---
title: k-Cosymplectic Geometry in Field Theories
url: https://www.emergentmind.com/topics/k-cosymplectic-geometry
type: topic
---

# k-Cosymplectic Geometry in Field Theories

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 \(x^1,\dots,x^k\) [1409.5604]. 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 [1305.3704]. 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 \(nk\)-dimensional distribution, with local coordinates adapted to independent variables, field coordinates, and polymomenta [1305.3704].

## 1. Standard definition and local normal form

A \(k\)-cosymplectic manifold is a smooth manifold of dimension \((k+1)n+k\) endowed with a family
\[
(\eta_\alpha,\omega_\alpha,\mathcal F)_{\alpha=1}^k
\]
where each \(\eta_\alpha\) is a closed \(1\)-form, each \(\omega_\alpha\) is a closed \(2\)-form, and \(\mathcal F\) is an \(nk\)-dimensional foliation, subject to the conditions
\[
\eta_1\wedge\cdots\wedge \eta_k\neq 0,
\]
\[
\eta_\alpha|_{T\mathcal F}=0,\qquad \omega_\alpha|_{T\mathcal F}=0,
\]
and
\[
\left(\bigcap_{\alpha=1}^k \ker \eta_\alpha\right)\cap \left(\bigcap_{\alpha=1}^k \ker \omega_\alpha\right)=\{0\},\qquad \dim\left(\bigcap_{\alpha=1}^k \ker\omega_\alpha\right)=k.
\]
These conditions generalize the cosymplectic pattern from one distinguished time form and one closed \(2\)-form to \(k\) time forms, \(k\) closed \(2\)-forms, and a characteristic foliation adapted to field-theoretic directions [1305.3704].

The canonical model is the stable cotangent bundle of \(k^1\)-covelocities,
\[
\mathbb{R}^k \times (T^1_k)^*Q \cong \mathbb{R}^k \times \underbrace{T^*Q\oplus\cdots\oplus T^*Q}_{k\ \text{copies}},
\]
with coordinates \((x^\alpha,q^i,p_i^\alpha)\), dimension
\[
\dim\big(\mathbb{R}^k \times (T^1_k)^*Q\big)=k+n(k+1),
\]
canonical \(1\)-forms
\[
\eta^\alpha=dx^\alpha,\qquad \theta^\alpha=p_i^\alpha\,dq^i,
\]
canonical \(2\)-forms
\[
\omega^\alpha=-d\theta^\alpha=dq^i\wedge dp_i^\alpha,
\]
and vertical distribution
\[
V=\operatorname{span}\left\{\frac{\partial}{\partial p_i^\alpha}\right\}.
\]
These satisfy \(d\eta^\alpha=0\) and \(d\omega^\alpha=0\), and they provide the model for the Darboux theorem in the theory [1409.5604].

The local normal form is correspondingly simple. Around every point there are coordinates \((x^\alpha,q^i,p_i^\alpha)\) such that
\[
\eta^\alpha = dx^\alpha,\qquad \omega^\alpha = dq^i\wedge dp_i^\alpha,\qquad V=\operatorname{span}\left\{\frac{\partial}{\partial p_i^\alpha}\right\}.
\]
Equivalently, in the survey notation one may use coordinates \((x_\alpha,y_i,z_{\alpha i})\) with
\[
\eta_\alpha = dx_\alpha,\qquad \omega_\alpha=\sum_{i=1}^n dy_i\wedge dz_{\alpha i},\qquad
T\mathcal F=\operatorname{span}\left\{\frac{\partial}{\partial z_{\alpha i}}\right\}.
\]
This local standardization is the direct multi-variable analogue of Darboux coordinates in ordinary cosymplectic geometry [1305.3704].

Associated with the structure are \(k\) Reeb vector fields \(\xi_1,\dots,\xi_k\), uniquely defined by
\[
i_{\xi_\alpha}\eta_\beta=\delta_{\alpha\beta},\qquad i_{\xi_\alpha}\omega_\beta=0.
\]
In canonical coordinates on \(\mathbb{R}^k\times (T^1_k)^*Q\), these are simply
\[
R_\alpha=\frac{\partial}{\partial x^\alpha}.
\]
They play the role of the distinguished base-space directions in the geometry [1305.3704].

