Abstract We examine what survives of symplectic geometry when the non-degeneracy of \(\omega\) is given up, and how far the algebra of observables can still be built. Non-degeneracy is what makes \(\omega^\flat:TM\to T^*M\) an isomorphism, and therefore what assigns to each observable \(g\) a single field \(X_g\) with \(\iota_{X_g}\omega=dg\); once it fails, \(X_g\) is determined only up to the kernel of \(\omega^\flat\) and the Poisson bracket loses its meaning. We show that if \(\omega\) has constant rank the kernel is a subbundle of \(TM\) which, because \(\omega\) is closed, is involutive and hence arises from a regular foliation. Taking the quotient of \(TM\) by the kernel directions, and of \(M\) by the leaves, yields a base which is again a symplectic manifold, and on it the Poisson brackets, the canonical transformations and the remaining structure are recovered. The second part takes up Abraham's intrinsic definition of a second-order equation [1] — a vector field \(X\) on \(TM\) with \(T\tau_M\circ X=\mathrm{Id}\) — exhibiting the rhombic diagram it obeys, the two distinct projections \(\tau_{TM}\) and \(T\tau_M\) of \(TTM\) onto \(TM\), and how the condition on the integral curves reduces, in local coordinates, to the classical second-order equation of motion.

Editorial note: this abstract is an addition of 2026. The seven pages that follow are a transcription of the 1993 manuscript; everything after them — the motivation, the account of how the work was done, the commentary, the glossaries and the references — was written in 2026.

Degeneracy of the Symplectic Form

The features that define the symplectic form on a manifold \(M\) are: closedness, skew-symmetry and non-degeneracy. The first two properties are both fundamental, and they come into play as follows: closedness and skew-symmetry will be essential for Darboux's Theorem [1] and for setting up an algebra of observables through the Poisson brackets. Skew-symmetry gives rise to conservation properties, which extend to degenerate forms. Non-degeneracy can be weakened in such a way that we still establish the good properties in more general cases.

Let us first look briefly at the importance of non-degeneracy.

Bilinear Form — Non-degeneracy \(\omega\) is non-degenerate if and only if: \[\omega(e_1,e_2)=0,\quad\forall\,e_2\in E\;\Longrightarrow\;e_1=0.\]

We define the linear map \(\omega^\flat:E\to E^*\) \((TM\to T^*M)\) by \(\omega^\flat(e)\cdot e'=\omega(e,e')\). Then \(\omega^\flat(e)=\iota_e\omega\) and \([\omega^\flat]=[\omega_{ij}]\). The non-degeneracy of \(\omega\) implies that the kernel of \(\omega^\flat\) is trivial; hence \(\omega^\flat\) is injective and, since \(\dim E=\dim E^*\), it is an isomorphism. A vector field \(X_g\) is therefore determined by the condition: for a given \(g:E\to\mathbb{R}\),

\omega^\flat(X_g)=\iota_{X_g}\omega=dg,\qquad\text{that is:}\quad [X_g]=\bigl[\omega_{ij}\bigr]^{-1\,t}[dg].

If the observable \(g\) does not depend explicitly on some variable, \(g=g(q^1,\dots,q^{2n})\), we shall have \(dg=\bigl(\tfrac{\partial g}{\partial q_i},0\bigr)^\top\) and then \(\omega\) may be degenerate,

\omega=\left[\begin{array}{c|c}\omega_{ij}&\\\hline&0\end{array}\right],

and then \(X_g\) will have 1 degree of freedom.

For example: \(g(u,y,z)=uy+c\), \(dg=y\,du+x\,dy=(y,u,0)\). We know, from Darboux's Theorem, that we can find a basis in which \(\omega\) takes the form:

\[\omega=\begin{bmatrix}0&1&0\\-1&0&0\\0&0&0\end{bmatrix}.\qquad\text{Hence}\quad X_g=(u,-y,0)\] there is no isomorphism because we have 1 degree of freedom

Check: \([x\;\;{-y}\;\;0]\begin{bmatrix}0&1&0\\-1&0&0\\0&0&0\end{bmatrix}=[y\;\;x\;\;0]\).

In order to set up an algebra of observables we have to remove this excess of possible fields. In our example the kernel of \(\omega\) consists of all vectors of the form \(\bigl(0,0,\lambda(u,y,z)\bigr)\), that is \(X(u,y,z)\tfrac{\partial}{\partial z}\). Applied to the observable \(g\), these vectors determine surfaces — lines, in this case — along which \(g\) is constant:

X\cdot g=X(u,y,z)\,\tfrac{\partial}{\partial z}\,g(u,y)=0.

So if we had another observable, also constant along \(z\), we would get

\[\{f,g\}=\mathcal{L}_{X_g}f=-\mathcal{L}_{X_f}g=0,\] were the Poisson brackets well defined

So, in order to define the brackets, we shall consider only the vectors perpendicular to these lines, since \(f\) and \(g\) depend on \(x\) and \(y\) alone.

\(X_g\cdot f=-X_f\cdot g=0\), but the lines are perpendicular to the \(x\)–\(y\) plane; for each \(dg\) there is no unique \(X_g\).

The leaves of the kernel \(\ker\omega^\flat=\langle\partial/\partial z\rangle\) are the vertical lines; over each level curve of \(g\) stands a vertical level surface, since \(g\) does not depend on \(z\)

Differential Geometry

\omega(x,y)=\omega^\flat(x)\cdot y=\iota_x\omega\cdot y;\qquad\ker\omega^\flat=\{X\mid\iota_X\omega\equiv 0\}.

\(\omega^\flat\) is a smooth vector bundle mapping and determines a submanifold of \(TM\). By exercise 1.6F c) (p. 51) in [1], if \(\omega\) has constant rank \(k\) — a necessary condition for defining the new algebra, as is easily seen from our linear example, since otherwise the number of "degrees of freedom" varies from point to point — then \(\ker\omega\) is a subbundle of \(TM\) (call it \(E\)); a subbundle is one where the base is the same but the fibres are smaller. Then, by Frobenius' Theorem [5], \(E\) is integrable if and only if it arises from a regular foliation.

Its image lies in \((T^*M)^E\), a manifold isomorphic to \(TM|_E\); \(\iota_X\omega\) is a subbundle of \(T^*M\).

Proof that \(E\) is integrable: let \(X,Y\in\ker\omega\). We check that \([X,Y]\in\ker\omega\):

\iota_{[X,Y]}\omega=\mathcal{L}_X\iota_Y\omega-\iota_Y\mathcal{L}_X\omega=\mathcal{L}_X\iota_Y\omega-\iota_Yd\iota_X\omega-\iota_Y\iota_X\,d\omega=0.

Note: We need \(d\omega=0\), which is a fundamental condition for setting up the algebra.

So we now know that \(E\) arises from a regular foliation, that is: "for each \(m_0\in M\) there is a local submanifold \(N\subset M\), called a leaf of the foliation, which contains \(m_0\) and whose Tangent bundle is exactly \(E\) restricted to \(N\)."

Definition — Foliation (after Abraham [1]) A foliation \(\mathcal{F}\) of class \(C^n\), \(n\geq 0\), and dimension \(p\) on an \(m\)-dimensional manifold \(M\) is a decomposition of \(M\) into disjoint connected subsets \(\mathcal{F}=\{L_\alpha\}_{\alpha\in A}\), called the leaves, such that every point of \(M\) has a neighbourhood \(U\) and a \(C^n\) coordinate system \((x,y):U\to\mathbb{R}^p\times\mathbb{R}^{m-p}\) for which, for each leaf \(L_\alpha\), the components of \(U\cap L_\alpha\) are described by the equations \(y_1=c_1,\;\dots,\;y_{m-p}=c_2\).

The chart \(\varphi=(x,y):U\to\mathbb{R}^p\times\mathbb{R}^{m-p}\) flattens the leaves: each \(L_\alpha\) becomes a level set \(y_1=c_1,\,\dots,\,y_{m-p}=c_2\)

We can pin down the leaves through the observables that stay constant on them: \(f=c_1\), \(g=c_2\), etc.

\[\varphi:U\to U'\times\mathbb{R}^{n-k},\qquad u\longmapsto\bigl(\varphi_1(u),f(u)\bigr)\] \(\bigl(\varphi_1(u),f(u),g(u),\dots\bigr)\), with \(c\in\mathbb{R}^{n-k}\)

With \(F=\bigl(f(u),\dots,h(u)\bigr)\): \(c\) is a regular value of \(F\) if \(df(u)\wedge\dots\wedge dh(u)\neq 0\) for \(u\in F^{-1}(c)\) — that is, the matrix \(DF\) is non-singular.

We can now see that, if \(df(u)\neq 0\) with \(u\in f^{-1}(c)\), then \(c\) is a regular value of \(f\) and \(f^{-1}(c)\) is a submanifold of \(M\) of codimension \(k\) — the dimension of the leaves is \(n-k\).

\(df(u)\neq 0\) is essential so that, applied to \(X\),

df(u)\cdot X=X\cdot f(u)=0

is not trivial and therefore determines the vectors of the kernel of \(\omega^\flat\), which are tangent to the leaves where \(f\) and \(g\) are constant.

\[\begin{cases}X\cdot f=0\\Y\cdot f=0\end{cases}\Longrightarrow\;[X,Y]\cdot f=0,\] since \([X,Y]\) is still a vector field tangent to the leaf.

So everything "fits together" rather well.

In order to establish intrinsically the "perpendicularity" glimpsed in the linear example we have no structure to fall back on —

only \(\omega\), since we have no metric. We can remove the unwanted fields by setting up an equivalence relation on \(TM\) and taking the quotient of \(TM\) by that relation. Two elements of \(TM\) are equivalent if

\sim\;:\quad \iota_{(X_1-X_2)}\,\omega(m)=0,

that is, we remove every component along the direction of the leaves:

The difference of the two vectors lies along the leaf

We also remove the leaves themselves. For that, the equivalence relation is defined by the constancy of the observable \(f\) on a leaf: all points \(u\in f^{-1}(c)\), where \(c\) is a regular value of \(f\), are equivalent.

The new Tangent bundle will be \(TM/N\). Its base is a symplectic manifold. On this manifold we can now set up an algebra of observables with Poisson brackets, canonical transformations and all the good properties.

Note: \(TM/N\) is obtained by the canonical projection \(M\to M/\mathcal{F}\), and the two reductions mentioned above are obtained simultaneously.

