---
title: Cosymplectic Marsden–Weinstein Process
url: https://www.emergentmind.com/topics/cosymplectic-marsden-weinstein-process
type: topic
---

# Cosymplectic Marsden–Weinstein Process

The cosymplectic Marsden–Weinstein process is the odd-dimensional analogue of symplectic Marsden–Weinstein reduction. In its classical Lie-group form, it starts from a cosymplectic manifold \((M,\omega,\eta)\), a symmetry action preserving the cosymplectic structure, and a momentum map; one then restricts to a momentum level set and quotients by an appropriate isotropy subgroup to obtain a reduced manifold carrying a descended cosymplectic structure. In the literature covered here, this process appears in several layers: Albert’s classical cosymplectic reduction as reviewed and refined for time-dependent Hamiltonian systems, a Mikami–Weinstein type theorem for cosymplectic groupoid actions, and higher-dimensional generalizations to \(k\)-polycosymplectic and \(q\)-cosymplectic geometry [2302.05827], [2410.05846].

## 1. Geometric setting

A cosymplectic manifold is a \((2n+1)\)-dimensional manifold \(M\) endowed with a closed \(1\)-form \(\eta\) and a closed \(2\)-form \(\omega\) such that
\[
d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.
\]
The associated Reeb vector field \(R\) is uniquely characterized by
\[
\iota_R\omega=0,\qquad \iota_R\eta=1.
\]
Equivalent formulations used in the literature include the bundle isomorphism
\[
\flat:TM\to T^*M,\qquad v_x\mapsto \iota_{v_x}\omega_x+(\iota_{v_x}\eta_x)\eta_x,
\]
and the decomposition
\[
TM=\ker \omega\oplus \ker \eta,
\]
with \(\ker\omega\) one-dimensional and \(\ker\eta\) of rank \(2n\) [2302.05827].

This structure is the standard geometric model for time-dependent Hamiltonian systems. If \(M=T\times P\), with \(T\) one-dimensional and \((P,\omega)\) symplectic, then pulling back \(\omega\) from \(P\) and a nonvanishing closed \(1\)-form \(\eta\) from \(T\) yields a cosymplectic structure. In Darboux coordinates,
\[
\omega=\sum_{i=1}^n dq^i\wedge dp_i,\qquad \eta=dt,\qquad R=\frac{\partial}{\partial t},
\]
and the evolution vector field
\[
E_h=R+X_h
\]
encodes the usual time-dependent Hamilton equations [2302.05827].

A complementary description emphasizes the foliation defined by \(\ker\eta\). Because \(\eta\) is closed, \(\ker\eta\) is integrable, and its leaves are symplectic with symplectic form \(\omega|_S\). Equivalently, a cosymplectic manifold carries a regular Poisson structure of corank \(1\), with symplectic foliation equal to the foliation of \(\ker\eta\), together with a transverse Poisson vector field, namely the Reeb field [2410.05846]. This leafwise symplectic viewpoint is central in the groupoid version of the reduction process.

## 2. Lie-group reduction on cosymplectic manifolds

For a Lie-group action \(\Phi:G\times M\to M\), the cosymplectic Marsden–Weinstein process begins with a cosymplectic action,
\[
\Phi_g^*\omega=\omega,\qquad \Phi_g^*\eta=\eta,
\]
and, in the connected case, infinitesimally
\[
\mathcal L_{\xi_M}\omega=0,\qquad \mathcal L_{\xi_M}\eta=0,\qquad \forall \xi\in\mathfrak g.
\]
A key restriction is that the fundamental vector fields lie in \(\ker\eta\),
\[
\iota_{\xi_M}\eta=0,\qquad \forall \xi\in\mathfrak g,
\]
so the symmetry does not move the time variable [2302.05827].

A cosymplectic momentum map is a map
\[
\mathbf J^\Phi:M\to \mathfrak g^*
\]
such that, for every \(\xi\in\mathfrak g\),
\[
\iota_{\xi_M}\omega=d\langle \mathbf J^\Phi,\xi\rangle=:dJ_\xi,\qquad RJ_\xi=0.
\]
The additional condition \(RJ_\xi=0\) is specific to the cosymplectic setting; it guarantees that the Reeb field is tangent to each momentum level set [2302.05827].

