Cosymplectic Marsden–Weinstein Process

• The cosymplectic Marsden–Weinstein process is the odd-dimensional analogue of symplectic reduction, applying to time-dependent Hamiltonian systems with a preserved time direction.
• It employs a momentum map and symmetry actions (via Lie groups or groupoids) that leave the closed 1- and 2-forms invariant, ensuring the reduced manifold inherits a cosymplectic structure and descended Reeb field.
• Higher-dimensional extensions generalize the process to k-polycosymplectic and q-cosymplectic settings, broadening its applications to multi-time dynamics and complex field theories.

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 (Lucas et al., 2023, Yonehara, 2024).

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 $k$0 is uniquely characterized by

$k$1

Equivalent formulations used in the literature include the bundle isomorphism

$k$2

and the decomposition

$k$3

with $k$4 one-dimensional and $k$5 of rank $k$6 (Lucas et al., 2023).

This structure is the standard geometric model for time-dependent Hamiltonian systems. If $k$7, with $k$8 one-dimensional and $k$9 symplectic, then pulling back $q$0 from $q$1 and a nonvanishing closed $q$2-form $q$3 from $q$4 yields a cosymplectic structure. In Darboux coordinates,

$q$5

and the evolution vector field

$q$6

encodes the usual time-dependent Hamilton equations (Lucas et al., 2023).

A complementary description emphasizes the foliation defined by $q$7. Because $q$8 is closed, $q$9 is integrable, and its leaves are symplectic with symplectic form $(2n+1)$0. Equivalently, a cosymplectic manifold carries a regular Poisson structure of corank $(2n+1)$1, with symplectic foliation equal to the foliation of $(2n+1)$2, together with a transverse Poisson vector field, namely the Reeb field (Yonehara, 2024). 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 $(2n+1)$3, the cosymplectic Marsden–Weinstein process begins with a cosymplectic action,

$(2n+1)$4

and, in the connected case, infinitesimally

$(2n+1)$5

A key restriction is that the fundamental vector fields lie in $(2n+1)$6,

$(2n+1)$7

so the symmetry does not move the time variable (Lucas et al., 2023).

A cosymplectic momentum map is a map

$(2n+1)$8

such that, for every $(2n+1)$9,

$M$0

The additional condition $M$1 is specific to the cosymplectic setting; it guarantees that the Reeb field is tangent to each momentum level set (Lucas et al., 2023).

In the refined formulation, the momentum map need not be $M$2-equivariant. Its defect is measured by the cocycle

$M$3

which yields an associated affine action $M$4 on $M$5. The relevant isotropy subgroup is then

$M$6

not necessarily the usual coadjoint stabilizer. If $M$7 is a weakly regular value and

$M$8

is a manifold with quotient projection

$M$9

a submersion, then there exists a unique cosymplectic manifold

$1$0

such that

$1$1

where $1$2 is the inclusion (Lucas et al., 2023).

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 $1$3 and $1$4; and the Reeb field projects to a reduced Reeb field $1$5 satisfying

$1$6

A key linear-algebra identity is

$1$7

which isolates the additional one-dimensional direction absent in ordinary symplectic reduction (Lucas et al., 2023).

The process also reduces dynamics. If $1$8 is $1$9-invariant, then the evolution vector field $\eta$0 is tangent to momentum levels, descends to the quotient, and there is a unique reduced Hamiltonian $\eta$1 such that

$\eta$2

Thus the reduction concerns not only the pair $\eta$3 but the full time-dependent Hamiltonian dynamics (Lucas et al., 2023).

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

$\eta$4

whose arrow manifold carries a multiplicative cosymplectic structure: $\eta$5 A basic structural fact is that all unit arrows lie in the same symplectic leaf $\eta$6, and

$\eta$7

is a symplectic subgroupoid (Yonehara, 2024).

The action datum is a left groupoid action with momentum map

$\eta$8

and action map

$\eta$9

The action is cosymplectic if $2$0 and the graph of the action is a Lagrangian-Legendrean submanifold in a suitable enlarged cosymplectic manifold. The condition $2$1 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 (Yonehara, 2024).

Under these hypotheses, if $2$2 is a cosymplectic, free and proper left $2$3-module and $2$4 is a regular value of $2$5, then

$2$6

is a cosymplectic manifold, where

$2$7

is the isotropy Lie group at $2$8 inside the symplectic subgroupoid (Yonehara, 2024).

The mechanism is leafwise. Because $2$9, each symplectic leaf $\omega$0 of the foliation $\omega$1 intersects $\omega$2 transversely. Proposition 3.2 of the paper shows that the symplectic subgroupoid $\omega$3 acts on each leaf $\omega$4 symplectically. One then applies ordinary Mikami–Weinstein reduction leafwise,

$\omega$5

obtaining a codimension-$\omega$6 foliation of the total reduced space. The Reeb field descends to $\omega$7, and the reduced forms are characterized by

$\omega$8

In this way the reduced cosymplectic structure is reconstructed from reduced leafwise symplectic geometry together with the descended transverse direction (Yonehara, 2024).

This theorem recovers Albert’s cosymplectic reduction for Lie-group actions as a special case by using the trivial $\omega$9-central extension

$$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$0

It therefore generalizes both Albert’s cosymplectic reduction and Mikami–Weinstein’s symplectic groupoid reduction (Yonehara, 2024).

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 $$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$1 by families of closed $$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$2- and $$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$3-forms.

Setting	Reduced space	Reduced structure
Cosymplectic Lie-group reduction (Lucas et al., 2023)	$$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$4	$$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$5
Cosymplectic groupoid reduction (Yonehara, 2024)	$$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$6	$$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$7
$$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$8-polycosymplectic reduction (Lucas et al., 2023)	$$d\eta=0,\qquad d\omega=0,\qquad \eta\wedge \omega^n\neq 0.$$9	$k$00
$k$01-cosymplectic reduction (Leok et al., 7 Sep 2025)	$k$02	$k$03

In $k$04-polycosymplectic geometry, the structure is a pair

$k$05

with closed components, $k$06, and $k$07. The main reduction theorem is obtained by passing to the fibred $k$08-polysymplectic manifold

$k$09

performing $k$10-polysymplectic reduction there, and identifying the quotient as again fibred. If $k$11 is a weak regular value and the relevant kernel conditions hold, then

$k$12

inherits a reduced $k$13-polycosymplectic structure characterized by

$k$14

The same framework yields reduction of Hamiltonian $k$15-vector fields and the reduced Hamilton–De Donder–Weyl equations (Lucas et al., 2023).

A closely related polycosymplectic theorem gives a necessary-and-sufficient criterion for polycosymplectic reduction and recovers Albert’s theorem as the case $k$16. In that setting the reduced forms are determined by

$k$17

and the reduction criterion is

$k$18

where $k$19 is the Reeb distribution. For ordinary cosymplectic reduction this condition is automatic, which explains why the $k$20 theory has the same clean form as Albert’s theorem (Andrés et al., 2023).

The $k$21-cosymplectic theory provides a different multitime extension. A $k$22-cosymplectic structure on a manifold of dimension $k$23 consists of a closed $k$24-form $k$25, closed $k$26-forms

$k$27

and a splitting

$k$28

with $k$29. For a restricted Hamiltonian action, if $k$30 is a weakly regular value and the quotient is smooth, then there exists a unique reduced $k$31-cosymplectic structure

$k$32

such that

$k$33

When $k$34, this is precisely ordinary cosymplectic reduction (Leok et al., 7 Sep 2025).

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 $k$35 is $k$36-invariant, then the reduced quotient carries a reduced Hamiltonian $k$37 and the projected dynamics is the reduced evolution field $k$38 (Lucas et al., 2023).

This dynamical interpretation is especially explicit on manifolds of the form

$k$39

When $k$40 is connected and $k$41, the momentum components are basic with respect to the projection to $k$42, so the momentum map is effectively time-independent. The reduction then preserves the explicit time factor: $k$43 This is one of the reasons the cosymplectic framework is natural for time-dependent Hamiltonian systems (Lucas et al., 2023).

The literature supplies several concrete applications. For the phase action of $k$44 on the two-level Schrödinger system $k$45, the momentum level

$k$46

reduces for $k$47 to

$k$48

with reduced forms

$k$49

For the $k$50-level system, one obtains

$k$51

In these examples, the reduced equilibria encode phase-orbit solutions of the Schrödinger equation (Lucas et al., 2023).

Field-theoretic generalizations produce equally explicit reduced systems. In the $k$52-polycosymplectic model of two coupled vibrating strings, the translation symmetry in the variable $k$53 has momentum map

$k$54

and the quotient $k$55 carries reduced structure

$k$56

The reduced Hamilton–De Donder–Weyl equations are then written explicitly in the reduced variables (Lucas et al., 2023).

The $k$57-cosymplectic framework is designed for multitime dynamics. Its basic fast-slow $k$58-cosymplectic model uses

$k$59

with Reeb fields $k$60, $k$61. In the constant-frequency case, reduction by an $k$62-symmetry removes the fast oscillator angle and leaves a reduced structure

$k$63

The same paper presents a variable-frequency version in which the reduced time sector again survives unchanged (Leok et al., 7 Sep 2025).

6. Limitations and adjacent frameworks

A major controversy concerns the scope of Albert’s cosymplectic theorem for time-dependent Hamiltonian systems. One paper argues that Albert’s condition

$k$64

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 $k$65, where $k$66 is a closed $k$67-form of corank $k$68 and $k$69 (Gutierrez-Sagredo et al., 2024). 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 $k$70 is already symplectic, but the paper does not formulate a cosymplectic Marsden–Weinstein theorem (Crooks et al., 2021). Likewise, exact symplectic/contact reduction on energy hypersurfaces studies compatibility between exact symplectic and contact reduction, not genuine reduction of a closed pair $k$71 with $k$72 (Lange et al., 20 Nov 2025).

The same caution applies to derived reduction. Derived symplectic reduction in algebraic geometry proves that

$k$73

carries a $k$74-shifted symplectic form, but the paper explicitly contains no cosymplectic geometry, no Reeb field, and no reduction theorem for pairs $k$75 (Pecharich, 2012). The same is true for the differential-geometric dg-groupoid approach to derived symplectic reduction, which constructs a reduced closed and nondegenerate derived $k$76-form but does not formulate a cosymplectic analogue (Sheshko, 15 May 2026). 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$77-polycosymplectic manifolds, $k$78-cosymplectic manifolds, or mechanical presymplectic replacements—and the extent to which time-dependence, multitime variables, or non-$k$79-equivariant momentum maps are built into the theorem (Lucas et al., 2023, Yonehara, 2024).