# Global realisation of magnetic fields as 1$\frac{1}{2}$D Hamiltonian systems

N. Duignan  School of Mathematics and Statistics, University of Sydney, NSW
2050, Australia  D. Perrella  D. Pfefferlé<EMAIL_ADDRESS>The
University of Western Australia, 35 Stirling Highway, Crawley WA 6009,
Australia

###### Abstract

The paper reviews the notion of $n+\frac{1}{2}$D non-autonomous Hamiltonian
systems, portraying their dynamics as the flow of the Reeb field related to a
closed two-form of maximal rank on a cosymplectic manifold, and naturally
decomposing into time-like and Hamiltonian components. The paper then
investigates the conditions under which the field-line dynamics of a
(tangential) divergence-free vector field on a connected compact three-
manifold (possibly with boundary) diffeomorphic to a trivial fibre bundle over
the circle can be conversely identified as a non-autonomous $1\frac{1}{2}$D
Hamiltonian system. Under the assumption that the field is transverse to a
global compact Poincaré section, an adaptation of Moser’s trick shows that all
such fields are locally-Hamiltonian. A full identification is established upon
further assuming that the Poincaré sections are planar, which crucially
implies (together with Dirichlet boundary conditions) that the cohomology
class of the generating closed one-forms on each section is constant. By
reviewing the classification of fibre bundles over the circle using the
monodromy representation, it is remarked that as soon as the Poincaré section
of an alleged field is diffeomorphic to a disk or an annulus, the domain is
necessarily diffeomorphic to a solid or hollow torus, and thus its field-line
dynamics can always be identified as a non-autonomous Hamiltonian system.

##  I Introduction

Magnetic confinement fusion devices are shaped like tori because such domains
are the “simplest” orientable manifolds in Euclidean three-space on which a
vector field, tangent to the boundary, has a chance of being nowhere zero [1].
The layout of the magnetic field-lines in a fusion device determines the
confinement [2, 3, 4]. The less chaotic their trajectories, the less heat and
particles fusion devices will lose. This is the qualitative reason why
configurations that exhibit nested toroidal flux-surfaces are sought after [5,
6, 7]. Within magneto-hydrostatics (MHS), ensuring that the magnetic field-
lines lie tangential to nested surfaces has been conjectured by Grad to be
impossible without symmetry by isometry [8]. Certainly, configurations
featuring three-dimensional flux-surfaces, as desired in stellarators, can
only be explained through extreme fine-tuning and optimisation.
Sensitivity/robustness to small changes is then an important area of
investigation. Many equilibrium solvers and design/optimisation codes assume
straight field-line coordinates and nested flux-surfaces as a starting point.

The magnetic field $B$ is considered to be a divergence-free vector field on a
domain of Euclidean three-space, $M\subseteq\mathbb{R}^{3}$. One then defines
the magnetic two-form through the natural volume-form $\mu$ as
$\beta=i_{B}\mu$. The magnetic field $B$ being divergence-free is equivalent
to the two-form $\beta$ being closed, $d\beta=0$. The latter statement is
looser, in the sense that it does not depend on the metric. How best to
express the closed two-form $\beta$ in order to highlight the features of the
field-lines of $B$ is the focus of this paper.

Estimating the degree of chaos in magnetic field-lines is hard other than
through numerical field-line tracing. There are plenty of examples of nowhere
vanishing vector fields on three-manifolds with extremely rich field-line
dynamics. The latter dynamics can be as complicated as desired, for example
with ABC-flows [9], Turing complete flows [10], knotted flows [11], etc.
Analytically, the tools of Hamiltonian perturbation theory are often employed.
For example, the Chirikov criterion predicts the onset of chaos when
individual resonances overlap [12]. The KAM theorem [13, 14, 15] is invoked to
assess the persistence of irrational surfaces under small perturbations [16,
17, 18, 19, 20]. Magnetic field-lines are thus often compared to non-
autonomous $1\frac{1}{2}$D Hamiltonian systems [21, 22, 23, 24], as
highlighted in the following direct quote from [25, Section 4.5 a)].