In the refined formulation, the momentum map need not be \(\mathrm{Ad}^*\)-equivariant. Its defect is measured by the cocycle
\[
\sigma(g):=\mathbf J^\Phi\circ \Phi_g-\mathrm{Ad}^*_{g^{-1}}\mathbf J^\Phi,
\]
which yields an associated affine action \(\Delta\) on \(\mathfrak g^*\). The relevant isotropy subgroup is then
\[
G_\mu^\Delta:=\{g\in G:\Delta_g(\mu)=\mu\},
\]
not necessarily the usual coadjoint stabilizer. If \(\mu\) is a weakly regular value and
\[
M_\mu^\Delta:=\mathbf J^{\Phi-1}(\mu)/G_\mu^\Delta
\]
is a manifold with quotient projection
\[
\pi_\mu:\mathbf J^{\Phi-1}(\mu)\to M_\mu^\Delta
\]
a submersion, then there exists a unique cosymplectic manifold
\[
(M_\mu^\Delta,\omega_\mu,\eta_\mu)
\]
such that
\[
\iota_\mu^*\omega=\pi_\mu^*\omega_\mu,\qquad
\iota_\mu^*\eta=\pi_\mu^*\eta_\mu,
\]
where \(\iota_\mu:\mathbf J^{\Phi-1}(\mu)\hookrightarrow M\) is the inclusion [2302.05827].

The proof mirrors the symplectic argument but includes the Reeb direction. The pulled-back forms are shown to be basic; closedness descends from closedness of \(\omega\) and \(\eta\); and the Reeb field projects to a reduced Reeb field \(R_\mu\) satisfying
\[
\iota_{R_\mu}\eta_\mu=1,\qquad \iota_{R_\mu}\omega_\mu=0.
\]
A key linear-algebra identity is
\[
\bigl(T_x(\mathbf J^{\Phi-1}(\mu))\bigr)^{\perp_\omega}
=
T_x(Gx)\oplus \langle R_x\rangle,
\]
which isolates the additional one-dimensional direction absent in ordinary symplectic reduction [2302.05827].

The process also reduces dynamics. If \(h\) is \(G\)-invariant, then the evolution vector field \(E_h\) is tangent to momentum levels, descends to the quotient, and there is a unique reduced Hamiltonian \(k_\mu\) such that
\[
\pi_\mu^*k_\mu=\iota_\mu^*h,\qquad
\pi_{\mu *}\bigl(E_h|_{\mathbf J^{\Phi-1}(\mu)}\bigr)=E_{k_\mu}.
\]
Thus the reduction concerns not only the pair \((\omega,\eta)\) but the full time-dependent Hamiltonian dynamics [2302.05827].

## 3. Groupoid formulation and leafwise reduction

The groupoid-generalized version is a Mikami–Weinstein type theorem for cosymplectic groupoid actions. A cosymplectic groupoid is a Lie groupoid
\[
(G_1\rightrightarrows G_0,\eta_{G_1},\omega_{G_1})
\]
whose arrow manifold carries a multiplicative cosymplectic structure:
\[
m^\ast\eta_{G_1}=\mathrm{pr}_1^\ast\eta_{G_1}+\mathrm{pr}_2^\ast\eta_{G_1}, \qquad
m^\ast\omega_{G_1}=\mathrm{pr}_1^\ast\omega_{G_1}+\mathrm{pr}_2^\ast\omega_{G_1}.
\]
A basic structural fact is that all unit arrows lie in the same symplectic leaf \(S_{G_1}\subset G_1\), and
\[
S_\mathcal G:=(S_{G_1}\rightrightarrows G_0)
\]
is a symplectic subgroupoid [2410.05846].

The action datum is a left groupoid action with momentum map
\[
\rho:M\to G_0
\]
and action map
\[
\Phi:G_1\; {}_s\times_\rho M\to M.
\]
The action is cosymplectic if \(d\rho(R)=0\) and the graph of the action is a Lagrangian-Legendrean submanifold in a suitable enlarged cosymplectic manifold. The condition \(d\rho(R)=0\) guarantees that the Reeb field preserves momentum levels; the Lagrangian-Legendrean graph condition is the odd-dimensional analogue of the Lagrangian action graph in the symplectic Mikami–Weinstein theorem [2410.05846].