## 2. Hamiltonian and Lagrangian formalisms

The Hamiltonian side is built on a \(k\)-cosymplectic Hamiltonian system
\[
(M,\eta^\alpha,\omega^\alpha,H),
\]
or, in canonical situations, on \(\mathbb{R}^k \times (T^1_k)^*Q\) with Hamiltonian
\[
H=H(x^\alpha,q^i,p_i^\alpha).
\]
The fundamental geometric field equation is written for a \(k\)-vector field
\[
X=(X_1,\dots,X_k)
\]
as
\[
\iota_{X_\alpha}\omega^\alpha = dH - \sum_{\alpha=1}^k R_\alpha(H)\,\eta^\alpha,\qquad
\eta^\alpha(X_\beta)=\delta^\alpha_\beta.
\]
In local coordinates, if
\[
X_\alpha = \frac{\partial}{\partial x^\alpha} + (X_\alpha)^i \frac{\partial}{\partial q^i} + (X_\alpha)_i^\beta \frac{\partial}{\partial p_i^\beta},
\]
then one obtains
\[
(X_\alpha)^i = \frac{\partial H}{\partial p_i^\alpha},\qquad
\sum_{\alpha=1}^k (X_\alpha)_i^\alpha = -\frac{\partial H}{\partial q^i}.
\]
For an integral section \(\psi(x)=(x^\alpha,q^i(x),p_i^\alpha(x))\), these become the Hamilton–De Donder–Weyl equations
\[
\frac{\partial q^i}{\partial x^\alpha} = \frac{\partial H}{\partial p_i^\alpha},\qquad
\sum_{\alpha=1}^k \frac{\partial p_i^\alpha}{\partial x^\alpha} = -\frac{\partial H}{\partial q^i}.
\]
This is the standard \(k\)-cosymplectic Hamiltonian encoding of first-order field equations [1409.5604].

The Lagrangian side is formulated on
\[
\mathbb{R}^k \times T^1_k Q
\]
with coordinates \((x^\alpha,q^i,v^i_\alpha)\) and Lagrangian
\[
L=L(x^\alpha,q^i,v^i_\alpha).
\]
The canonical tensors are
\[
J^\alpha = \frac{\partial}{\partial v^i_\alpha}\otimes dq^i,\qquad
\Delta = \sum_{\alpha=1}^k v^i_\alpha \frac{\partial}{\partial v^i_\alpha},\qquad
\Delta_\alpha = v^i_\alpha \frac{\partial}{\partial v^i_\alpha}.
\]
From \(L\) one defines the Poincaré–Cartan forms
\[
\theta_L^\alpha = dL\circ J^\alpha,\qquad \omega_L^\alpha=-d\theta_L^\alpha,
\]
with coordinate expressions
\[
\theta_L^\alpha = \frac{\partial L}{\partial v^i_\alpha}\,dq^i,\qquad
\omega_L^\alpha = dq^i \wedge d\!\left(\frac{\partial L}{\partial v^i_\alpha}\right),
\]
and the energy
\[
E_L=\Delta(L)-L.
\]
The geometric Euler–Lagrange equation is
\[
\iota_{X_\alpha}\omega_L^\alpha = dE_L + \frac{\partial L}{\partial x^\alpha}\,dx^\alpha,\qquad
dx^\alpha(X_\beta)=\delta^\alpha_\beta,
\]
which yields the classical first-order PDEs
\[
\sum_{\alpha=1}^k \frac{\partial}{\partial x^\alpha} \left(\frac{\partial L}{\partial v^i_\alpha}\right) - \frac{\partial L}{\partial q^i} =0.
\]
Regularity is expressed by invertibility of the Hessian
\[
\left(\frac{\partial^2 L}{\partial v^i_\alpha \partial v^j_\beta}\right),
\]
and for a hyperregular Lagrangian the Legendre map
\[
FL(x^\alpha,q^i,v^i_\alpha)=\left(x^\alpha,q^i,\frac{\partial L}{\partial v^i_\alpha}\right)
\]
is a global diffeomorphism. In that case the Hamiltonian and Lagrangian formalisms are equivalent, with
\[
H = E_L\circ FL^{-1},\qquad FL^*\theta^\alpha = \theta_L^\alpha,\qquad FL^*\omega^\alpha = \omega_L^\alpha
\]
[1409.5604].