✦ ✦ ✦

Second-Order Equation Defined Intrinsically

Definition — Abraham [1] A second-order equation on a manifold \(M\) is a vector field \(X\) on \(TM\) such that \(T\tau_M\circ X\) is the identity on \(TM\).

Remark. The condition above was not new in 1993: it is the non-homogeneous half of the definition of a spray, in circulation since [9] and given its textbook treatment in [8], and it reaches the present work through Abraham [1]. What is the author's own is the road to it. The spray literature arrives at the condition through the homogeneity of the geodesic flow, and the Lagrangian school through the canonical vertical endomorphism of \(TTM\) [10, 11]; the pages that follow arrive at it by noticing that \(TTM\) carries two distinct projections onto \(TM\), \(\tau_{TM}\) and \(T\tau_M\), and by demanding that they agree on the image of \(X\) — which is exactly what the rhombic diagram, and the coincidence of the curves ① and ②, say. The author reached that formulation independently, and it is that road, and not a reading of the published account, that these pages record.

Before going on it is worth defining the operations involved in the definition of a second-order equation: \(\tau_M:TM\to M\) is the projection of the Tangent bundle onto its base, which is a differentiable manifold; \(T\tau_M:TTM\to TM\) is the tangent map between the vector bundles \(TTM\) and \(TM\). A tangent map is a particular case of a vector bundle mapping, that is, a \(C^\infty\) fiber-preserving map — the diagram of the projections commutes — which is linear on each fibre:

f_*X(n)=\bigl(Tf\circ X\circ f^{-1}\bigr)(n)

A tangent map \(Tf\) is a vector bundle mapping: it preserves the fibres and is linear on each of them

The initial definition means that "if \(X\) is a second-order equation on \(M\) we have the following commutative rhombic diagram (except in the cycles involving only 2 arrows)":

Rhombic diagram — it commutes except in the cycles involving only two arrows. \(X\) is a second-order equation \(\iff T\tau_M\circ X=\mathrm{Id}_{TM}\)

The same diagram in local coordinates

\begin{aligned}T(\tau_{M\varphi})(u',e,e_1,e_2)&=\Bigl((\tau_{M\varphi})(u',e),\;D(\tau_{M\varphi})(u',e)\cdot(e_1,e_2)\Bigr)\\[.3em]&=\left(u',\begin{bmatrix}1&0\\0&0\end{bmatrix}\begin{bmatrix}e_1\\e_2\end{bmatrix}\right)=(u',e_1)\end{aligned}

The canonical projection \(\tau_{TM}\) "flattens" the fibres onto the base \(TM\), whereas the projection \(T\tau_M\) projects only the "horizontal" part of the fibre and the base \(M\).

The two projections seen on the manifolds: \(\tau_{TM}\) flattens the whole fibre and returns \(c\); \(T\tau_M\) keeps only the "horizontal" part and returns \((\tau_M\circ c)'\)

The same scheme level by level, with \(c\) an integral curve of \(X\) and \(c'=X\circ c\): \(X\) is second order if and only if ① \(=\) ②, that is \(c=(\tau_M\circ c)'\)

\(X\) is a second-order equation on \(M\) if and only if the curves ① and ② coincide. Then

(\tau_M\circ c)'(t)=c(t),\qquad (\tau_M\circ c)''(t)=X\circ(\tau_M\circ c)'(t)=(X\circ c)(t)=c'(t),

since the curve \(c(t)=(\tau_M\circ c)'(t)\) is an integral curve of the vector field \(X\).

We make a change of notation \((\tau_M\circ c)\to c\), and for the base integral curve of \(X\), \(c(t)\), we get:

c''(t)=X\bigl(c'(t)\bigr).

We could not have established this equality had we not observed that \(X\) determines the curve \(c'(t)\) — previously \((\tau_M\circ c)'=c\).

Locally

\begin{aligned} (c'')_\varphi:\quad & T^2\varphi\circ c''(t)=T\bigl(T\varphi\circ c'\bigr)(t,1)=(c'_\varphi)'(t)\\[.35em] X(c'(t))\big|_\varphi:\quad & T^2\varphi\circ X\circ T\varphi^{-1}\circ T\varphi\circ c'(t)=X_\varphi\circ c'_\varphi(t) \end{aligned}

hence \((c'_\varphi)'=X_\varphi\bigl(c'_\varphi(t)\bigr)\), that is

\begin{aligned}&\Bigl(u(c(t)),\;e(c'(t)),\;\tfrac{d}{dt}u(c(t)),\;\tfrac{d}{dt}e(c'(t))\Bigr)\\[.3em]={}&\Bigl(u(c(t)),\;e(c'(t)),\;X_1\bigl(u(c(t)),e(c'(t))\bigr),\;X_2\bigl(u(c(t)),e(c'(t))\bigr)\Bigr)\end{aligned}

since \(u(c'(t))=u(c(t))\) by \((\tau_M)_\varphi:(u,e)\mapsto u\) — on both sides of the rhombic diagram.

From what has been said it follows easily that:

(*)\quad\begin{cases}\bigl(T\tau_M\circ X\bigr)(u,e)=T\tau_M\bigl(u,e,X_1(u,e),X_2(u,e)\bigr)=\bigl(u,X_1(u,e)\bigr)\\[.5em]\bigl(\tau_{TM}\circ X\bigr)(u,e)=\tau_{TM}\bigl(u,e,X_1(u,e),X_2(u,e)\bigr)=(u,e)\end{cases}

hence \(X_1(u,e)=e\), so that the identity in the rhombic diagram holds. Then, finally:

\[\begin{cases}\dfrac{d}{dt}\,u(c(t))=X_1\bigl(u(c(t)),e(c'(t))\bigr)=e(c'(t))\\[1em]\dfrac{d}{dt}\,e(c'(t))=X_2\bigl(u(c(t)),e(c'(t))\bigr)\end{cases}\] \[\Longleftrightarrow\qquad\begin{cases}\dfrac{d}{dt}\,u(c(t))=e(c'(t))\\[1em]\dfrac{d^2}{dt^2}\,u(c(t))=X_2\bigl(u(c(t)),e(c'(t))\bigr)\end{cases}\] \(c'(t)=\bigl(c(t),\,\dot{c}(t)\bigr)\)

And we have a second-order equation defined intrinsically.

(*) This means that, when one makes the "passages" — tangent and projection — between \(M\leftrightarrow TM\leftrightarrow TTM\), there is a single "route", by the tangent and by the projection, between \(c\) and \(c''\), such that \[\begin{aligned}c''_\varphi=X_\varphi\bigl(c'_\varphi(t)\bigr)\;&\iff\;(c'_\varphi)'=X_\varphi\bigl(c'_\varphi(t)\bigr)'\\[.2em]&\overset{?}{\iff}\;(c_\varphi)''=X_\varphi\bigl(c'_\varphi(t)\bigr)\;\iff\;(c_\varphi)''=X_\varphi\bigl((c_\varphi)'(t)\bigr)\end{aligned}\] since \(c''_\varphi=T^2\varphi\circ c''(t)=T\bigl(T\varphi\circ c'\bigr)(t,1)\overset{?}{=}T^2(\varphi\circ c)(t,1)=(c_\varphi)''\). That is: \[\begin{aligned}&\Bigl(u(c(t)),\;\tfrac{du}{dt}(c(t)),\;\tfrac{d}{dt}u(c(t)),\;\tfrac{d^2}{dt^2}u(c(t))\Bigr)\\[.3em]={}&\Bigl(u(c(t)),\;e(c(t)),\;X_1\bigl(u(c(t)),e(c(t))\bigr),\;X_2\bigl(u(c(t)),e(c(t))\bigr)\Bigr)\end{aligned}\]

Note: What matters here is the second-order equation of motion, which we obtained in local coordinates starting from our intrinsic definition.

· · ·

Luís Ferreira

IST n.º 31727 · 2 / 2 / 93

Motivation

A mechanical system is described by its observables — position, energy, momentum — but a list of measurable quantities is not yet physics. What turns it into physics is a bracket: a rule that takes two observables and returns a third. Given the energy \(H\), the bracket returns the rate of change of everything else, \(df/dt=\{f,H\}\), and the observables it sends to zero are the conserved ones. The same bracket decides which changes of description are legitimate — the canonical transformations are exactly those that preserve it — and it is the structure that canonical quantization carries over into the commutator [1, 2, 3]. Lose the bracket and one is left with a stage and no play.

The bracket is built from the symplectic form: \(\{f,g\}=\omega(X_f,X_g)\), where the Hamiltonian field \(X_g\) is obtained by inverting \(\omega^\flat\) — and that inversion is precisely what non-degeneracy grants. When \(\omega\) degenerates, \(X_g\) is determined only up to the kernel of \(\omega^\flat\), and altering \(X_g\) by a kernel vector \(k\) alters \(\{f,g\}\) by \(df(k)\). The bracket is therefore well defined only on those observables whose differentials annihilate the kernel: the ones that are constant along the invisible directions.

Degenerate forms are no curiosity. They arise whenever the description carries more variables than the system has degrees of freedom — degenerate Lagrangians, whose Legendre transform cannot be inverted, and systems with constraints [3, 4]. There the kernel directions are the gauge directions, displacements that change the description and not the state, so removing them is not a technical repair but the physically correct move. That is what the first part does: constant rank makes the kernel a subbundle, closedness makes it involutive, Frobenius turns it into a foliation [5], and a double quotient leaves a symplectic base on which the algebra of observables works again. The second part then defines the equation of motion itself intrinsically, as a vector field on \(TM\) [1], and recovers from it the classical second-order equation in local coordinates.

Editorial note: the manuscript ends on the previous page. From here to the end, everything was written in 2026.

The route I took

I did not set out to weaken anything. I set out to write down the algebra of observables, and found that I could not.

The construction is short enough to state in a line: an observable ought to determine a motion, and it does so through \(\omega\) — one solves \(\iota_X\omega=dg\) for \(X\).

Read that equation as a question, and notice that the two directions \(\omega\) takes in are not doing the same job. The second one I choose, and I may choose it every possible way: it is a trial step, a what if I moved this way. The first is the unknown — the single motion I am after, fixed once and for all. So the equation says: find the motion whose pairing with any trial step whatsoever records how fast \(g\) changes along that step. What is peculiar about this, and what I did not appreciate at the time, is that I never get to look at that motion directly. I know it only by what it answers to every question I am able to put to it. Everything in Hamiltonian mechanics leans on that single inversion, and the inversion works because \(\omega\) is non-degenerate.

