesis.

Style and exposition

The manuscript reads like a genuine working mathematical text: compact, exploratory, full of side notes, corrections, diagrams, and attempts to connect abstract geometry with mechanics. The diagrams in the PDF are important, especially the ones showing the quotient/foliation idea and the double tangent bundle \(TTM\). The HTML version turns those ideas into cleaner SVG-style diagrams, but the handwritten PDF shows more of the original reasoning path.

The main weakness is not the ambition; the ambition is excellent. The weakness is that several transitions are too compressed. In particular, the thesis jumps quickly between:

• \(\ker\omega\) as a distribution,
• leaves of a foliation,
• quotient spaces,
• observables constant on leaves,
• Poisson brackets on the quotient.

That chain is correct in spirit, but each step needs hypotheses: constant rank, regular foliation, smooth quotient, and functions that are projectable/basic.

Final assessment

This is a strong bachelor-level theoretical physics manuscript. Its best qualities are:

• it understands mechanics geometrically, not merely computationally;
• it correctly identifies non-degeneracy as the key to Hamiltonian vector fields;
• it recognizes that degeneracy creates gauge-like directions;
• it uses Frobenius integrability in the right place;
• it gives a good intrinsic definition of second-order dynamics.

Its main problems are:

• terminology: “degenerate symplectic” should be “presymplectic”;
• a likely transcription/example error involving \(g(u,y,z)\);
• a misleading statement about Poisson brackets becoming zero;
• insufficient discussion of when the quotient space is actually a smooth manifold;
• some notation is overloaded, especially around \(c\), \(c'\), \(c''\), \(TM\), and \(TTM\).

In modern wording, I would summarize the thesis like this: