k-Cosymplectic Geometry in Field Theories

• k-Cosymplectic geometry is a framework extending time-dependent mechanics to multi-time field theories, characterized by k closed 1-forms, k closed 2-forms, and an integrable foliation.
• It provides unified Hamiltonian and Lagrangian formulations through local Darboux coordinates, leading to field equations like the Hamilton–De Donder–Weyl equations.
• The structure interconnects with k-symplectic, multisymplectic, and cocontact geometries, supporting applications in various physical field theories and reduction methods.

k-cosymplectic geometry is the finite-dimensional geometric framework on $\mathbb{R}^k \times (T^1_k)^*Q$ and $\mathbb{R}^k \times T^1_k Q$ that extends symplectic and cosymplectic mechanics to first-order classical field theories with explicit dependence on the base variables $x^1,\dots,x^k$ (León et al., 2014). It is the natural multi-time analogue of ordinary cosymplectic geometry from time-dependent mechanics, and it stands to $k$-symplectic geometry as the non-autonomous theory stands to the autonomous one (Cappelletti-Montano et al., 2013). In the standard field-theoretic formulation, a $k$-cosymplectic manifold has dimension $(k+1)n+k$ and carries $k$ closed $1$-forms, $k$ closed $2$-forms, and an integrable $\mathbb{R}^k \times T^1_k Q$0-dimensional distribution, with local coordinates adapted to independent variables, field coordinates, and polymomenta (Cappelletti-Montano et al., 2013).

1. Standard definition and local normal form

A $\mathbb{R}^k \times T^1_k Q$1-cosymplectic manifold is a smooth manifold of dimension $\mathbb{R}^k \times T^1_k Q$2 endowed with a family

$\mathbb{R}^k \times T^1_k Q$3

where each $\mathbb{R}^k \times T^1_k Q$4 is a closed $\mathbb{R}^k \times T^1_k Q$5-form, each $\mathbb{R}^k \times T^1_k Q$6 is a closed $\mathbb{R}^k \times T^1_k Q$7-form, and $\mathbb{R}^k \times T^1_k Q$8 is an $\mathbb{R}^k \times T^1_k Q$9-dimensional foliation, subject to the conditions

$x^1,\dots,x^k$0

$x^1,\dots,x^k$1

and

$x^1,\dots,x^k$2

These conditions generalize the cosymplectic pattern from one distinguished time form and one closed $x^1,\dots,x^k$3-form to $x^1,\dots,x^k$4 time forms, $x^1,\dots,x^k$5 closed $x^1,\dots,x^k$6-forms, and a characteristic foliation adapted to field-theoretic directions (Cappelletti-Montano et al., 2013).

The canonical model is the stable cotangent bundle of $x^1,\dots,x^k$7-covelocities,

$x^1,\dots,x^k$8

with coordinates $x^1,\dots,x^k$9, dimension

$k$0

canonical $k$1-forms

$k$2

canonical $k$3-forms

$k$4

and vertical distribution

$k$5

These satisfy $k$6 and $k$7, and they provide the model for the Darboux theorem in the theory (León et al., 2014).

The local normal form is correspondingly simple. Around every point there are coordinates $k$8 such that

$k$9

Equivalently, in the survey notation one may use coordinates $k$0 with

$k$1

This local standardization is the direct multi-variable analogue of Darboux coordinates in ordinary cosymplectic geometry (Cappelletti-Montano et al., 2013).

Associated with the structure are $k$2 Reeb vector fields $k$3, uniquely defined by

$k$4

In canonical coordinates on $k$5, these are simply

$k$6

They play the role of the distinguished base-space directions in the geometry (Cappelletti-Montano et al., 2013).

2. Hamiltonian and Lagrangian formalisms

The Hamiltonian side is built on a $k$7-cosymplectic Hamiltonian system

$k$8

or, in canonical situations, on $k$9 with Hamiltonian

$(k+1)n+k$0

The fundamental geometric field equation is written for a $(k+1)n+k$1-vector field

$(k+1)n+k$2

as

$(k+1)n+k$3

In local coordinates, if

$(k+1)n+k$4

then one obtains

$(k+1)n+k$5

For an integral section $(k+1)n+k$6, these become the Hamilton–De Donder–Weyl equations

$(k+1)n+k$7

This is the standard $(k+1)n+k$8-cosymplectic Hamiltonian encoding of first-order field equations (León et al., 2014).