I wanted to see how much of the structure survived if that assumption were weakened, so I wrote down the smallest example I could think of in which it fails: three variables, an observable that ignores one of them, and the matrix of \(\omega\) with a row and a column of zeros. The answer came at once. The field is no longer unique; I could add to it any multiple of one particular direction and nothing in the equation would notice.

That could have been the end of it — a formalism that breaks under a weaker hypothesis. What made me go on was noticing what the extra freedom actually was. The directions I was free to add were exactly the directions along which the observables I cared about stayed constant. They were not new physics. They were the same situation written twice.

So the question changed. If at each point there is a whole family of directions the form cannot see, do those families fit together into surfaces, or do they twist as one moves along? I checked it the only way I knew: take two fields lying in the invisible directions and ask whether their bracket is invisible too. It is — and the computation shows why. The one term that could have spoiled it is the term carrying \(d\omega\), and \(\omega\) is closed. That was the point at which the whole thing turned from an obstruction into a construction: closure, which I had been treating as a hypothesis inherited from the symplectic case, was doing the essential work. Frobenius then supplies the surfaces.

After that the argument writes itself. Sliding along one of those surfaces changes nothing measurable, so the points of a surface should not be counted as different. Collapse each one to a point. What remains has no invisible directions — not because I repaired the form, but because I removed what it could not see — and the bracket I could not build upstairs is available downstairs, on precisely the quantities that were constant along the surfaces to begin with.

The first half in five steps: what the form does, what goes wrong, what the wrongness means, why it is organised, and what to do about it

The second half

The second half began from a different irritation. Acceleration is the rate of change of velocity, and on a manifold I could not differentiate twice without coordinates creeping back in through the door I had just closed. The way out is to give velocities a space of their own: a point of \(TM\) is a position together with a velocity, a motion is a curve drawn in \(TM\), and an equation of motion is an arrow at every such pair, saying where to go next.

But most such fields are not equations of motion, and it took me a while to say why. A curve in \(TM\) moves two things at once, the position and the velocity, and nothing whatever forces the second to be the derivative of the first. A curve whose velocity slot reads five while its position stands still is a perfectly good curve and a nonsensical motion.

Looking for what does force it, I noticed that \(TTM\) comes back down to \(TM\) in two different ways. One of them remembers the pair the arrow is attached to. The other pushes the arrow down to \(M\) and reads it there as a velocity. I had been treating these as the same map, and they are not — they keep different halves of the same object. Once that was clear the definition was unavoidable: a motion is a field on which the two readings agree. The rhombic diagram is nothing but the bookkeeping for that statement, and the two curves I marked ① and ② are the two readings applied to one and the same integral curve. In coordinates the condition becomes \(\ddot u = X_2(u,\dot u)\), which is where I had started.

Abraham states the same definition, and states it first. What these pages record is the road by which I arrived at it, which was not his: the two projections and the diagram came before the name did.

The second half in three: give velocities a space, notice that most curves in it are not motions, and demand that the two readings agree

Editorial note: written in 2026, recalling work done in 1993.

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 order, and it may be read instead of the seven pages above or alongside them.

1. What the form measures

Begin with units, which settle the matter faster than any philosophy. A point of a phase space is a position together with a momentum. Ask for the length of a direction there and you are asking to add metres to kilogram-metres per second: the question has no answer. But take two directions, one carrying a little position and the other a little momentum, and multiply them, and the result is metres × kg·m/s — action. That combination is dimensionally sound, and it is essentially the only one that is. An instrument on a phase space therefore cannot sensibly be handed a single direction. The smallest thing it can be handed is two, one to pair against the other.

What it returns is the area of the parallelogram those two directions span, signed, so that swapping them reverses it. In coordinates \(\omega=\sum_i dq^i\wedge dp_i\) says exactly this: take the parallelogram, cast its shadow on each \((q^i,p_i)\) plane, measure the area of each shadow, add them up. And the property that looks like an odd axiom stops looking odd. Hand the instrument the same direction twice and the parallelogram is flat, so the area is zero. Antisymmetry is not a stipulation; it is what area means.

So the quantity being measured is action, position against momentum. Symplectic geometry is the geometry of action in the way that Riemannian geometry is the geometry of length, and the difference between the two subjects is already visible here, in what their instruments will and will not accept.

So far that is a statement about units, and it would be a thin thing if it stopped there. It does not. On a cotangent bundle there is a 1-form nobody has to choose, \(\theta=\sum_i p_i\,dq^i\), and \(\omega=-d\theta\). Stokes' theorem then says

\iint_\Sigma\omega=-\oint_{\partial\Sigma}p\,dq,

and the right-hand side is the action accumulated along the boundary. The area the instrument measures inside a patch of phase space is the action of a walk once round its rim. The smallest case makes it concrete: a rectangle of sides \(\Delta q\) and \(\Delta p\) has \(\omega\)-area \(\Delta q\,\Delta p\), and going round its boundary the vertical edges contribute nothing, since \(dq=0\) along them, while the lower edge gives \(p_0\,\Delta q\) and the upper one, traversed backwards, \(-(p_0+\Delta p)\Delta q\) — in sum \(-\Delta p\,\Delta q\). The same number, up to the sign the orientation fixes.

The slogan therefore has a consequence that can be checked against physics. The old quantization condition is \(\oint p\,dq=nh\), and \(h\) carries the dimensions of action: what it discretises is exactly the area \(\omega\) measures, so that a quantum state occupies a cell of area \(h\) in phase space. Planck's constant is a quantum of symplectic area. Riemannian geometry measures length and ordinary quantum mechanics has no quantum of length; symplectic geometry measures action, and there is one.

One qualification. \(\theta\) exists globally only where \(\omega\) is exact, which on a cotangent bundle it always is — the case of mechanics. On a compact symplectic manifold it never is, and there the action of a surface is defined only up to the periods of \(\omega\). Those periods are what the integrality condition of geometric quantization is about.

The contrast with Riemannian geometry has a further consequence worth stating early, because the wrong picture of it is easy to carry through everything that follows. A metric has a local invariant: measure lengths and angles inside a small patch and you can tell a sphere from a plane without ever leaving the patch. That invariant is curvature. A symplectic form has nothing of the kind. Darboux's theorem says that around every point there are coordinates in which \(\omega\) takes one fixed constant shape, so any two symplectic manifolds of the same dimension are indistinguishable close up. There is no symplectic curvature — not a curvature that happens to vanish, but no such quantity to be defined. Phase spaces are manifolds on account of their shape in the large, not their bending: the angle of a pendulum returns to where it started, so its phase space is a cylinder and not a plane. Where curvature does enter mechanics it enters from elsewhere — the kinetic energy is a metric on the space of configurations, and it is that metric, downstairs, whose geodesics appear in the second half of this work under the name of a spray.

The phase space of the pendulum. On the left, the plane of the angle \(\theta\) and the momentum \(p\), with the level sets of \(H=p^2/2-\cos\theta\): closed orbits about the centre, where the pendulum swings; open ones above and below, where it goes over the top; and between them the separatrix, through the saddle at \(\theta=\pm\pi\) where the pendulum stands inverted. The field runs along these curves and never across them. On the right, the same curves once \(\theta=-\pi\) and \(\theta=\pi\) are recognised as one state: the plane rolls into the cylinder \(T^*S^1\), on which the swinging orbits are loops that bound and the turning ones are loops that go round. Every patch of it is flat; the whole is not a plane.

2. How a quantity becomes a motion

An observable is a function on the space of states: energy, a coordinate, a component of momentum. Mechanics needs each observable to generate a motion — the energy generates time evolution, a momentum generates translation — and the equation that does this is the one the manuscript writes on its first page,

\iota_X\omega=dg.

Read it as a question rather than a formula. On the right, \(dg\) is the steepness meter of the observable: hand it a direction and it reports how fast \(g\) changes if you move that way. On the left, \(X\) has been slotted into the first place of the instrument and the second left open, so what remains also eats one direction and returns a number. The equation asks for the direction \(X\) whose pairing with anything at all reproduces the rate of change of \(g\) in that direction.

This is why a two-slot instrument is needed and not a one-slot one, quite apart from the units. To pass from \(dg\), which eats a direction, to \(X\), which is a direction, requires a dictionary between the two kinds of object — and a dictionary is precisely a machine with two slots, since filling one leaves something of the other kind. A metric is such a dictionary too. The only difference is that a metric is symmetric, and that single difference decides everything that follows.

With a metric, the same recipe produces the gradient, which climbs straight across the level sets of \(g\). With \(\omega\), the answer is turned through a quarter turn and runs along them. The proof is one line: \(dg(X)=\omega(X,X)=0\), because the instrument returns zero on a repeated direction. So the motion an observable generates leaves that observable unchanged. Energy is conserved because \(\omega\) is antisymmetric — not as a consequence of some further principle, but as a restatement of what the instrument is.

Because what the instrument returns is an area, the equation can be read with no calculus at all. Take a step in some direction; the observable changes by exactly the area of the parallelogram that step spans together with the motion. The two-slot instrument is not comparing a motion against a direction — it is measuring the figure the two of them make.

Worked example — the free particle Take \(\mathbb{R}^2\) with coordinates \(q,p\), the form \(\omega=dq\wedge dp\) and the observable \(H=p^2/2\). Writing \(X=A\,\partial_q+B\,\partial_p\) gives \(\iota_X\omega=A\,dp-B\,dq\), while \(dH=p\,dp\); so \(A=p\) and \(B=0\), and the motion is \(X=p\,\partial_q\), that is \(\dot q=p,\ \dot p=0\) — the free particle.

Now put the equation to its two obvious questions. Step in the momentum direction, \(Y=\partial_p\): the parallelogram spanned by \(X=(p,0)\) and \(Y=(0,1)\) is a rectangle of area \(p\), and indeed \(dH(Y)=p\). Push the momentum by one unit and the energy rises by the area whose base is the speed of the motion. Step in the position direction, \(Y=\partial_q\): now the trial step lies along \(X\) itself, the parallelogram collapses to a segment and encloses nothing — and \(dH(\partial_q)=0\), since the energy of a free particle does not depend on where it is. The one-line proof that an observable is constant along its own motion is that picture: two directions laid on top of one another enclose no area.

In canonical coordinates, unwinding the contraction turns the one line into \(\dot q^i=\partial g/\partial p_i\) and \(\dot p_i=-\partial g/\partial q^i\): Hamilton's equations, when \(g\) is the energy. The coordinate-free equation is not an elegant repackaging of the coordinate ones. It is the statement of which the coordinate ones are a shadow.