> The equation of a field line can be cast in a Hamiltonian form. Hamiltonian
> theory guarantees the existence of magnetic surfaces in devices with a
> symmetry direction: an ideal, axisymmetric tokamak; a helically symmetric
> straight stellarator; etc. In non-symmetric devices, on the other hand,
> magnetic surfaces do not exist rigorously everywhere because ergodic regions
> of finite volume exist […]. In practice, we can get away with considering
> approximate magnetic surfaces provided the perturbations away from symmetry
> are not too large. In principle, one should either measure the surfaces
> experimentally or follow field lines on a computer to demonstrate the
> existence of suitable surfaces.

The justification in the physics literature is local, gauge dependent and
requires explicit coordinates [26, 17]. A variant of Darboux’s theorem can be
used in Euclidean three-space to show that a non-zero magnetic field can be
represented in _Clebsch form_ , $B=\nabla f\times\nabla g$, in a neighbourhood
of any point. The result is local in nature, and there are topological
obstructions to realising Clebsch representations globally.

Under the assumption that the magnetic field is transverse to the level sets
of a circle-valued _time_ or _angle_ function, we show using Moser’s trick
[27] under what circumstances a magnetic field can be identified with a
$1\frac{1}{2}$D Hamiltonian system. The procedure is coordinate-free and
global. This identification holds on compact three-manifolds diffeomorphic to
a product $S^{1}\times\Sigma$. This includes the domains of interest to
magnetic confinement such as the solid torus and the hollow torus. There is an
opportunity to extend the results to non-trivial bundles, for example
generated by excluding linked loops from a toroidal domain as in heliotrons
[6] or in the CNT device [28], or by removing loops in integer multiples of
the generating class of first homology.

The paper is organised as follows. In Section II.1, an example of a _Beltrami_
magnetic field on a hollow torus is shown to possess a Poincaré section that
cannot purely be the level sets of the toroidal angle or poloidal angle, but a
combination of both. In Sections II.2 and II.3, direct methods to express the
magnetic field in the so-called double-Clebsch form are explored. The first
method is local and coordinate-dependent, and the second is global and
coordinate-free relying on the so-called Weyl gauge. The drawback of these
direct methods is that they only apply on specific domains. In Section III, we
review the notion of non-autonomous Hamiltonian systems in the canonical
setting and reveal their relation to cosymplectic manifolds. Necessary
conditions for dynamics on cosymplectic manifolds to admit a Hamiltonian
representation are determined. In Section IV, we specialise to the case of
divergence-free fields on domains of $\mathbb{R}^{3}$ with tangential boundary
conditions and present a procedure involving Moser’s trick to decompose the
magnetic two-form into a symplectic form on the fibres and the wedge product
of a Hamiltonian function and the derivative of the circle-valued projection.
Section V discusses the results and hints to possible extensions.

##  II Motivating example and direct methods

The main strategy to relate field-line equations to Hamiltonian systems is to
arrive via coordinate transformations at the commonly adopted _double-Clebsch_
representation of the magnetic field on a toroidal domain of $\mathbb{R}^{3}$,

$\displaystyle
B=\nabla\psi_{t}\times\nabla\theta+\nabla\varphi\times\nabla\psi_{p}(\psi_{t},\theta,\varphi),$

where $(\psi_{t},\theta,\varphi)$ are known as _toroidal flux coordinates_ in
the literature [25, 26]; $\psi_{t}$ is a radial variable (measuring the
toroidal flux when the field-lines are nested), $\theta$ is a poloidal angle
wrapping around the short way of the torus and $\varphi$ is a toroidal angle
travelling around the long way. The function $\psi_{p}$ yields the poloidal
flux when field-lines are nested. In this picture, $\psi_{t}$ is seen as the
canonical action paired to the angle $\theta$ and $\psi_{p}$ is identified as
a “time-dependent” Hamiltonian function.

There are two distinct questions when proceeding in this way; how are these
coordinates (globally) recovered for an arbitrary magnetic field, and do they
necessarily lead to a Hamiltonian representation of the field-line dynamics ?