The Lagrangian side is formulated on

$(k+1)n+k$9

with coordinates $k$0 and Lagrangian

$k$1

The canonical tensors are

$k$2

From $k$3 one defines the Poincaré–Cartan forms

$k$4

with coordinate expressions

$k$5

and the energy

$k$6

The geometric Euler–Lagrange equation is

$k$7

which yields the classical first-order PDEs

$k$8

Regularity is expressed by invertibility of the Hessian

$k$9

and for a hyperregular Lagrangian the Legendre map

$1$0

is a global diffeomorphism. In that case the Hamiltonian and Lagrangian formalisms are equivalent, with

$1$1

(León et al., 2014).

3. Hamilton–Jacobi theory, symmetries, and conservation laws

The Hamilton–Jacobi problem in the $1$2-cosymplectic framework is formulated on

$1$3

by means of a section

$1$4

of the bundle $1$5, where locally

$1$6

and each $1$7 is closed along the fibers in the sense that

$1$8

Given an integrable Hamiltonian $1$9-vector field $k$0, one defines the induced $k$1-vector field

$k$2

on $k$3. The central theorem states that the following are equivalent: if $k$4 is an integral section of $k$5, then $k$6 is an integral section of $k$7; and

$k$8

If locally $k$9, then the Hamilton–Jacobi relation is written in the compact form

$2$0

and one can set $2$1 because subtracting a function of $2$2 alone does not change the Hamiltonian field equations (León et al., 2013).

The symmetry theory developed for almost-standard $2$3-cosymplectic manifolds $2$4 isolates a class of structure-preserving symmetries adapted to Hamiltonian field theory. A diffeomorphism $2$5 is a $2$6-cosymplectic Noether symmetry if

$2$7

and in the preferred formulation one further requires

$2$8

so that $2$9 acts fiberwise over $\mathbb{R}^k \times T^1_k Q$00. The infinitesimal version is a vector field $\mathbb{R}^k \times T^1_k Q$01 such that

$\mathbb{R}^k \times T^1_k Q$02

The condition $\mathbb{R}^k \times T^1_k Q$03 means that $\mathbb{R}^k \times T^1_k Q$04 has no $\mathbb{R}^k \times T^1_k Q$05 component (Marrero et al., 2010).

A conservation law is a map

$\mathbb{R}^k \times T^1_k Q$06

such that, for every solution $\mathbb{R}^k \times T^1_k Q$07,

$\mathbb{R}^k \times T^1_k Q$08

Equivalently, for every integrable $\mathbb{R}^k \times T^1_k Q$09-vector field $\mathbb{R}^k \times T^1_k Q$10 solving the Hamiltonian equations,

$\mathbb{R}^k \times T^1_k Q$11

If $\mathbb{R}^k \times T^1_k Q$12 is an infinitesimal $\mathbb{R}^k \times T^1_k Q$13-cosymplectic Noether symmetry, then locally one can write

$\mathbb{R}^k \times T^1_k Q$14

where $\mathbb{R}^k \times T^1_k Q$15 locally and $\mathbb{R}^k \times T^1_k Q$16, and these functions satisfy

$\mathbb{R}^k \times T^1_k Q$17

This is the $\mathbb{R}^k \times T^1_k Q$18-cosymplectic version of Noether’s theorem: a symmetry preserving the geometric data and the Hamiltonian produces a conserved $\mathbb{R}^k \times T^1_k Q$19-current whose divergence vanishes along solutions. The same paper exhibits this mechanism for quadratic Hamiltonians on $\mathbb{R}^k \times T^1_k Q$20 and, for the $\mathbb{R}^k \times T^1_k Q$21-dimensional wave equation with $\mathbb{R}^k \times T^1_k Q$22, identifies the standard momentum components as a conserved current (Marrero et al., 2010).

4. Singular theories, Lie algebroids, and reduction

The regular nondegeneracy assumptions of $\mathbb{R}^k \times T^1_k Q$23-cosymplectic geometry do not cover singular field theories. For that purpose the theory of $\mathbb{R}^k \times T^1_k Q$24-precosymplectic manifolds replaces the nondegenerate $\mathbb{R}^k \times T^1_k Q$25-forms by closed forms of possibly smaller rank. A $\mathbb{R}^k \times T^1_k Q$26-precosymplectic manifold has dimension

$\mathbb{R}^k \times T^1_k Q$27

and carries

$\mathbb{R}^k \times T^1_k Q$28

such that each $\mathbb{R}^k \times T^1_k Q$29 and $\mathbb{R}^k \times T^1_k Q$30 is closed, $\mathbb{R}^k \times T^1_k Q$31, $\mathbb{R}^k \times T^1_k Q$32 is an integrable $\mathbb{R}^k \times T^1_k Q$33-dimensional distribution, and