3. The bracket, and why the paper wants an algebra

Once every observable names a motion, two observables can be paired: let one generate a motion and ask how fast the other changes along it. That is the Poisson bracket, \(\{f,g\}=\omega(X_f,X_g)\), and with it the observables stop being a mere list of measurable quantities and become an algebraic structure in their own right.

Four things live in that structure, and none of them survives without it. The evolution of any observable is \(df/dt=\{f,H\}\), so the bracket is not a description of the motion but the motion itself. An observable is conserved exactly when its bracket with the energy vanishes, which is where conservation laws come from. The changes of description that deserve to be called legitimate are the ones that preserve the bracket, and they are called canonical for that reason. And the bracket is the structure that canonical quantisation carries over into the commutator, which is why a classical theory without a well-defined bracket has nowhere to be quantised to.

One property of the bracket is worth isolating, because it will be the hinge of the whole first half. The Jacobi identity — the statement that makes the observables a Lie algebra rather than merely an antisymmetric product — holds precisely because \(d\omega=0\). Closure of the form is not a technical hypothesis inherited from the definition. It is the reason the algebra closes.

4. Where it breaks

Everything above rests on being able to solve \(\iota_X\omega=dg\) for \(X\), and that inversion is possible because \(\omega\) is non-degenerate: no direction is invisible to it. The paper's first move is to ask what survives when that assumption is dropped, and its answer is a deliberately small example — three variables, an observable that ignores one of them, and a form whose matrix has a row and a column of zeros.

The failure is immediate and it is not subtle. The field is no longer unique. One may add to it any multiple of the direction the form cannot see and the equation is satisfied just as well, because the addition contributes nothing when paired with anything. What the observable names is no longer a motion but a whole family of motions, and the bracket, which was built by following one of them, gives a different answer depending on which member is chosen. The margin of that page puts it more bluntly than the text: for each \(dg\) there is no unique \(X_g\).

A degenerate form is not a pathology invented to make the problem interesting. It is what one is handed by any description with more variables than the system has degrees of freedom — by a Lagrangian whose Legendre transform cannot be inverted, or by a system with constraints. The afterword takes that up; here it is enough to say that the case being studied is the ordinary one, not the exotic one.

5. What the extra freedom actually is

The turn in the argument comes from looking at the ambiguity rather than lamenting it. The directions that may be added freely are exactly the directions along which the observables in question do not change. Slide along one of them and every measurement returns what it returned before. Two points joined by such a direction are not two states; they are one state, written twice.

That observation converts an obstruction into a programme. If the ambiguity lies entirely in directions along which nothing measurable varies, then it is the description that is redundant, not the physics that is undetermined. And the bracket, which was ill-defined in general, is perfectly well defined on the observables whose differentials annihilate those directions: altering \(X_g\) by an invisible vector \(k\) alters \(\{f,g\}\) by \(df(k)\), which vanishes for exactly such observables. The class of quantities for which mechanics still works is therefore identified before any construction is carried out.

6. Why the blindness is organised

What remains is to show that the redundancy has a shape. Two hypotheses do the work, and the paper is careful about both.

The first is constant rank: the number of invisible directions must be the same at every point. The manuscript justifies this in the language of the example — otherwise the number of degrees of freedom would vary from place to place — and it is what makes the kernel a subbundle rather than a collection of unrelated subspaces.

Counter-example — a rank that jumps The hypothesis is easy to read as a technical convenience. It is not. On \(\mathbb{R}^4\) with coordinates \((x,y,z,w)\) take

\omega=dx\wedge dy+z\,dz\wedge dw.

It is closed — \(d\omega=dz\wedge dz\wedge dw=0\) — so it is a presymplectic form of exactly the kind the work studies. Its matrix is block diagonal with determinant \(z^2\): wherever \(z\neq0\) the form is non-degenerate and the kernel is zero, and on the wall \(z=0\) the rank falls to \(2\) and the kernel jumps to the plane spanned by \(\partial_z\) and \(\partial_w\). The invisible directions appear out of nowhere along that wall and are gone on either side of it.

Nothing that follows could then even be stated. The union of the kernels is not a subbundle, so it is not a distribution and Frobenius has nothing to apply to; with no leaves there is no leaf space and no base; and \(TM/N\) is not a bundle, since dividing each fibre by a subspace whose dimension changes from point to point does not leave fibres of a common dimension. Constant rank is not making the argument tidier. It is what gives the argument objects to be about.

The second is the one that matters more. Given a field of invisible directions, do they fit together into surfaces, or do they twist as one moves along? The test is to take two fields lying in the kernel and ask whether their bracket lies there too, and the computation is the one on page 2:

\iota_{[X,Y]}\omega=\mathcal{L}_X\iota_Y\omega-\iota_Yd\iota_X\omega-\iota_Y\iota_X\,d\omega=0.

Each of the first two terms vanishes because \(X\) and \(Y\) are invisible. The third would not vanish for any old form, and it is the one that carries \(d\omega\). Closure kills it. Frobenius' theorem then converts the algebraic statement into a geometric one: the invisible directions are tangent to a family of surfaces which fill the space, a foliation, and each surface is a leaf.

It is worth pausing on what has just happened, because the paper does not pause. Closure entered the definition of a symplectic form as one of three conditions, alongside antisymmetry and non-degeneracy. Non-degeneracy has been dropped. Antisymmetry gave conservation. And closure, which might have looked like the least consequential of the three, turns out to be the one holding the repair together: without it the redundancy would not organise itself, and there would be nothing to quotient by.

7. The double quotient

Now the construction. Two elements of \(TM\) are declared equivalent when their difference is invisible, \(\iota_{(X_1-X_2)}\omega=0\), which removes the components along the leaves. And the leaves themselves are removed by declaring equivalent the points on which the admissible observables agree — all the points of \(f^{-1}(c)\), for \(c\) a regular value. What results is written \(TM/N\), and its base is again a symplectic manifold: on it nothing is invisible, and the bracket, the canonical transformations and the rest of the apparatus are available once more.

The move deserves to be stated in the way the narrative states it, because it is easy to misread as a repair of the form. Nothing was repaired. What the form could not see was removed, and what remains is a space on which there is nothing it cannot see. The observables that survive the passage are precisely the ones that were constant along the leaves to begin with — which is to say, the ones for which the bracket already made sense.

Two honest limits. The construction is local: the space of leaves carries a topology always, but it is a manifold only under further hypotheses, and the paper works where Frobenius works. And constant rank is a hypothesis, not a theorem; the afterword shows a case of real importance in which it fails.

8. What an equation of motion is

The second half of the paper begins from an irritation of a different kind. Acceleration is the rate of change of velocity, and on a manifold one cannot differentiate twice without coordinates creeping back in. The remedy is to give velocities a space of their own. A point of \(TM\) is a position together with a velocity; a motion becomes a curve drawn in \(TM\); and an equation of motion becomes a vector field there, an arrow at every such pair saying where to go next.

But most fields on \(TM\) are not equations of motion, and saying why takes some care. A curve in \(TM\) moves two things at once, the position slot and the velocity slot, and nothing whatever forces the second to be the derivative of the first. A curve whose velocity slot reads five while its position stands still is a perfectly good curve and a nonsensical motion. The condition that excludes it is the content of Abraham's definition, and the geometry that expresses it is the observation the manuscript is built on.

The observation is that \(TTM\) comes back down to \(TM\) in two different ways. One of them, \(\tau_{TM}\), remembers the pair the arrow is attached to and forgets the arrow: it flattens the whole fibre onto its base point. The other, \(T\tau_M\), pushes the arrow down to \(M\) and reads it there as a velocity, keeping only what the manuscript calls the horizontal part. These are different maps, and treating them as one is the mistake that makes the second-order condition look mysterious. Demanding that they agree on the image of \(X\),

T\tau_M\circ X=\mathrm{Id}_{TM},

is exactly the demand that the velocity slot hold the derivative of the position slot. The rhombic diagram on page 5 is the bookkeeping for that sentence, and the two curves marked ① and ② on page 6 are the two readings applied to one and the same integral curve: the field is a second-order equation when, and only when, they coincide.

The last page pays this off in coordinates. The system marked \((*)\) computes both projections in local terms and forces \(X_1(u,e)=e\) — the first slot of the field must be the velocity itself, which is the coordinate form of the whole condition. What is left is

\frac{d^2}{dt^2}u(c(t))=X_2\bigl(u(c(t)),e(c'(t))\bigr),

Newton's second law, obtained without ever having chosen coordinates and only then written in them.

9. What the two halves have to do with each other

They are not two papers bound together, though the manuscript does not argue the connection. Both halves are about doing mechanics without coordinates and finding out what the geometry will and will not supply. The first finds that the geometry supplies less than expected when the form degenerates, and repairs the loss by removing redundancy. The second finds that the geometry supplies more than expected — that the distinction between a curve in \(TM\) and a motion is already written into the two projections of \(TTM\), waiting to be noticed.

And the two meet in the place the afterword describes. A degenerate Lagrangian is what produces a degenerate form; the Legendre transform, which carries velocities to momenta, is the bridge between the space where the second half works and the space where the first half does. A reader who wants to see where all of it leads should turn there.

Editorial note: this commentary is an addition of 2026 and is no part of the 1993 manuscript. Where it says what the manuscript does not, it says so.

Afterword — where this leads

The first part of this work assumes a degenerate \(\omega\) and asks what can be rescued. What is rescued is the bracket, so the first thing to say is what the bracket is for; and then where the situation occurs at all, because it is not a curiosity of the formalism: gauge theory is the main place in physics where exactly this happens, and the construction carried out on pages 2 to 4 is the geometric skeleton of how physicists deal with it.

Quantum mechanics, and what the bracket is for

The reason for insisting on the algebra of observables, through all the trouble the first part takes to recover it, is that the algebra is what quantum mechanics inherits. Classical mechanics rests on an assumption so natural that it is seldom stated: the system is somewhere. There is a point of phase space, and at that point every observable has a value, all of them at once. Quantum mechanics gives that up. What it does not give up is the structure built on top of it.