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

The Hamilton–Jacobi problem in the \(k\)-cosymplectic framework is formulated on
\[
\mathbb{R}^k \times (T^1_k)^*Q
\]
by means of a section
\[
\gamma=(\gamma^1,\dots,\gamma^k)
\]
of the bundle \((T^1_k)^*Q\to \mathbb{R}^k\times Q\), where locally
\[
\gamma^\alpha(x,q)=\gamma^\alpha_i(x,q)\,dq^i,
\]
and each \(\gamma^\alpha\) is closed along the fibers in the sense that
\[
\frac{\partial \gamma^\alpha_i}{\partial q^j} = \frac{\partial \gamma^\alpha_j}{\partial q^i}.
\]
Given an integrable Hamiltonian \(k\)-vector field \(Z\), one defines the induced \(k\)-vector field
\[
Z^\gamma = T\pi \circ Z \circ \gamma
\]
on \(\mathbb{R}^k\times Q\). The central theorem states that the following are equivalent: if \(\psi\) is an integral section of \(Z^\gamma\), then \(\gamma\circ\psi\) is an integral section of \(Z\); and
\[
(TQ)^*\left[d(H\circ \gamma_x)\right] +\sum_{\alpha=1}^k i(R_\alpha)\, d\gamma^\alpha = 0.
\]
If locally \(\gamma^\alpha=dW^\alpha\), then the Hamilton–Jacobi relation is written in the compact form
\[
\frac{\partial W^\alpha}{\partial x^\alpha} + H\left(x^\alpha,q^i,\frac{\partial W^\alpha}{\partial q^i}\right) = K(x),
\]
and one can set \(K=0\) because subtracting a function of \(x\) alone does not change the Hamiltonian field equations [1304.3360].

The symmetry theory developed for almost-standard \(k\)-cosymplectic manifolds \(\mathbb{R}^k\times M\) isolates a class of structure-preserving symmetries adapted to Hamiltonian field theory. A diffeomorphism \(\Phi\) is a \(k\)-cosymplectic Noether symmetry if
\[
\Phi^*\omega^A = \omega^A,\qquad \Phi^*\eta^A = \eta^A,\qquad \Phi^*H = H,
\]
and in the preferred formulation one further requires
\[
\Phi^* t^A = t^A,
\]
so that \(\Phi\) acts fiberwise over \(\mathbb{R}^k\). The infinitesimal version is a vector field \(Y\) such that
\[
\mathcal{L}_Y \omega^A = 0,\qquad i(Y)\eta^A = 0,\qquad \mathcal{L}_Y H = 0.
\]
The condition \(i(Y)\eta^A=0\) means that \(Y\) has no \(\partial/\partial t^A\) component [1009.2703].

A conservation law is a map
\[
F=(F_1,\dots,F_k):\mathbb{R}^k\times M \to \mathbb{R}^k
\]
such that, for every solution \(v\),
\[
\operatorname{Div}(F\circ v)=0,\qquad
\sum_{A=1}^k \frac{\partial}{\partial t^A}\Big(F_A\circ v\Big)=0.
\]
Equivalently, for every integrable \(k\)-vector field \(X=(X_1,\dots,X_k)\) solving the Hamiltonian equations,
\[
\sum_{A=1}^k \mathcal{L}(X_A)F_A = 0.
\]
If \(Y\) is an infinitesimal \(k\)-cosymplectic Noether symmetry, then locally one can write
\[
F_A=i(Y)\theta^A-\xi_A,
\]
where \(\omega^A=d\theta^A\) locally and \(\mathcal{L}_Y\theta^A=d\xi_A\), and these functions satisfy
\[
\sum_{A=1}^k \mathcal{L}(X_A)F_A = 0.
\]
This is the \(k\)-cosymplectic version of Noether’s theorem: a symmetry preserving the geometric data and the Hamiltonian produces a conserved \(k\)-current whose divergence vanishes along solutions. The same paper exhibits this mechanism for quadratic Hamiltonians on \(\mathbb{R}^k\times (T^1_k)^*Q\) and, for the \(3\)-dimensional wave equation with \(k=4\), identifies the standard momentum components as a conserved current [1009.2703].