$\mathbb{R}^k \times T^1_k Q$34

In Darboux coordinates

$\mathbb{R}^k \times T^1_k Q$35

one has

$\mathbb{R}^k \times T^1_k Q$36

Reeb vector fields still exist, but they are not unique; locally they may be written as

$\mathbb{R}^k \times T^1_k Q$37

To recover dynamics one applies a constraint algorithm, generating a sequence

$\mathbb{R}^k \times T^1_k Q$38

that stabilizes at a final constraint manifold $\mathbb{R}^k \times T^1_k Q$39 where solutions of the field equations exist (Gràcia et al., 2018).

A second major extension replaces the tangent and cotangent bundles by a Lie algebroid $\mathbb{R}^k \times T^1_k Q$40 and its dual $\mathbb{R}^k \times T^1_k Q$41. The standard spaces $\mathbb{R}^k \times T^1_k Q$42 and $\mathbb{R}^k \times T^1_k Q$43 are replaced by

$\mathbb{R}^k \times T^1_k Q$44

together with the prolongation Lie algebroids

$\mathbb{R}^k \times T^1_k Q$45

The resulting theory defines Poincaré–Cartan $\mathbb{R}^k \times T^1_k Q$46-sections $\mathbb{R}^k \times T^1_k Q$47, $\mathbb{R}^k \times T^1_k Q$48-sections $\mathbb{R}^k \times T^1_k Q$49, Hamiltonian Liouville $\mathbb{R}^k \times T^1_k Q$50-sections $\mathbb{R}^k \times T^1_k Q$51, canonical $\mathbb{R}^k \times T^1_k Q$52-sections $\mathbb{R}^k \times T^1_k Q$53, a generalized Legendre transformation

$\mathbb{R}^k \times T^1_k Q$54

and Hamiltonian and Lagrangian field equations with anchor and structure-function terms. When $\mathbb{R}^k \times T^1_k Q$55, the standard $\mathbb{R}^k \times T^1_k Q$56-cosymplectic theory is recovered, and when $\mathbb{R}^k \times T^1_k Q$57, one recovers Lie algebroid mechanics (Diego et al., 2010).

Reduction theory introduces an additional layer. In the $\mathbb{R}^k \times T^1_k Q$58-polycosymplectic setting one has a pair $\mathbb{R}^k \times T^1_k Q$59 of $\mathbb{R}^k \times T^1_k Q$60-valued closed forms with

$\mathbb{R}^k \times T^1_k Q$61

and the key equivalence states that

$\mathbb{R}^k \times T^1_k Q$62

This allows a Marsden–Weinstein reduction theory for $\mathbb{R}^k \times T^1_k Q$63-polycosymplectic manifolds and, in the $\mathbb{R}^k \times T^1_k Q$64-cosymplectic framework, a $\mathbb{R}^k \times T^1_k Q$65-cosymplectic to $\mathbb{R}^k \times T^1_k Q$66-cosymplectic geometric reduction that can reduce space-time variables (Lucas et al., 2023).

5. Relations to $\mathbb{R}^k \times T^1_k Q$67-symplectic, multisymplectic, and cocontact geometries

Within field theory, $\mathbb{R}^k \times T^1_k Q$68-cosymplectic geometry occupies the non-autonomous side of a closely related family of formalisms. The book-length treatment of the subject makes the relation explicit: $\mathbb{R}^k \times T^1_k Q$69-symplectic geometry treats first-order field theories whose Hamiltonian or Lagrangian do not depend explicitly on the base coordinates, while $\mathbb{R}^k \times T^1_k Q$70-cosymplectic geometry incorporates the base variables $\mathbb{R}^k \times T^1_k Q$71 into the geometry and is therefore suited to non-autonomous theories. In the autonomous case, when the $\mathbb{R}^k \times T^1_k Q$72-cosymplectic Hamiltonian $\mathbb{R}^k \times T^1_k Q$73 does not depend on $\mathbb{R}^k \times T^1_k Q$74, the system reduces to a $\mathbb{R}^k \times T^1_k Q$75-symplectic one, and the same reduction holds on the Lagrangian side (León et al., 2014).

The same source also places the theory in relation with multisymplectic geometry. For trivial bundles $\mathbb{R}^k \times T^1_k Q$76, the multisymplectic multimomentum bundle becomes diffeomorphic to

$\mathbb{R}^k \times T^1_k Q$77

and the multisymplectic Hamiltonian and Lagrangian forms decompose into the $\mathbb{R}^k \times T^1_k Q$78-cosymplectic data

$\mathbb{R}^k \times T^1_k Q$79

In this trivial-bundle case, the local Hamilton–De Donder–Weyl equations in the multisymplectic formalism reduce to the same local PDEs as in the $\mathbb{R}^k \times T^1_k Q$80-cosymplectic formalism (León et al., 2014).