The translation is short. An observable, a function on \(M\), becomes a self-adjoint operator; the values it can take, classically the values of the function, become the spectrum of the operator; a state, classically a point, becomes a unit vector, and the value of an observable in a state becomes an expectation. The bracket becomes the commutator, \(\{f,g\}\mapsto[\hat f,\hat g]/i\hbar\), which is what turns the canonical bracket of a coordinate and its momentum into \([\hat q,\hat p]=i\hbar\). What is worth noticing is the line that does not change: classically \(df/dt=\{f,H\}\), quantum mechanically \(d\hat A/dt=[\hat A,\hat H]/i\hbar\). The equation of motion is the same equation, with the same \(H\), written in the other bracket. Nothing is added; something is taken away.

What is taken away is exactly what the commentary measured at the outset. A classical state is a point, and a point encloses no area; a quantum state cannot enclose less than \(h\). That one sentence is the uncertainty principle: \(\Delta q\,\Delta p\ge\hbar/2\) follows from the commutator and says that squeezing a state in one direction widens it in the other, because the area has a floor. Read backwards it is also why \(h\) had to be smuggled into statistical mechanics before quantum mechanics existed — the number of states in a region of phase space is its volume divided by \(h^n\), and without a grain the entropy of a gas comes out infinite. And it is the whole content of the old Bohr–Sommerfeld rule \(\oint p\,dq=nh\): the permitted orbits are those enclosing a whole number of cells.

The dictionary is not perfect, and its failures are precise rather than vague. To the product \(qp\) there correspond both \(\hat q\hat p\) and \(\hat p\hat q\), which differ by \(i\hbar\); a symmetric choice can be made, but it is a choice and not a translation. Worse, Groenewold and van Hove proved that no quantization can be exact on all polynomials in \(q\) and \(p\) and irreducible at the same time: beyond degree two the correspondence must break. And geometric quantization, the most careful attempt to do this without coordinates, requires \([\omega]/2\pi\hbar\) to be an integral class — a condition on the periods of \(\omega\). Not every symplectic manifold can be quantized at all, and the obstruction is topological.

For a degenerate \(\omega\) a question of order then arises, and it is the one the rest of this afterword turns on. One may reduce first: build the space of leaves, as pages 2 to 4 do, and quantize that. Or one may quantize first and impose the constraints on the states afterwards, which is Dirac's route [3] — the physical states are those annihilated by the constraint operators, \(\hat C_a\psi=0\), the quantum counterpart of admitting only those observables the kernel cannot see. The first is the cleaner and is usually impossible, since the reduced space is seldom something one can work on; the second keeps the coordinates and pays for them. The two do not agree in general, and the theorems that say when they do carry hypotheses that gravity does not meet.

One thing should be said plainly, since the rest of this is a story of structure carrying over intact. All of the above says how a state evolves and what probabilities to assign to a measurement. None of it says why a measurement yields one outcome and not another. That question is untouched by anything in this work, and by anything the bracket carries.

Gauge theories

A gauge theory is a description with redundancy built in: many mathematical configurations stand for one and the same physical state. That redundancy is precisely what makes the form degenerate. The directions \(\omega\) cannot see are the directions in which the description changes and the physics does not — the gauge directions. Everything else follows the pattern of the preceding pages.

The kernel of \(\omega\) is what Dirac calls the first-class constraints [3]. That they close under the Poisson bracket — the property from which the name is taken — is not an extra fact to be verified case by case: it is the computation on page 2, \(\iota_{[X,Y]}\omega=\mathcal{L}_X\iota_Y\omega-\iota_Yd\iota_X\omega-\iota_Y\iota_Xd\omega=0\), with the last term vanishing because \(\omega\) is closed. Involutivity and first class are the same statement in two vocabularies. Frobenius then gives the foliation, whose leaves are the gauge orbits; the admissible observables — those constant along the leaves — are the gauge-invariant ones; and the quotient is what is called the reduced, or physical, phase space.

Maxwell's theory is the smallest honest example. Since \(\dot A_0\) never appears in the Lagrangian, the momentum conjugate to \(A_0\) vanishes identically: the Legendre transform is not invertible, its image is a submanifold of phase space, and the form pulled back to that submanifold is closed and degenerate — the situation of page 2, arrived at from physics rather than from an example built by hand. Requiring the constraint to be preserved in time produces Gauss's law, \(\nabla\!\cdot\mathbf{E}=0\); the flow of its Hamiltonian field is \(A_i\mapsto A_i+\partial_i\lambda\), the gauge transformation; and the quotient by those orbits leaves the transverse field — two polarisations, where the description started with the four components of \(A_\mu\). Yang–Mills theory is the same account with a non-abelian orbit.

Two things the treatment given here does not cover should be said. First, the constraint surface has to be found and not assumed: one starts from a degenerate Lagrangian, imposes a constraint, demands that it be preserved, obtains further constraints, and iterates until the process closes — the presymplectic form of that algorithm is due to Gotay, Nester and Hinds [4]. Second, not all degeneracy is gauge. Dirac's second-class constraints do not lie in the kernel, and they are handled the other way round: one restricts to them, the form becomes non-degenerate there, and what results is the Dirac bracket rather than a quotient. The construction of page 4 covers the first-class half.

General relativity

Gravity was put into this form early — by Dirac in the late 1950s, which is why the constraint machinery exists at all, and by Arnowitt, Deser and Misner [12], whose formulation is still the standard one. The spatial metric is the configuration variable, and the constraints are first class: one generates the diffeomorphisms of the slice, the other its deformation in time. The total Hamiltonian is itself a sum of constraints, which is where the difficulty with time in quantum gravity begins.

The 3+1 split. Spacetime is cut into spatial slices; the metric on a slice is the configuration variable and its momentum is built from the extrinsic curvature. The lapse \(N\) and the shift \(N^i\) say how one slice sits inside the next, and are pure gauge: choosing them differently relabels the same spacetime.

The count is the one the first part performs, done on gravity. At each point of the slice the phase space has twelve dimensions — six for the spatial metric, six for its momentum. There are four constraints, one scalar and three vector, and, being first class, each costs two dimensions: one for restricting to the surface on which it vanishes, one for quotienting by the motion it generates.

\underbrace{12}_{q_{ij},\;\pi^{ij}}\;-\;2\times\underbrace{4}_{\text{constraints}}\;=\;4\;=\;2\times\underbrace{2}_{\text{polarisations}}

What is left is two field degrees of freedom with a momentum apiece: the two polarisations of a gravitational wave. The arithmetic of pages 2 to 4 is, here, the count of what gravity actually carries.

It is worth saying plainly what the difficulty with time is, now that the previous section has said what quantization does. Since the total Hamiltonian is a sum of constraints, Dirac's route requires the physical states to be annihilated by it, \(\hat H\psi=0\) — the Wheeler–DeWitt equation. What is missing from it is the right-hand side. In every other theory the Hamiltonian is the operator that moves a state through time; here it is the operator a physical state must be killed by, and so nothing evolves. This is not an embarrassment to be patched over. It is, in quantum form, what the glossary says in classical form: nothing distinguishes the energy from any other observable except the convention that its flow is called time. Take away the background time and the convention has nothing left to stand on.

Here the hypothesis this work depends on is the one that fails. The reduced space — the space of geometries — is not a manifold: at a geometry with a symmetry the gauge group does not act freely, the rank is not constant, and the quotient is stratified rather than smooth. The remark on page 2, that constant rank is needed because otherwise the number of degrees of freedom varies from point to point, is exactly the obstruction, met in the one case where it matters most. That is why gravity is not quantised by first reducing and then quantising the reduced space, but the other way round: the constraints are imposed on states.

\(\mathrm{Riem}(\Sigma)\) is the space of all metrics on a slice, and the orbits of \(\mathrm{Diff}(\Sigma)\) are the sets of metrics differing only by a relabelling of points — so one orbit is one geometry. At a metric with a symmetry the group does not act freely, the orbit is smaller and the rank drops: the failure that the counter-example of §6 arranges by hand, here occurring on its own in the case that matters most.

The best known attempt to make that second route work is Ashtekar's, from 1986 [13]. It is not a new framework but a change of variables inside this one: a self-dual connection in place of the spatial metric, with a densitised triad as its momentum. The constraints become polynomial, which is what made them look quantisable at all, and general relativity acquires the shape of an \(SU(2)\) gauge theory — which is what let the methods of the previous section be brought to bear on it, and what loop quantum gravity was built upon [15]. The connection so obtained is complex, and recovering real Lorentzian gravity from it requires reality conditions that were never made to work comfortably; Barbero's real connection [14], carrying the parameter now named after him and Immirzi, is what the subject uses today. The price is specific: with a real parameter the scalar constraint acquires a term in the extrinsic curvature, carrying a factor that vanishes only in the self-dual case, so that the polynomial form which motivated the change is lost — while the connection variables, the gauge structure and the rest of the framework are kept. A regularisation making that constraint into a well-defined operator was given later; whether it has the right classical limit, and so whether it describes gravity, is still argued. The programme is not a finished theory.

What the programme did obtain is worth stating, because it touches the first section of this commentary. In the loop representation the operators for area and for volume have discrete spectra: an area comes in multiples set by \(\ell_P^2\) and the Barbero–Immirzi parameter, so that a surface cannot be given an arbitrarily small area, much as a state cannot be given an arbitrarily small cell. The parallel is tempting and should be resisted. The area quantised there is area in space, measured by the metric; the area of the first section is area in phase space, measured by \(\omega\). They carry different dimensions and come from different places, and that both should turn out granular is a suggestion and not a consequence. Nothing in this work implies it.

Editorial note: this afterword is an addition of 2026 and is no part of the 1993 manuscript, which stops with the second-order equation.

Glossary — formal statements

Precise statements of the terms the work uses. \(M\) is a smooth manifold of dimension \(n\), everything is \(C^\infty\), and the sign conventions are the ones of the manuscript. A plain-language gloss of the same terms follows in the next section; it is there, and not here, that the terms in the body link to.

Manifolds and maps

manifold — A set \(M\) with a maximal atlas of charts \(\varphi_\alpha:U_\alpha\to\mathbb{R}^n\), homeomorphisms onto open sets covering \(M\), whose transition maps \(\varphi_\beta\circ\varphi_\alpha^{-1}\) are \(C^\infty\) where defined. Hausdorff and second countable are assumed throughout; \(n=\dim M\).

chart — A pair \((U,\varphi)\) with \(U\subset M\) open and \(\varphi\) a diffeomorphism of \(U\) onto an open set of \(\mathbb{R}^n\). The components \(u^1,\dots,u^n\) of \(\varphi\) are the local coordinates.

homeomorphism — A bijection continuous with a continuous inverse. Two spaces so related are indistinguishable as topological spaces.

diffeomorphism — A bijection smooth with a smooth inverse — the isomorphisms of the smooth category, as homeomorphisms are of the topological one.

isomorphism — A morphism admitting a two-sided inverse of the same kind: linear for vector spaces, smooth for manifolds, fibrewise linear over \(\mathrm{id}_M\) for vector bundles. \(\omega^\flat\) is one precisely when \(\omega\) is non-degenerate, which is the hinge of the whole first half.