It is useful to note that the toroidal component of the magnetic field,

$\displaystyle
B^{\varphi}=B\cdot\nabla\varphi=(\nabla\psi_{t}\times\nabla\theta)\cdot\nabla\varphi=\frac{1}{\sqrt{g}},$
(1)

is inversely proportional to the Jacobian determinant of the coordinate
transformation. By the inverse function theorem, the transformation is
(locally) invertible if and only if $B^{\varphi}\neq 0$ is nowhere zero,
namely the magnetic field admits (global) Poincaré sections given by the level
sets of $\varphi$, also known as poloidal cross-sections. In the literature of
magnetic confinement fusion, the assumptions around Poincaré sections are
often taken for granted. There are examples of devices where the toroidal
component changes sign, for example in the Reversed-Field Pinch (RFP) [29].
The magnetic fields in such devices cannot admit a global Poincaré section,
preventing the identification of a forward “time” direction and therefore the
strict association of the field-line dynamics with a Hamiltonian system. While
it is possible to glue coordinate patches in which magnetic fields are
expressed as above and a consistent forward time-direction is found, the
procedure is bound to fail under global perturbations.

Below is an example of a nowhere-vanishing force-free (Beltrami) magnetic
field whose toroidal and poloidal components vanish at different locations,
which seems to suggest that the identification with a Hamiltonian system might
(globally) fail. After an elementary redefinition of the angles via a global
coordinate transformation, it is shown that the field actually admits a global
Poincaré section, and is therefore genuinely Hamiltonian, albeit with respect
to non-standard poloidal and toroidal angles. This example highlights part of
the challenges pertaining to Poincaré sections. The rest of the paper will be
concerned about deeper obstructions.

###  II.1 A Lundquist-Beltrami field with “full” rotational transform

Eigenfields of the curl operator, depicting Beltrami fields or linear force-
free fields, were actively studied in astrophysical contexts [30, 31] (see
also [32, 33]). Such vector fields are relevant in plasma physics and magnetic
confinement fusion through a process called _Taylor relaxation_ [34], whereby
the magnetic energy is dissipated on faster timescales than magnetic helicity.
As such, Beltrami states are minimisers of a quadratic energy functional, and
thus solutions of an elliptic variational problem. They are also special cases
of magneto-hydrostatics (MHS) solutions (also called MHD equilibrium).
Examples of Beltrami fields can easily be constructed in simple slab or
cylindrical geometries. The question is whether the field-line dynamics can be
cast as a non-autonomous Hamiltonian system. As with the elementary ABC-flows
[9], the identification is not generally expected to hold. An important
obstruction is highlighted, but then resolved, in the following example.

Let $J_{0},J_{1}:\mathbb{R}\to\mathbb{R}$ denote the Bessel functions of order
$0$ and order $1$ respectively. For $0<r_{0}<r_{1}<\infty$, consider the
annulus

$A_{r_{0},r_{1}}=\\{(x,y)\in\mathbb{R}^{2}:r_{0}\leq\sqrt{x^{2}+y^{2}}\leq
r_{1}\\}.$ (2)

Then, on the _periodic co-axial cylinder_ $M=S^{1}\times A_{r_{0},r_{1}}$,
where
$S^{1}={\left.\raisebox{1.99997pt}{$\mathbb{R}$}\middle/\raisebox{-1.99997pt}{$2\pi\mathbb{Z}$}\right.}$,
the vector field

$B=\frac{J_{1}(r)}{r}\frac{\partial}{\partial\theta}\Big{|}_{r,\varphi}-J_{0}(r)\frac{\partial}{\partial\varphi}\Big{|}_{r,\theta}=-\nabla(rJ_{1}(r))\times\nabla\theta+\nabla
J_{0}(r)\times\nabla\varphi,$ (3)

satisfies, with respect to the metric
$g=d\varphi^{2}+dr^{2}+r^{2}d\theta^{2}$,

$\nabla\times B=-B,\qquad B\cdot n=0\quad\text{(on {\hbox{\partial M}})},$

where $n$ denotes the outward unit normal on $M$ and $(\varphi,r,\theta)$ are
coordinates coming from the projection $\varphi:M\to S^{1}$ and usual polar
coordinates. The vector field $B$ is the same as that considered by Lundquist
[30] but restricted to a different domain.