Under these hypotheses, if \(M\) is a cosymplectic, free and proper left \(\mathcal G\)-module and \(\xi\in \rho(M)\) is a regular value of \(\rho\), then
\[
(S_\mathcal G)_\xi\backslash \rho^{-1}(\xi)
\]
is a cosymplectic manifold, where
\[
(S_\mathcal G)_\xi=s^{-1}(\xi)\cap t^{-1}(\xi)\subset S_{G_1}
\]
is the isotropy Lie group at \(\xi\) inside the symplectic subgroupoid [2410.05846].

The mechanism is leafwise. Because \(d\rho(R)=0\), each symplectic leaf \(S_i\) of the foliation \(\ker\eta\) intersects \(\rho^{-1}(\xi)\) transversely. Proposition 3.2 of the paper shows that the symplectic subgroupoid \(S_\mathcal G\) acts on each leaf \(S_i\) symplectically. One then applies ordinary Mikami–Weinstein reduction leafwise,
\[
S_i^\xi:=(S_\mathcal G)_\xi\backslash (\rho|_{S_i})^{-1}(\xi),
\]
obtaining a codimension-\(1\) foliation of the total reduced space. The Reeb field descends to \(R^\xi\), and the reduced forms are characterized by
\[
\pi^\ast\eta^\xi=\eta,\qquad \pi^\ast\omega^\xi=\omega.
\]
In this way the reduced cosymplectic structure is reconstructed from reduced leafwise symplectic geometry together with the descended transverse direction [2410.05846].

This theorem recovers Albert’s cosymplectic reduction for Lie-group actions as a special case by using the trivial \(\mathbb R\)-central extension
\[
T^\ast G\times \mathbb R \simeq G\times \mathfrak g^\ast\times \mathbb R \rightrightarrows \mathfrak g^\ast.
\]
It therefore generalizes both Albert’s cosymplectic reduction and Mikami–Weinstein’s symplectic groupoid reduction [2410.05846].

## 4. Higher-dimensional extensions

Several later works generalize the cosymplectic Marsden–Weinstein process to multi-time or multi-form settings. These extensions preserve the same basic scheme—momentum level, quotient, descended forms—but replace the single pair \((\eta,\omega)\) by families of closed \(1\)- and \(2\)-forms.

| Setting | Reduced space | Reduced structure |
|---|---|---|
| Cosymplectic Lie-group reduction [2302.05827] | \(M_\mu^\Delta=\mathbf J^{\Phi-1}(\mu)/G_\mu^\Delta\) | \(\iota_\mu^*\omega=\pi_\mu^*\omega_\mu,\ \iota_\mu^*\eta=\pi_\mu^*\eta_\mu\) |
| Cosymplectic groupoid reduction [2410.05846] | \((S_\mathcal G)_\xi\backslash \rho^{-1}(\xi)\) | \(\pi^\ast\eta^\xi=\eta,\ \pi^\ast\omega^\xi=\omega\) |
| \(k\)-polycosymplectic reduction [2302.09037] | \(M_\mu:=J^{-1}(\mu)/G_\mu\) | \(\pi_\mu^*\tau_\mu=j_\mu^*\tau,\ \pi_\mu^*\omega_\mu=j_\mu^*\omega\) |
| \(q\)-cosymplectic reduction [2509.05998] | \(M_\mu^\Delta=J^{\Phi-1}(\mu)/G_\mu^\Delta\) | \(i_\mu^*\Omega=\pi_\mu^*\Omega_\mu,\ i_\mu^*\lambda_i=\pi_\mu^*\lambda_{i\mu}\) |

In \(k\)-polycosymplectic geometry, the structure is a pair
\[
(\tau,\omega),\qquad \tau\in \Omega^1(M,\mathbb R^k),\quad \omega\in\Omega^2(M,\mathbb R^k),
\]
with closed components, \(\operatorname{rank}\ker\omega=k\), and \(\ker\omega\cap\ker\tau=0\). The main reduction theorem is obtained by passing to the fibred \(k\)-polysymplectic manifold
\[
(\mathbb R^k\times M,\ \boldsymbol\Omega:=\operatorname{pr}_M^*\omega + du\wedge \operatorname{pr}_M^*\tau),
\]
performing \(k\)-polysymplectic reduction there, and identifying the quotient as again fibred. If \(\mu\) is a weak regular value and the relevant kernel conditions hold, then
\[
M_\mu:=J^{-1}(\mu)/G_\mu
\]
inherits a reduced \(k\)-polycosymplectic structure characterized by
\[
\pi_\mu^*\tau_\mu = j_\mu^*\tau,\qquad \pi_\mu^*\omega_\mu = j_\mu^*\omega.
\]
The same framework yields reduction of Hamiltonian \(k\)-vector fields and the reduced Hamilton–De Donder–Weyl equations [2302.09037].

