Sympletic Geometry

Por Luís Matos Ferreira agosto 26, 2026

Instituto Superior Técnico

Symplectic
Geometry

Luís Ferreira

IST no. 31727 · 2 / 2 / 93

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.

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 abstract, this motivation and the references are additions of 2026. Everything that follows is a transcription of the 1993 manuscript.

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: \(g(u,y,z)=uy+\mathrm{const.}\), \(dg=y\,du+x\,dy=(y,u,0)\). 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 \(X_g\).

Axes x, y and z with the vertical lines of the kernel and the level surfaces of 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 φ carries the neighbourhood U of M onto the level sets y = const. — 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:

Two vectors of TM differing by a vector along the leaf — 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 \(TM\).

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)

Rhombic diagram: TTM on top, two copies of TM at the sides, M at the base — 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)":

Vector bundle mapping: f carries M into N and Tf carries TM into TN preserving the fibres — 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 rhombic diagram in local coordinates — 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 of TTM onto TM, drawn on the manifolds — 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 \((\tau_M\circ c)'\)

Another scheme: TTM, TM and M stacked, the interval I and the routes ① and ② — 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

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.

References [2]–[7] are editorial additions of 2026, given as entry points to the setting of the work; the manuscript itself cites only Abraham [1].