Hausdorff, second countable — Distinct points admit disjoint neighbourhoods; the topology has a countable base. Both are assumed of manifolds throughout, and neither is automatic — the leaf space of a foliation may fail the first.

submanifold — \(N\subset M\) such that each \(p\in N\) lies in a chart \((U,\varphi)\) of \(M\) with \(\varphi(U\cap N)=\varphi(U)\cap(\mathbb{R}^k\times\{0\})\). Then \(\dim N=k\).

submersion — A smooth \(f:M\to N\) with \(T_mf\) surjective at every \(m\); preimages of points are then submanifolds of codimension \(\dim N\).

immersion, immersed submanifold — \(T_mf\) injective at every \(m\). An immersed submanifold carries its own manifold structure and need not have the subspace topology, which is why the leaves of a foliation are immersed rather than embedded.

codimension — \(\operatorname{codim}N=\dim M-\dim N\).

regular value — \(c\) is a regular value of \(F:M\to\mathbb{R}^m\) when \(dF_u\) is surjective for every \(u\in F^{-1}(c)\). Then \(F^{-1}(c)\) is a submanifold of codimension \(m\) — the fact used on page 3 with \(F=(f,\dots,h)\).

Bundles

vector bundle — A surjective submersion \(\pi:E\to M\) whose fibres \(E_m=\pi^{-1}(m)\) carry vector space structures, locally trivial: every \(m\) has a neighbourhood \(U\) and a diffeomorphism \(\pi^{-1}(U)\to U\times F\) commuting with the projections and linear on each fibre.

base, total space, fibre — \(M\), \(E\) and \(E_m\) respectively, for \(\pi:E\to M\). Which manifold is the base is a property of the bundle, not of the manifold: \(M\) is the base of \(\tau_M:TM\to M\), while \(TM\) is the base of both projections of \(TTM\). On page 4 the words its base are applied to \(TM/N\) and denote the leaf space \(M/\mathcal{F}\), not \(M\); see leaf space below for the hypotheses under which that quotient is a manifold at all.

section — \(\sigma:M\to E\) smooth with \(\pi\circ\sigma=\mathrm{id}_M\). The sections form \(\Gamma(E)\); \(\Gamma(TM)\) is the set of vector fields.

subbundle — \(E'\subset E\) with \(E'_m\subset E_m\) a subspace of dimension independent of \(m\), such that \(\pi|_{E'}\) makes \(E'\) a vector bundle. Constant rank is exactly what is needed for \(\ker\omega\) to be one.

tangent bundle — \(TM=\bigsqcup_{m\in M}T_mM\) with \(\tau_M(m,e)=m\); a chart \((u^i)\) of \(M\) induces the chart \((u^i,e^i)\) of \(TM\), so \(\dim TM=2\dim M\).

cotangent bundle — \(T^*M=\bigsqcup_{m\in M}(T_mM)^*\), with the induced charts \((u^i,p_i)\).

iterated tangent bundle — \(TTM=T(TM)\), of dimension \(4\dim M\). It carries two distinct projections onto \(TM\): \(\tau_{TM}\), its own bundle projection, and \(T\tau_M\), the tangent map of \(\tau_M\). Their difference is the subject of the second half of the work.

projection \(\tau_M\) — \(\tau_M:TM\to M\), \((m,e)\mapsto m\); a surjective submersion and a vector bundle projection.

tangent map — For \(f:M\to N\), \(Tf:TM\to TN\), \((m,e)\mapsto(f(m),df_m(e))\). It is functorial, \(T(g\circ f)=Tg\circ Tf\), and \(T\mathrm{id}=\mathrm{id}\).

pushforward — For \(f\) a diffeomorphism and \(X\in\Gamma(TM)\), \(f_*X=Tf\circ X\circ f^{-1}\in\Gamma(TN)\) — the formula quoted on page 5.

vector bundle mapping — \(\phi:E\to E'\) covering \(f:M\to M'\), that is \(\pi'\circ\phi=f\circ\pi\), with \(\phi|_{E_m}:E_m\to E'_{f(m)}\) linear for every \(m\). Fiber-preserving is the first condition alone.

vertical subbundle — \(V=\ker T\tau_M\subset TTM\), the vectors tangent to the fibres of \(\tau_M\). It is canonical; a horizontal complement \(H\) with \(TTM=V\oplus H\) is not — choosing one is precisely choosing a connection. The phrase "horizontal part", in the text, means the component that \(T\tau_M\) records.

connection — A choice of horizontal complement \(H\) with \(TTM=V\oplus H\), equivalently a rule for differentiating sections along curves. It is extra data: the manifold does not supply one.

curvature — The failure of a connection's horizontal distribution to be involutive; for the Levi-Civita connection of a metric, the Riemann tensor. A symplectic form admits no such invariant — by Darboux there is nothing local to measure.

Forms and the operations on them

vector field — \(X\in\Gamma(TM)\); in a chart \(X=X^i\,\partial/\partial u^i\). Equivalently a derivation of \(C^\infty(M)\).

2-form — \(\omega\in\Gamma(\Lambda^2T^*M)\): for each \(m\), a bilinear alternating \(\omega_m:T_mM\times T_mM\to\mathbb{R}\), smooth in \(m\). Remark. In canonical coordinates \(\omega=\sum_i dq^i\wedge dp_i\) returns the sum of the signed areas of the projections of the parallelogram spanned by \(u\) and \(v\) onto the planes \((q^i,p_i)\). Since the pairing is between a coordinate and its conjugate momentum, \(\omega(u,v)\) carries the dimensions of action; pairing a direction with itself would require adding quantities of different dimensions, which is the physical counterpart of \(\omega(u,u)=0\). A metric on phase space can always be had by choosing scales — a mass and a frequency, say — but nothing in the phase space itself supplies them.

skew-symmetry — \(\omega(u,v)=-\omega(v,u)\) for all \(u,v\), equivalently \(\omega(u,u)=0\).

closed — \(d\omega=0\). By Cartan's formula this is what makes \(\iota_{[X,Y]}\omega\) reduce as it does on page 2, and hence what makes \(\ker\omega\) involutive.

exact — \(\omega=d\theta\) for some \(\theta\). Exact implies closed, since \(d^2=0\); the converse holds locally, by the Poincaré lemma, and globally only when \(H^2_{\mathrm{dR}}(M)\) permits. On \(T^*M\) one has \(\omega=-d\theta\) with \(\theta=\sum_ip_i\,dq^i\) the tautological form; on a compact symplectic manifold \(\omega\) is never exact.

Stokes' Theorem — \(\int_\Sigma d\alpha=\oint_{\partial\Sigma}\alpha\). Applied to \(\omega=-d\theta\) it turns the area of a patch of phase space into the action along its boundary.

period — The integral of a closed form over a cycle; equivalently the value of \([\omega]\in H^2_{\mathrm{dR}}(M)\) on a homology class. All periods vanish exactly when the form is exact.

non-degenerate — \(\omega^\flat_m\) is injective for every \(m\): \(\omega_m(u,v)=0\) for all \(v\) forces \(u=0\). A non-degenerate alternating form exists only in even dimension.

rank — \(\operatorname{rank}\omega_m=\dim T_mM-\dim\ker\omega_m\), always even. Constant rank means this number is independent of \(m\).

kernel — \(\ker\omega_m=\{u\in T_mM:\iota_u\omega_m=0\}\), and \(\ker\omega=\bigsqcup_m\ker\omega_m\), a subbundle of \(TM\) when the rank is constant.

flat map — \(\omega^\flat:TM\to T^*M\), \(\omega^\flat(u)=\iota_u\omega\). A vector bundle mapping over \(\mathrm{id}_M\); an isomorphism precisely when \(\omega\) is non-degenerate, with inverse written \(\omega^\sharp\).

interior product — \((\iota_X\omega)(Y)=\omega(X,Y)\); an antiderivation of degree \(-1\) on forms.

Lie derivative — \(\mathcal{L}_X=\frac{d}{dt}\big|_{t=0}(\phi^X_t)^*\), with \(\phi^X_t\) the flow of \(X\). Cartan's formula: \(\mathcal{L}_X=d\circ\iota_X+\iota_X\circ d\).

Lie bracket — \([X,Y]f=X(Yf)-Y(Xf)\), equivalently \([X,Y]=\mathcal{L}_XY\). It makes \(\Gamma(TM)\) a Lie algebra.

Foliations

distribution — A subbundle \(E\subset TM\) of constant rank \(k\): a \(k\)-plane \(E_m\subset T_mM\) at each point, varying smoothly.

involutive — \(X,Y\in\Gamma(E)\Rightarrow[X,Y]\in\Gamma(E)\).

integrable — Through every point there passes an integral submanifold \(N\), that is \(T_pN=E_p\) for all \(p\in N\).

Frobenius' Theorem — A distribution is integrable if and only if it is involutive; equivalently, each point admits a chart in which \(E\) is spanned by \(\partial/\partial x^1,\dots,\partial/\partial x^k\). The flat chart is the one drawn on page 3.

foliation — A partition of \(M\) into connected immersed submanifolds of dimension \(p\) such that each point has a chart \((x,y):U\to\mathbb{R}^p\times\mathbb{R}^{m-p}\) in which the components of each leaf meeting \(U\) are the sets \(y=\text{const.}\) This is the definition quoted from Abraham on page 3.

leaf — A maximal connected integral submanifold of the distribution; the classes of the partition.

leaf space — \(M/\mathcal{F}\) with the quotient topology. It need not be Hausdorff, and is a manifold only under extra hypotheses on the foliation; the constructions of page 4 are therefore local.