More recent work extends the conservative framework to non-conservative field theories. $\mathbb{R}^k \times T^1_k Q$81-cocontact geometry combines $\mathbb{R}^k \times T^1_k Q$82-cosymplectic space-time variables with $\mathbb{R}^k \times T^1_k Q$83-contact dissipation variables. Its canonical model is

$\mathbb{R}^k \times T^1_k Q$84

with Darboux coordinates $\mathbb{R}^k \times T^1_k Q$85 and canonical local form

$\mathbb{R}^k \times T^1_k Q$86

The paper introducing this structure emphasizes that it extends $\mathbb{R}^k \times T^1_k Q$87-cosymplectic geometry by allowing dissipation via contact-type forms $\mathbb{R}^k \times T^1_k Q$88, and illustrates the formalism with a nonlinear damped wave equation with time-dependent forcing (Rivas, 2022).

The scope of the theory is correspondingly broad. The $\mathbb{R}^k \times T^1_k Q$89-cosymplectic approach has been used as a geometric language for electrostatics, the wave equation, Laplace equation, sine-Gordon equation, Ginzburg-Landau equation, massive scalar field, harmonic maps, and Maxwell equations, all within the finite-dimensional formalism on $\mathbb{R}^k \times T^1_k Q$90 and $\mathbb{R}^k \times T^1_k Q$91 (León et al., 2014).

6. Alternative usages of the term and current directions

The term “$\mathbb{R}^k \times T^1_k Q$92-cosymplectic” also appears in a different, non-field-theoretic sense in the study of log symplectic manifolds. In that usage, a $\mathbb{R}^k \times T^1_k Q$93-cosymplectic structure on a $\mathbb{R}^k \times T^1_k Q$94-dimensional manifold $\mathbb{R}^k \times T^1_k Q$95 is a family

$\mathbb{R}^k \times T^1_k Q$96

consisting of $\mathbb{R}^k \times T^1_k Q$97 closed $\mathbb{R}^k \times T^1_k Q$98-forms $\mathbb{R}^k \times T^1_k Q$99 and one closed $x^1,\dots,x^k$00-form $x^1,\dots,x^k$01 such that

$x^1,\dots,x^k$02

Locally, any such manifold admits coordinates

$x^1,\dots,x^k$03

for which

$x^1,\dots,x^k$04

In the paper on partitionable log symplectic manifolds, these structures arise on intersections of divisor hypersurfaces as the residue geometry induced by the ambient log symplectic form (Lanius, 2016).

This alternative usage should be distinguished from the field-theoretic $x^1,\dots,x^k$05-cosymplectic structure with $x^1,\dots,x^k$06 closed $x^1,\dots,x^k$07-forms, $x^1,\dots,x^k$08 closed $x^1,\dots,x^k$09-forms, and a polarization or vertical distribution. The two notions are related by a common “cosymplectic directions plus symplectic directions” intuition, but they are not the same formal object. The log-symplectic usage generalizes the ordinary cosymplectic pattern by taking $x^1,\dots,x^k$10, whereas the field-theoretic usage generalizes time-dependent Hamiltonian mechanics to $x^1,\dots,x^k$11 independent variables (Lanius, 2016).

A further active direction is suggested by locally conformally cosymplectic geometry. In the $x^1,\dots,x^k$12 case, an LCC manifold $x^1,\dots,x^k$13 satisfies

$x^1,\dots,x^k$14

with closed Lee form $x^1,\dots,x^k$15, and supports a Hamiltonian and Hamilton–Jacobi theory based on the Lichnerowicz–de Rham differential $x^1,\dots,x^k$16. The authors explicitly remark that locally conformal $x^1,\dots,x^k$17-cosymplectic geometry is not yet developed in the literature and indicate it as a natural future direction (Ateşli et al., 2022).

Taken together, these developments present $x^1,\dots,x^k$18-cosymplectic geometry as a central structure in the differential-geometric study of first-order classical field theories, with a mature Hamiltonian and Lagrangian formalism, Hamilton–Jacobi and Noether theories, singular and reduced variants, and a network of relations to $x^1,\dots,x^k$19-symplectic, multisymplectic, contact, cocontact, and log-symplectic geometries (León et al., 2014).