Let $z_{i,1}$ denote the first positive zero of $J_{i}$, $i\in\\{0,1\\}$. As
well-known, $0<z_{0,1}<z_{1,1}$, and so we may set

$r_{0}=z_{0,1},\qquad r_{1}=z_{1,1}.$

The field-lines of $B$ satisfy the following ODEs

$\displaystyle\dot{r}(t)$ $\displaystyle=0,$ $\displaystyle\dot{\theta}(t)$
$\displaystyle=\frac{J_{1}(r(t))}{r(t)},$ $\displaystyle\dot{\varphi}(t)$
$\displaystyle=-J_{0}(r(t)),$

yielding excessively simple dynamics. Any function of the coordinate $r$ forms
an invariant or _flux-function_. The pitch of the field-lines in these
coordinates is constant, and equal to the _winding number_ , also known as
_rotational transform_ [25],

$\displaystyle\iota(r)=\frac{\dot{\theta}}{\dot{\varphi}}=\frac{B\cdot\nabla\theta}{B\cdot\nabla\varphi}=-\frac{J_{1}(r)}{rJ_{0}(r)}.$

We would like to see if the field-lines dynamics can be described as a non-
autonomous $1\frac{1}{2}$D Hamiltonian system. The necessary condition that
$B$ is nowhere zero is met, as shown later in Equation 4. An obstacle is
encountered in that neither the $\varphi$ coordinate nor the $\theta$
coordinate serve as an angle against which the field $B$ is positive. In other
words, the magnetic field has a rotational transform ranging from $0$ to
$\infty$. As well-known, writing $z_{0,2}$ for the second positive zero of
$J_{0}$, we have $z_{0,1}<z_{1,1}<z_{0,2}$ and $J_{0}|_{(z_{0,1},z_{0,2})}<0$.
On the other hand, $J_{1}|_{(0,z_{1,1})}>0$. So,
$J_{0}(z_{0,1})=0=J_{0}(z_{0,2})$ and $J_{1}(z_{1,1})=0$. Therefore,

$\displaystyle\\{x\in M:B\cdot\nabla\varphi|_{x}>0\\}$
$\displaystyle=M\setminus\partial_{0}M,$ $\displaystyle\\{x\in
M:B\cdot\nabla\theta|_{x}>0\\}$ $\displaystyle=M\setminus\partial_{1}M,$

where $\partial_{0}M$ is the inner connected component of $\partial M$,

$\partial_{0}M=S^{1}\times\\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=r_{0}^{2}\\},$

and $\partial_{1}M$ is the outer connected component of $\partial M$,

$\partial_{1}M=S^{1}\times\\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=r_{1}^{2}\\}.$

That is, neither $B\cdot\nabla\varphi>0$ nor $B\cdot\nabla\theta>0$ on the
whole of $M$. In fact, even if one introduces smooth functions $f,g$ on $M$,
and considers angle functions $\theta^{\prime},\varphi^{\prime}:M\to S^{1}$
such that

$\displaystyle\nabla\theta^{\prime}$
$\displaystyle=\nabla(\theta+f)=\nabla\theta+\nabla f,$
$\displaystyle\nabla\varphi^{\prime}$
$\displaystyle=\nabla(\varphi+g)=\nabla\varphi+\nabla g,$

then one finds that neither $B\cdot\nabla\theta^{\prime}>0$ nor
$B\cdot\nabla\varphi^{\prime}>0$. Indeed, one has

$B\cdot\nabla\theta^{\prime}|_{\partial_{1}M}=B\cdot\nabla\theta|_{\partial_{1}M}+B\cdot\nabla
f|_{\partial_{1}M}=B\cdot\nabla f|_{\partial_{1}M},$

and

$B\cdot\nabla\varphi^{\prime}|_{\partial_{0}M}=B\cdot\nabla\varphi|_{\partial_{0}M}+B\cdot\nabla
g|_{\partial_{0}M}=B\cdot\nabla g|_{\partial_{0}M}.$

But, for any function $h\in C^{\infty}(M)$ and any $i\in\\{0,1\\}$, because
$h|_{\partial_{i}M}$ attains a maximum on $\partial_{i}M$ and therefore has a
critical point on $\partial_{i}M$, the fact that $B$ is tangent to
$\partial_{i}M$ implies that $B\cdot\nabla h|_{\partial_{i}M}$ vanishes at
some point in $\partial_{i}M$. Therefore, modifying the angle functions
$\theta,\varphi$ by smooth functions $f,g$ is futile.