## 4. Singular theories, Lie algebroids, and reduction

The regular nondegeneracy assumptions of \(k\)-cosymplectic geometry do not cover singular field theories. For that purpose the theory of \(k\)-precosymplectic manifolds replaces the nondegenerate \(2\)-forms by closed forms of possibly smaller rank. A \(k\)-precosymplectic manifold has dimension
\[
k(n+1)+n-l,\qquad 1\le l< nk,
\]
and carries
\[
(\eta^\alpha,\omega^\alpha,V),\qquad \alpha=1,\dots,k,
\]
such that each \(\eta^\alpha\) and \(\omega^\alpha\) is closed, \(\operatorname{rank}\omega^\alpha=2r_\alpha\), \(V\) is an integrable \(nk\)-dimensional distribution, and
\[
\eta^\alpha|_V=0,\qquad \omega^\alpha|_{V\times V}=0,\qquad
\dim\Big(\bigcap_{\alpha=1}^k \ker \omega^\alpha\Big)\ge k.
\]
In Darboux coordinates
\[
(x^\alpha, y^i, y_i^\alpha, z^j)
\]
one has
\[
\eta^\alpha = dx^\alpha,\qquad \omega^\alpha = dy^i\wedge dy_i^\alpha,\qquad
V=\left\langle \frac{\partial}{\partial y_i^\alpha},\,\frac{\partial}{\partial z^j}\right\rangle.
\]
Reeb vector fields still exist, but they are not unique; locally they may be written as
\[
R_\alpha = \frac{\partial}{\partial x^\alpha}+D_\alpha^j\frac{\partial}{\partial z^j}.
\]
To recover dynamics one applies a constraint algorithm, generating a sequence
\[
M=M_0 \supset M_1 \supset M_2 \supset \cdots
\]
that stabilizes at a final constraint manifold \(M_f\) where solutions of the field equations exist [1812.08487].

A second major extension replaces the tangent and cotangent bundles by a Lie algebroid \(E\to Q\) and its dual \(E^*\). The standard spaces \(\mathbb{R}^k\times \bigoplus^k TQ\) and \(\mathbb{R}^k\times \bigoplus^k T^*Q\) are replaced by
\[
\mathbb{R}^k\times \bigoplus_{A=1}^k E,\qquad
\mathbb{R}^k\times \bigoplus_{A=1}^k E^*,
\]
together with the prolongation Lie algebroids
\[
\mathcal{T}^{E}\!\left(\mathbb{R}^k\times \bigoplus^k E\right),\qquad
\mathcal{T}^{E}\!\left(\mathbb{R}^k\times \bigoplus^k E^*\right).
\]
The resulting theory defines Poincaré–Cartan \(1\)-sections \(\Theta_L^A\), \(2\)-sections \(\Omega_L^A=-d\Theta_L^A\), Hamiltonian Liouville \(1\)-sections \(\Theta^A\), canonical \(2\)-sections \(\Omega^A=-d\Theta^A\), a generalized Legendre transformation
\[
\operatorname{Leg}_L(t^A,q^i,y_A^a)=\left(t^A,q^i,\frac{\partial L}{\partial y_A^a}\right),
\]
and Hamiltonian and Lagrangian field equations with anchor and structure-function terms. When \(E=TQ\), the standard \(k\)-cosymplectic theory is recovered, and when \(k=1\), one recovers Lie algebroid mechanics [1001.1177].

Reduction theory introduces an additional layer. In the \(k\)-polycosymplectic setting one has a pair \((\tau,\omega)\) of \(\mathbb{R}^k\)-valued closed forms with
\[
\operatorname{rank}\bigl(\ker\omega\bigr)=k,\qquad
\ker\tau\cap \ker\omega = 0,
\]
and the key equivalence states that
\[
(M,\tau,\omega)\text{ is \(k\)-polycosymplectic}
\quad\Longleftrightarrow\quad
\bigl(\mathbb{R}^k\times M,\ \Omega:=pr_M^*\omega + du\wedge pr_M^*\tau\bigr)
\text{ is \(k\)-polysymplectic.}
\]
This allows a Marsden–Weinstein reduction theory for \(k\)-polycosymplectic manifolds and, in the \(k\)-cosymplectic framework, a \(k\)-cosymplectic to \(l\)-cosymplectic geometric reduction that can reduce space-time variables [2302.09037].