A closely related polycosymplectic theorem gives a necessary-and-sufficient criterion for polycosymplectic reduction and recovers Albert’s theorem as the case \(k=1\). In that setting the reduced forms are determined by
\[
\pi_\mu^*\omega_\mu^a=j_\mu^*\omega^a,\qquad \pi_\mu^*\eta_\mu^a=j_\mu^*\eta^a,
\]
and the reduction criterion is
\[
\mathcal R_x\oplus \mathfrak g_{\mu\,M}(x)= \mathfrak g_M(x)^\omega\cap \mathfrak g_M(x)^{\omega\omega},
\]
where \(\mathcal R_x\) is the Reeb distribution. For ordinary cosymplectic reduction this condition is automatic, which explains why the \(k=1\) theory has the same clean form as Albert’s theorem [2303.16037].

The \(q\)-cosymplectic theory provides a different multitime extension. A \(q\)-cosymplectic structure on a manifold of dimension \(2n+q\) consists of a closed \(2\)-form \(\Omega\), closed \(1\)-forms
\[
\vec\lambda=(\lambda_1,\dots,\lambda_q),
\]
and a splitting
\[
TM=\mathcal R\oplus \xi
\]
with \(\ker\Omega=\mathcal R=\operatorname{Span}\{R_1,\dots,R_q\}\). For a restricted Hamiltonian action, if \(\mu\) is a weakly regular value and the quotient is smooth, then there exists a unique reduced \(q\)-cosymplectic structure
\[
(M_\mu^\Delta,\Omega_\mu,\vec\lambda_\mu)
\]
such that
\[
i_\mu^*\Omega=\pi_\mu^*\Omega_\mu,\qquad i_\mu^*\lambda_i=\pi_\mu^*\lambda_{i\mu}.
\]
When \(q=1\), this is precisely ordinary cosymplectic reduction [2509.05998].

## 5. Dynamical role and applications

The cosymplectic Marsden–Weinstein process is not only a structural theorem; it is a reduction procedure for nonautonomous Hamiltonian dynamics. In the basic Lie-group formulation, it preserves the time-dependent Hamiltonian picture: if \(h\) is \(G\)-invariant, then the reduced quotient carries a reduced Hamiltonian \(k_\mu\) and the projected dynamics is the reduced evolution field \(E_{k_\mu}\) [2302.05827].

This dynamical interpretation is especially explicit on manifolds of the form
\[
M=T\times P.
\]
When \(T\) is connected and \(RJ_\xi=0\), the momentum components are basic with respect to the projection to \(P\), so the momentum map is effectively time-independent. The reduction then preserves the explicit time factor:
\[
\mathbf J^{\Phi-1}(\mu)\simeq T\times \pi_P(\mathbf J^{\Phi-1}(\mu)),\qquad
M_\mu^\Delta\simeq T\times P_\mu^\Delta.
\]
This is one of the reasons the cosymplectic framework is natural for time-dependent Hamiltonian systems [2302.05827].

The literature supplies several concrete applications. For the phase action of \(SO_2\simeq U_1\) on the two-level Schrödinger system \((\mathbb R\times \mathbb C^2,\omega_S,\eta_S)\), the momentum level
\[
\mathbf J^{\Phi-1}(\mu)=\mathbb R\times A_\mu,\qquad A_\mu\simeq S^3,
\]
reduces for \(\mu\neq 0\) to
\[
\mathbf J^{\Phi-1}(\mu)/G_\mu \simeq \mathbb R\times S^2,
\]
with reduced forms
\[
\eta_\mu=dt,\qquad
\omega_\mu=\mu\sin(2\varphi)\,d\varphi\wedge d(\theta_1-\theta_2).
\]
For the \(n\)-level system, one obtains
\[
M_\mu^\Delta\simeq \mathbb R\times \mathbb{P}\mathbb C^n.
\]
In these examples, the reduced equilibria encode phase-orbit solutions of the Schrödinger equation [2302.05827].