Figure 1: Field-lines and flux-surfaces of the Beltrami field of Equation 3,
represented in toroidal geometry (a non-isometric embedding). On the (grey)
inner boundary, the (black) field-lines are purely poloidal and on the (red)
outer boundary, the (red) field-lines are purely toroidal. On an intermediate
flux-surface, (blue/orange) field-lines feature irrational winding. The green
surfaces are level-sets of the $\phi=\theta+\varphi$ circle-valued function.
The field-lines are everywhere transverse to these surfaces, so that they
constitute global Poincaré sections, which is the necessary and sufficient
requirement to identify the field-lines $B$ as a non-autonomous Hamiltonian
system.

This predicament is remedied by considering a new angle111Any integer
combination of $\theta$ and $\varphi$ produces a circle-valued function by the
group structure of $S^{1}$, whereas general $\mathbb{R}$-linear combinations
do not yield a function $M\to S^{1}$. $\phi=\theta+\varphi:M\to S^{1}$. To see
this, first note that

$B\cdot\nabla\phi=\frac{J_{1}(r)}{r}-J_{0}(r).$

From the fact that $J_{0}|_{(z_{0,1},z_{0,2})}<0$ and
$J_{1}|_{(0,z_{1,0})}>0$, it follows that for $r\in[z_{0,1},z_{1,1}]$, one has
that $J_{1}(r)/r-J_{0}(r)>0$. Thus,

$B\cdot\nabla\phi>0.$ (4)

In fact, let us consider the coordinate transformation $F:M\to M$ given by

$\displaystyle(\varphi,r,\theta)\mapsto(\theta+\varphi,r,\theta).$

Then, the contravariant components of the magnetic field in these new
coordinates are computed as

$\displaystyle B^{\phi}=B\cdot\nabla\phi$
$\displaystyle=\frac{J_{1}(r)}{r}-J_{0}(r)>0,$ $\displaystyle
B^{r}=B\cdot\nabla r$ $\displaystyle=0,$ $\displaystyle
B^{\theta}=B\cdot\nabla\theta$ $\displaystyle=\frac{J_{1}(r)}{r},$

so that

$B=\frac{J_{1}(r)}{r}\frac{\partial}{\partial\theta}\Big{|}_{r,\phi}+\left(\frac{J_{1}(r)}{r}-J_{0}(r)\right)\frac{\partial}{\partial\phi}\Big{|}_{r,\theta}=-\nabla(J_{0}+rJ_{1})\times\nabla\theta+\nabla
J_{0}\times\nabla\phi,$

One can then identify the flow of $B$ as the dynamics of a non-autonomous
Hamiltonian system by following the approach in Section III.

In the remainder of the paper, we consider in higher generality what
representing a magnetic field as a non-autonomous Hamiltonian system entails
and show that diffeomorphisms such as the one above can be found in a wide
variety of cases. A direct consequence of our investigation applied to the
case of a hollow torus or solid torus is the following list of equivalent
statements, presented here as amuse-bouches. To set up, let $D^{2}$ denote the
(closed) unit disk in $\mathbb{R}^{2}$.

######  Main Conclusion.

Let $M$ be a compact connected manifold with boundary, $\mu$ a volume-form on
$M$, and $B$ a divergence-free vector field on $M$ tangent to $\partial M$.
Let $\Sigma\in\\{D^{2},A_{1,2}\\}$. Then, the following are equivalent.

  1. 1.

$B$ has a Poincaré section diffeomorphic to $\Sigma$;

  2. 2.

$M$ is diffeomorphic to $S^{1}\times\Sigma$ and there exists a closed one-form
$\eta\in\Omega^{1}(M)$ such that $\eta(B)>0$;

  3. 3.