## 5. Relations to \(k\)-symplectic, multisymplectic, and cocontact geometries

Within field theory, \(k\)-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: \(k\)-symplectic geometry treats first-order field theories whose Hamiltonian or Lagrangian do not depend explicitly on the base coordinates, while \(k\)-cosymplectic geometry incorporates the base variables \(x^\alpha\) into the geometry and is therefore suited to non-autonomous theories. In the autonomous case, when the \(k\)-cosymplectic Hamiltonian \(H\) does not depend on \(x^\alpha\), the system reduces to a \(k\)-symplectic one, and the same reduction holds on the Lagrangian side [1409.5604].

The same source also places the theory in relation with multisymplectic geometry. For trivial bundles \(E=\mathbb{R}^k\times Q\), the multisymplectic multimomentum bundle becomes diffeomorphic to
\[
\mathbb{R}^k \times (\mathbb{R}\times (T^1_k)^*Q),
\]
and the multisymplectic Hamiltonian and Lagrangian forms decompose into the \(k\)-cosymplectic data
\[
\eta^\alpha = dx^\alpha,\qquad \omega^\alpha \leftrightarrow \text{components of }\Omega.
\]
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 \(k\)-cosymplectic formalism [1409.5604].

More recent work extends the conservative framework to non-conservative field theories. \(k\)-cocontact geometry combines \(k\)-cosymplectic space-time variables with \(k\)-contact dissipation variables. Its canonical model is
\[
M = \mathbb{R}^k \times \bigoplus\nolimits^k TQ \times \mathbb{R}^k
\]
with Darboux coordinates \((t^\alpha,q^i,p_i^\alpha,z^\alpha)\) and canonical local form
\[
\tau^\alpha = dt^\alpha,\qquad \eta^\alpha = dz^\alpha - p_i^\alpha\,dq^i.
\]
The paper introducing this structure emphasizes that it extends \(k\)-cosymplectic geometry by allowing dissipation via contact-type forms \(\eta^\alpha\), and illustrates the formalism with a nonlinear damped wave equation with time-dependent forcing [2210.09166].

The scope of the theory is correspondingly broad. The \(k\)-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 \(\mathbb{R}^k \times (T^1_k)^*Q\) and \(\mathbb{R}^k \times T^1_k Q\) [1409.5604].

## 6. Alternative usages of the term and current directions

The term “\(k\)-cosymplectic” also appears in a different, non-field-theoretic sense in the study of log symplectic manifolds. In that usage, a \(k\)-cosymplectic structure on a \(k+2\ell\)-dimensional manifold \(M\) is a family
\[
(\alpha_i,\beta)
\]
consisting of \(k\) closed \(1\)-forms \(\alpha_i\in\Omega^1(M)\) and one closed \(2\)-form \(\beta\in\Omega^2(M)\) such that
\[
\left(\bigwedge_i \alpha_i\right)\wedge \beta^\ell \neq 0.
\]
Locally, any such manifold admits coordinates
\[
(s_1,\dots,s_k,p_1,q_1,\dots,p_n,q_n)
\]
for which
\[
\alpha_j = ds_j,\qquad \beta = \sum dp_k\wedge dq_k.
\]
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 [1605.03854].

This alternative usage should be distinguished from the field-theoretic \(k\)-cosymplectic structure with \(k\) closed \(1\)-forms, \(k\) closed \(2\)-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 \(k=1\), whereas the field-theoretic usage generalizes time-dependent Hamiltonian mechanics to \(k\) independent variables [1605.03854].

A further active direction is suggested by locally conformally cosymplectic geometry. In the \(k=1\) case, an LCC manifold \((M,\eta,\Omega,\Theta)\) satisfies
\[
d\eta=\Theta\wedge\eta,\qquad d\Omega=2\Theta\wedge\Omega,
\]
with closed Lee form \(\Theta\), and supports a Hamiltonian and Hamilton–Jacobi theory based on the Lichnerowicz–de Rham differential \(d_\Theta\). The authors explicitly remark that locally conformal \(k\)-cosymplectic geometry is not yet developed in the literature and indicate it as a natural future direction [2205.13329].

Taken together, these developments present \(k\)-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 \(k\)-symplectic, multisymplectic, contact, cocontact, and log-symplectic geometries [1409.5604].

Source: https://www.emergentmind.com/topics/k-cosymplectic-geometry