Field-theoretic generalizations produce equally explicit reduced systems. In the \(k\)-polycosymplectic model of two coupled vibrating strings, the translation symmetry in the variable \(q^1+q^2\) has momentum map
\[
J=(p_1^1+p_2^1,\ p_1^2+p_2^2),
\]
and the quotient \(J^{-1}(\mu)/G\) carries reduced structure
\[
\tau_\mu = dt\otimes e_1 + dx\otimes e_2,\qquad
\omega_\mu = dq\wedge dp^1\otimes e_1 + dq\wedge dp^2\otimes e_2.
\]
The reduced Hamilton–De Donder–Weyl equations are then written explicitly in the reduced variables [2302.09037].

The \(q\)-cosymplectic framework is designed for multitime dynamics. Its basic fast-slow \(2\)-cosymplectic model uses
\[
\lambda_1=dt,\qquad \lambda_2=d\tau,\qquad \Omega=dq\wedge dp+dQ\wedge dP,
\]
with Reeb fields \(R_1=\partial_t\), \(R_2=\partial_\tau\). In the constant-frequency case, reduction by an \(S^1\)-symmetry removes the fast oscillator angle and leaves a reduced structure
\[
\Omega_{\mathrm{red}}=dQ\wedge dP,\qquad
\lambda_{1,\mathrm{red}}=dt,\qquad
\lambda_{2,\mathrm{red}}=d\tau.
\]
The same paper presents a variable-frequency version in which the reduced time sector again survives unchanged [2509.05998].

## 6. Limitations and adjacent frameworks

A major controversy concerns the scope of Albert’s cosymplectic theorem for time-dependent Hamiltonian systems. One recent paper argues that Albert’s condition
\[
\eta(\xi_M)=0,\qquad \forall \xi\in\mathfrak g,
\]
is too restrictive for general symmetric time-dependent dynamics, because many natural symmetries mix space and time and therefore have a nonzero time component. In that analysis, cosymplectic reduction is said to be “not appropriate” for the reduction of general symmetric time-dependent Hamiltonian systems, and the proposed replacement is a Marsden–Weinstein theory for mechanical presymplectic structures \((\omega,\mathcal R)\), where \(\omega\) is a closed \(2\)-form of corank \(1\) and \(\ker\omega=\langle \mathcal R\rangle\) [2411.11997]. This does not negate the classical cosymplectic theorem; it limits its dynamical range.

A related point is that neighboring reduction theories are sometimes described as “cosymplectic” only in a loose or indirect sense. In “symplectic reduction along a submanifold,” the phrase “Poisson transversal (or sometimes a cosymplectic submanifold)” is used for a special case in which the preimage \(\mu^{-1}(S)\) is already symplectic, but the paper does not formulate a cosymplectic Marsden–Weinstein theorem [2107.03198]. Likewise, exact symplectic/contact reduction on energy hypersurfaces studies compatibility between exact symplectic and contact reduction, not genuine reduction of a closed pair \((\eta,\Omega)\) with \(d\eta=0\) [2511.16607].

The same caution applies to derived reduction. Derived symplectic reduction in algebraic geometry proves that
\[
\bigl(X\times_{\mathfrak g^*}^h \mathcal O\bigr)/G
\]
carries a \(0\)-shifted symplectic form, but the paper explicitly contains no cosymplectic geometry, no Reeb field, and no reduction theorem for pairs \((\eta,\omega)\) [1205.6519]. The same is true for the differential-geometric dg-groupoid approach to derived symplectic reduction, which constructs a reduced closed and nondegenerate derived \(2\)-form but does not formulate a cosymplectic analogue [2605.16226]. These theories are methodologically suggestive, but they are not themselves instances of the cosymplectic Marsden–Weinstein process.

In this sense, the expression “cosymplectic Marsden–Weinstein process” names a family of reduction procedures with a stable core: momentum constraints are imposed on a cosymplectic or cosymplectic-type structure, orbit directions are quotiented out, the Reeb direction must survive in a controlled way, and the reduced quotient inherits the same geometric type. What varies across the literature is the precise category—Lie groups, Lie groupoids, \(k\)-polycosymplectic manifolds, \(q\)-cosymplectic manifolds, or mechanical presymplectic replacements—and the extent to which time-dependence, multitime variables, or non-\(\mathrm{Ad}^*\)-equivariant momentum maps are built into the theorem [2302.05827], [2410.05846].

Source: https://www.emergentmind.com/topics/cosymplectic-marsden-weinstein-process