Mechanics

observable — \(f\in C^\infty(M)\).

action — Two objects share the name. (i) The functional \(S[\gamma]=\int L\,dt\) on paths, whose critical points are the physical trajectories: \(\delta S=0\), which is Hamilton's principle and yields the Euler–Lagrange equations. Whether a critical point is a minimum depends on the interval — it is one up to the first conjugate point and a saddle beyond it, the index of the second variation counting the conjugate points with multiplicity (Morse). It is never a maximum for \(L=T-V\) with \(T\) positive definite, since high-frequency variations raise \(S\) without bound. (ii) The abbreviated action \(S_0=\int p\,dq\), stationary at fixed energy (Maupertuis), which is the quantity \(\omega\) measures: \(\iint_\Sigma\omega=-\oint_{\partial\Sigma}p\,dq\) by Stokes. The two are joined by \(S=\int(p\,dq-H\,dt)\). This work uses only (ii); no variational principle appears in it.

symplectic manifold — \((M,\omega)\) with \(\omega\) a 2-form that is closed and non-degenerate; then \(\dim M=2n\) and \(\omega^n\) is a volume form.

presymplectic manifold — \((M,\omega)\) with \(\omega\) closed, degeneracy allowed. Constant rank is usually added, and is assumed throughout the first half of this work.

Hamiltonian — \(H\in C^\infty(M)\), the observable whose field is taken to generate time. For a Lagrangian system it is the Legendre transform \(H=p_i\dot q^i-L\), read on \(T^*M\); it coincides with the energy when the constraints are time-independent and \(L=T-V\) with \(T\) quadratic in the velocities, and not in general. Nothing distinguishes it geometrically from any other observable — the construction \(\iota_X\omega=dg\) is the same for all of them — and when \(\mathbb{F}L\) is not invertible it is defined only modulo the constraints, \(H+\lambda^a\phi_a\).

Hamiltonian vector field — \(X_f\) defined by \(\iota_{X_f}\omega=df\). Unique when \(\omega\) is non-degenerate; determined only modulo \(\ker\omega\) otherwise, and existing only if \(df\) annihilates \(\ker\omega\) Antisymmetry gives \(df(X_f)=\omega(X_f,X_f)=0\), so \(X_f\) is tangent to the level sets of \(f\) — the geometric form of the conservation of \(f\) along its own flow, and the point at which the construction parts company with the gradient of a metric.

Poisson bracket — \(\{f,g\}=\omega(X_f,X_g)=\mathcal{L}_{X_g}f=-\mathcal{L}_{X_f}g\) — the sign convention of the manuscript. It is bilinear and skew, satisfies the Leibniz rule in each argument, and satisfies the Jacobi identity precisely because \(d\omega=0\).

algebra of observables — \(C^\infty(M)\) with the pointwise product and the Poisson bracket: a Poisson algebra, and a Lie algebra under the bracket alone.

canonical transformation — A diffeomorphism \(\phi\) with \(\phi^*\omega=\omega\); equivalently \(\{f\circ\phi,g\circ\phi\}=\{f,g\}\circ\phi\) for all \(f,g\).

degrees of freedom — \(n=\tfrac12\dim M\) for a symplectic \(M\). In the degenerate case the useful count is \(\tfrac12\operatorname{rank}\omega\), which is why constant rank is imposed.

Darboux's Theorem — Around each point of a symplectic \(2n\)-manifold there are coordinates with \(\omega=\sum_{i=1}^{n}dq^i\wedge dp_i\). In the presymplectic case of constant rank \(2k\) the normal form is \(\omega=\sum_{i=1}^{k}dq^i\wedge dp_i\), the remaining coordinates spanning \(\ker\omega\) — which is the matrix written on page 1.

Legendre transform — The fibre derivative \(\mathbb{F}L:TM\to T^*M\) of a Lagrangian \(L\), \(\mathbb{F}L(v)\cdot w=\frac{d}{dt}\big|_{0}L(v+tw)\). \(L\) is regular when \(\mathbb{F}L\) is a local diffeomorphism; when it is not, \(\omega_L=-d(\mathbb{F}L)^*\theta\) is presymplectic.

constraint — A submanifold \(N\subset M\) of a symplectic \((M,\omega)\), typically the common zero set of functions \(\phi_1,\dots,\phi_r\) having \(0\) as a regular value. The pullback \(\iota_N^*\omega\) is closed but in general degenerate — a presymplectic form, which is where the first half of this work applies.

first class — A family of constraints \(\phi_a\) whose brackets close among themselves, \(\{\phi_a,\phi_b\}=f_{ab}^{\;c}\phi_c\), and so vanish on the constraint surface. It is the involutivity of \(\ker\omega\) in the vocabulary of mechanics.

gauge direction — For a presymplectic \((N,\omega)\), an element of \(\ker\omega\); the leaves of the resulting foliation are the gauge orbits.

Curves and second-order equations

integral curve — \(c:I\to M\) with \(\dot c(t)=X(c(t))\) for all \(t\in I\).

tangent lift — For \(c:I\to M\), the curve \(c':I\to TM\), \(c'(t)=(c(t),\dot c(t))=Tc(t,1)\). Iterating gives \(c'':I\to TTM\).

base integral curve — \(\tau_M\circ v\) for \(v\) an integral curve of a vector field on \(TM\).

second-order equation — \(X\in\Gamma(TTM)\) with \(T\tau_M\circ X=\mathrm{id}_{TM}\). Equivalently every integral curve \(v\) satisfies \((\tau_M\circ v)'=v\). In induced coordinates \(X(u,e)=\bigl(u,e,e,X_2(u,e)\bigr)\), so the base integral curves satisfy \(\ddot u=X_2(u,\dot u)\).

geodesic — A curve whose velocity is parallel along itself for a given connection; for the Levi-Civita connection of a metric, a locally length-extremising curve. The base integral curves of a spray are geodesics.

spray — A second-order equation whose components are homogeneous of degree two in the velocities, \(X_2(u,\lambda e)=\lambda^2X_2(u,e)\). Its base integral curves are the geodesics of a connection; dropping homogeneity leaves the semispray, which is the object defined here.

vertical endomorphism — The canonical \((1,1)\)-tensor \(J\) on \(TM\) with \(J(\partial/\partial u^i)=\partial/\partial e^i\) and \(J(\partial/\partial e^i)=0\). With \(\Delta=e^i\,\partial/\partial e^i\) the Liouville field, \(X\) is a second-order equation if and only if \(JX=\Delta\).

Editorial note: this glossary and the informal one that follows are additions of 2026, and are no part of the 1993 manuscript.

Glossary — in plain language

The same terms in plain language, for a reader who is not a differential geometer. Terms in the body of the work link here at their first appearance; the formal statement of each is in the preceding section, and the definitions given inside the work itself, in the boxes, take precedence over both.

Manifolds and maps

manifold — A space that, seen close up, looks like ordinary flat space but need not look like it as a whole — a sphere, a torus, the set of all positions of a pendulum. It is where the states of the system live.

chart, local coordinates — A way of naming the points of one patch of a manifold by lists of numbers. Charts are the price of global shape rather than of curvature — a flat torus still needs several of them, a curved paraboloid is covered by one — and no single chart need cover everything, so a statement only counts as intrinsic if it survives a change of chart.

diffeomorphism — A smooth change of variables that can be undone smoothly. Two manifolds related by one are the same manifold under different names.

isomorphism — A dictionary that works in both directions and loses nothing on the way. ω♭ is one while ω can see every direction; when it cannot, the dictionary has words with no translation, and that is where this work begins.

submanifold — A subset of a manifold which is itself a manifold, the way a surface sits inside space.

codimension — How many dimensions a submanifold is missing relative to what surrounds it: a surface in three-space has codimension one.

regular value — A value c of a function at which the differential does not vanish. Regular values are exactly the ones whose level sets are honest submanifolds.

Bundles

bundle, vector bundle — A manifold with a vector space glued to each of its points, all of the same size and varying smoothly from point to point.

base — The manifold underneath a bundle — the one the vector spaces are attached to. Which manifold that is depends on the bundle and not on the manifold: \(M\) is the base of \(TM\), but \(TM\) is itself the base of \(TTM\). The word carries its full weight on page 4, where the work says of \(TM/N\) that its base is a symplectic manifold. The base meant there is not the original \(M\) but what is left of it once every point indistinguishable from another has been collapsed to a single one — the space of leaves. That new floor, and not the one the work started on, is where the algebra of observables works again; in physical language it is the reduced, or physical, space of states, one point per state rather than one point per description.

fibre — The vector space sitting over one particular point of the base.

section — A choice of one vector in each fibre, varying smoothly. A vector field is a section of the tangent bundle.

subbundle — A bundle inside another, over the same base, with smaller fibres.

tangent bundle, TM — All the velocities of all possible motions through all points of M, gathered into a single manifold. It has twice the dimension of M: a point of it is a position together with a velocity.

cotangent bundle, T*M — The same, with covectors — things that eat velocities and return numbers — in place of velocities. It is where momenta live.

iterated tangent bundle, TTM — The tangent bundle of TM: where the velocities of curves drawn in TM live, and therefore where accelerations live. Four times the dimension of M.

projection, τ_M — The map that forgets the vector and keeps the point it was attached to.

tangent map, Tf — The derivative of a map, seen as a map between the tangent bundles: it carries velocities to velocities.

pushforward, f∗X — A vector field carried from one manifold to another by a tangent map.

vector bundle mapping, fiber-preserving — A map between bundles that sends each fibre into a fibre — never mixing up the points underneath — and is linear on each of them.

vertical and horizontal part — A vector in TTM has a piece that moves the base point, the horizontal one, and a piece that moves only inside the fibre, the vertical one. The two projections of TTM keep different pieces, and that difference is the subject of the second half of this work.

connection — A choice of what shall count as moving straight from one fibre to the next. Nothing supplies it; one has to be picked, and different picks are different geometries.

curvature — What you find when going round a small loop does not bring you back unchanged. It belongs to a metric or to a connection, never to a manifold on its own — and a symplectic form has none at all.

Forms and the operations on them

vector field — An arrow attached to every point, varying smoothly. Its integral curves are the motions it generates.

2-form — An instrument that eats two directions and returns the signed area of the parallelogram they span. Hand it the same direction twice and the parallelogram is flat, so the answer is zero — which is all that "changes sign when the two are swapped" amounts to. On a phase space the two directions it pairs are a position and a momentum, so the area it measures carries the units of action. That is what a phase space has instead of a way of measuring length.

skew-symmetry — The rule ω(u,v) = −ω(v,u). It forces ω(u,u) = 0: no vector pairs with itself.

closed form, closedness — A form whose exterior derivative vanishes, dω = 0. Here it is what makes the invisible directions fit together into surfaces, and it is also what Darboux's theorem needs.

exact form — A form that is the derivative of something. Every exact form is closed; the converse holds patch by patch but not always in the large, and the discrepancy is a fact about the shape of the space rather than about the form.

Stokes' Theorem — What happens inside a region is the sum of what happens along its edge. It is what turns the area ω measures into the action accumulated round the boundary.

period — What you get by integrating a closed form round a loop that does not bound anything. The periods all vanish exactly when the form is the derivative of something.

non-degeneracy — No vector is invisible to ω: if ω(u,·) gives zero on everything, then u itself is zero. It is what allows ι_Xω = dg to be solved for X.

rank — The number of directions ω actually sees. Full rank is non-degeneracy; constant rank is the assumption that this number does not jump from point to point.

kernel — The directions ω does not see — the vectors that give zero against everything.

flat map, ω♭ — The map that turns a vector into a covector by feeding it to ω. Non-degeneracy is precisely the statement that it can be inverted.

interior product, ι_Xω — The result of feeding X into the first slot of ω and leaving the other slot open.

Lie derivative, L_X — The rate at which a quantity changes as it is dragged along the flow of a vector field.

Lie bracket, [X,Y] — The vector field measuring the failure of two flows to commute — go along X then Y, come back the other way, and see where you land.

Foliations

distribution — A choice, at each point, of a subspace of directions, varying smoothly. The kernel of ω is one.

involutive — A distribution closed under the Lie bracket: bracket two fields lying in it and you stay in it.

integrable — Tangent to a family of submanifolds that fills the space — the directions really are the directions of surfaces.

Frobenius' Theorem — Involutive and integrable are the same thing. It is what turns an algebraic condition on brackets into actual surfaces.

foliation — A decomposition of a manifold into parallel-looking submanifolds, like the pages of a book.

leaf — One of those submanifolds — one page of the book.

leaf space, quotient — What remains when each leaf is collapsed to a single point.

Mechanics

observable — A measurable quantity: a function on the space of states. Energy, position, momentum.

action — Two things go by this name. Usually it means the total accumulated along a path, and the rule that picks the true motion out of all the possible ones. That rule is traditionally called least action, which misleads: the true motion makes the action stationary, not minimal — flat to first order, the way a mountain pass is flat, and for a long enough journey it is a pass rather than a valley floor. The other meaning is the one this work uses: action as the quantity a position and a momentum multiply to, which is what ω measures, and of which Planck's constant is one quantum. Stokes' Theorem joins the two — the area inside a patch is the action accumulated round its edge.

symplectic manifold — A manifold carrying a 2-form that is closed and non-degenerate — the phase space of a classical system.

presymplectic — Closed but degenerate. This is the case the first half of the work is about.

Hamiltonian, H — The energy, written as a function of position and momentum. It is the whole of the dynamics in one function: hand it over and the motion follows, with no forces to be supplied on the side. And it is not privileged. Every observable generates a motion by the same rule — momentum generates translation, angular momentum rotation — and the energy is simply the one whose motion we have agreed to call the passing of time. In a theory with no background time, such as general relativity, that agreement cannot be made, and the Hamiltonian turns out to be a constraint rather than a generator.

Hamiltonian vector field — The field attached to an observable by ι_{X_g}ω = dg. Its flow is the motion that the observable generates: for the energy, the motion is time itself. The same recipe with a metric in place of ω would give the gradient, which climbs straight across the level sets; because ω changes sign when its two arguments are swapped, the answer here is turned a quarter turn and runs along them instead — which is why an observable is conserved by the motion it generates.

Poisson bracket — The rule that takes two observables to a third, {f,g} = ω(X_f,X_g). It gives the evolution df/dt = {f,H}, and it is the structure quantization turns into the commutator.

algebra of observables — The observables together with the bracket, treated as an algebraic structure in its own right.

canonical transformation — A change of description that leaves ω, and hence the bracket, alone.

degrees of freedom — The number of independent directions the system can actually move in. When ω degenerates, the number of candidate fields exceeds it, and that is the problem this work starts from.

Darboux's Theorem — Locally, all symplectic forms of the same dimension look alike: coordinates always exist in which ω takes one standard constant shape. Symplectic geometry has no local invariant, no curvature.

Legendre transform — The passage from velocities to momenta. When it cannot be inverted the Lagrangian is degenerate and a presymplectic form is what one is left with.

constraint — A relation the states are required to satisfy, cutting the description down to a subset of itself.

first class — The physicist's name for constraints whose brackets stay among themselves. It is the same condition as the involutivity of the kernel: one word from mechanics, one from geometry, one statement.

gauge direction — A direction in which the description changes and the physical state does not. These are the directions the quotient throws away.

integral curve — A curve whose velocity at each instant is the value of the field there — the motion the field generates.

Curves and second-order equations

tangent lift, c′ — Given a curve in M, the curve drawn in TM by its successive velocities.

base integral curve — The shadow in M of an integral curve living upstairs in TM.

second-order equation — A vector field on TM whose integral curves are tangent lifts, so that in coordinates it reads ü = f(u,u̇). Newton's law, written without choosing coordinates.

geodesic — The straightest curve available, once a connection has been chosen. For the metric hidden in the kinetic energy, the path of a free motion.

spray, semispray — The same condition as it appears in differential geometry. A spray adds a homogeneity requirement, which singles out the geodesics.

vertical endomorphism — The canonical operator on TTM through which the Lagrangian school states the same second-order condition.

Editorial note: an addition of 2026, no part of the 1993 manuscript.

Glossary — pictures

A dozen of the terms are better drawn than described. The captions link back to the informal glossary; the pictures are of the general notions, not of the particular objects of the work, which are drawn in their own place in the body.

A manifold is a space that carries charts: each patch is named by coordinates, and where two patches overlap the two namings differ by a smooth transition map

Base, fibre, section and subbundle — the picture the whole first part of the work leans on

The tangent bundle of the circle: a point together with a velocity, and the two of them assembled into a cylinder

Frobenius' Theorem in a picture: an involutive field of planes is integrable and gives a foliation; a twisting one gives none

The Lie bracket is the gap by which the quadrilateral fails to close — which is why "the bracket stays inside E" is a condition about surfaces

The Hamiltonian field of an observable runs along its level sets: conservation, and the reason the Poisson bracket measures it

At a regular value the level set is a submanifold, and the level sets stack up as a foliation; at a critical value they cross and the stack breaks down

The tangent map carries a curve, and with it its velocity — assembled over all points it gives the pushforward of a field

Non-degeneracy, rank and kernel at a single point: what the form sees, and what it does not

A foliation and its leaf space: each leaf becomes one point, and the quotient is what page 4 constructs

A vector field, its integral curves and its flow — along which the Lie derivative measures change

A canonical transformation preserves ω, and on a two-dimensional symplectic manifold that is exactly preserving area

Editorial note: an addition of 2026, no part of the 1993 manuscript.

References

[1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., Benjamin/Cummings, Reading, Mass., 1978.

[2] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Graduate Texts in Mathematics 60, Springer, New York, 1989.

[3] P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York, 1964.

[4] M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac–Bergmann theory of constraints, Journal of Mathematical Physics 19 (1978), 2388–2399.

[5] R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd ed., Applied Mathematical Sciences 75, Springer, New York, 1988.

[6] P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel, Dordrecht, 1987.

[7] A. Weinstein, The local structure of Poisson manifolds, Journal of Differential Geometry 18 (1983), 523–557.

[8] S. Lang, Introduction to Differentiable Manifolds, Interscience (John Wiley & Sons), New York, 1962 — chapter on sprays.

[9] W. Ambrose, R. S. Palais and I. M. Singer, Sprays, Anais da Academia Brasileira de Ciências 32 (1960), 163–178.

[10] J. Klein, Espaces variationnels et mécanique, Annales de l'Institut Fourier 12 (1962), 1–124.

[11] C. Godbillon, Géométrie différentielle et mécanique analytique, Hermann, Paris, 1969.

[12] R. Arnowitt, S. Deser and C. W. Misner, The dynamics of general relativity, in Gravitation: An Introduction to Current Research, ed. L. Witten, Wiley, New York, 1962.

[13] A. Ashtekar, New variables for classical and quantum gravity, Physical Review Letters 57 (1986), 2244–2247.

[14] J. F. Barbero G., Real Ashtekar variables for Lorentzian signature space-times, Physical Review D 51 (1995), 5507–5510.

[15] C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge, 2004.

References [2]–[15] are editorial additions of 2026: [2]–[7] as entry points to the setting of the work; [8]–[11] as the lineage the second-order condition comes from, [8] and [9] for the spray route and [10] and [11] for the route through the vertical endomorphism; [12]–[15] for the afterword. The manuscript itself cites only Abraham [1].

The manuscript

The seven pages as they were written: by hand, in Portuguese, in February 1993. Each thumbnail opens the page at full size, where the handwriting can be read and the original drawings can be seen as they were drawn — they are what the figures in this edition were made from, and in several places they say things the transcription cannot, in the margins and in the second colour of pen.

Manuscript page 1: degeneracy of the symplectic form, the flat map, and the example in which it fails

Manuscript page 2: the kernel of omega, the Poisson brackets, and the proof that the kernel is integrable

Manuscript page 3: the definition of a foliation, the chart that flattens the leaves, and regular values

Manuscript page 4: the two quotients and the reduced bundle TM/N

Manuscript page 5: the definition of a second-order equation, the rhombic diagram and its local form

Manuscript page 6: the two projections of TTM drawn on the manifolds, and the scheme by levels

Manuscript page 7: the system (*) and the equations of motion in local coordinates

Editorial note: the images are hosted alongside this document and pinned to a single revision of it, so that what a page shows here cannot change under it. The transcription in the pages above is of this same document.

SymplecticGeometry

Contents

Degeneracy of the Symplectic Form

Differential Geometry

Second-Order Equation Defined Intrinsically

Locally

Motivation

The route I took

The second half

The argument, explained

1. What the form measures

2. How a quantity becomes a motion

3. The bracket, and why the paper wants an algebra

4. Where it breaks

5. What the extra freedom actually is

6. Why the blindness is organised

7. The double quotient

8. What an equation of motion is

9. What the two halves have to do with each other

Afterword — where this leads

Quantum mechanics, and what the bracket is for

Gauge theories

General relativity

Glossary — formal statements

Manifolds and maps

Bundles

Forms and the operations on them

Foliations

Mechanics

Curves and second-order equations

Glossary — in plain language

Manifolds and maps

Bundles

Forms and the operations on them

Foliations

Mechanics

Curves and second-order equations

Glossary — pictures

References

The manuscript

Symplectic
Geometry