The field-lines of $B$ can be represented as a non-autonomous $1\frac{1}{2}$D
Hamiltonian system on $\Sigma$.

We clarify and recall the meaning of Item 3 in Section III and Item 1 in
Section IV.

###  II.2 Local flux-coordinates and Hamiltonian representation

There are several local procedures to derive flux-coordinates as in Equation
1. Following [35], let $(u^{1},u^{2},u^{3})$ be local coordinates of a domain
of $\mathbb{R}^{3}$ with volume form $\mu=\sqrt{g}du^{1}\wedge du^{2}\wedge
du^{3}$. First, the magnetic field is expressed in a _covariant_
representation as

$\displaystyle B=B^{i}\frac{\partial}{\partial
u^{i}}\implies\beta=i_{B}\mu=\sqrt{g}B^{1}du^{2}\wedge
du^{3}+\sqrt{g}B^{2}du^{3}\wedge du^{1}+\sqrt{g}B^{3}du^{1}\wedge du^{2}$

with $B^{i}=B\cdot\nabla u^{i}$. The condition that the magnetic field is
divergence-free (equivalently that the two-form $\beta$ is closed) reads

$\displaystyle\partial_{i}(\sqrt{g}B^{i})=0.$

We wish to establish a coordinate transformation
$(u^{1},u^{2},u^{3})\mapsto(q,p,t)$ defined by

$\displaystyle q$ $\displaystyle=Q(u^{1},u^{2},u^{3})=u^{1},$ $\displaystyle
p$ $\displaystyle=P(u^{1},u^{2},u^{3}),$ $\displaystyle t$
$\displaystyle=T(u^{1},u^{2},u^{3})=u^{3},$

such that the magnetic two-form has exactly the following expression

$\displaystyle\beta=dP\wedge dQ+dT\wedge dH=-\partial_{2}Hdu^{2}\wedge
du^{3}+(\partial_{3}P+\partial_{1}H)du^{3}\wedge
du^{1}-\partial_{2}Pdu^{1}\wedge du^{2}$

for some functions $H$ and $P$ of $(u^{1},u^{2},u^{3})$.

By identification, the following relations between the functions $P$, $H$ and
the covariant components of the magnetic field are obtained

$\displaystyle\partial_{2}H$ $\displaystyle=-\sqrt{g}B^{1},$
$\displaystyle\partial_{2}P$ $\displaystyle=-\sqrt{g}B^{3},$
$\displaystyle\partial_{3}P+\partial_{1}H$ $\displaystyle=\sqrt{g}B^{2}.$

If the first and second conditions are satisfied, then the derivative of the
third with respect to $u^{2}$ is automatically verified. Defining

$\displaystyle P(u^{1},u^{2},u^{3})$
$\displaystyle=-\int_{a}^{u^{2}}\sqrt{g}B^{3}(u^{1},v,u^{3})dv+c_{1}(u^{1},u^{3})$
$\displaystyle H(u^{1},u^{2},u^{3})$
$\displaystyle=-\int_{a}^{u^{2}}\sqrt{g}B^{1}(u^{1},v,u^{3})dv+c_{2}(u^{1},u^{3}),$

then, writing
$\sqrt{g}B^{2}(u^{1},u^{2},u^{3})=\sqrt{g}B^{2}(u^{1},a,u^{3})+\int_{a}^{u^{2}}\partial_{2}(\sqrt{g}B^{2})(u^{1},v,u^{3})dv$,
the double-Clebsch representation of the magnetic field is achieved by
asserting that

$\displaystyle\partial_{3}c_{1}(u^{1},u^{3})+\partial_{1}c_{2}(u^{1},u^{3})=\sqrt{g}B^{2}(u^{1},a,u^{3}).$

This is possible, for example, by letting $c_{2}=0$ and by setting

$\displaystyle c_{1}(u^{1},u^{3})=\int_{b}^{u^{3}}\sqrt{g}B^{2}(u^{1},a,v)dv.$

The coordinate transformation is valid provided that the Jacobian matrix is
invertible, or equivalently that the top-form

$\displaystyle dP\wedge dQ\wedge dT=\beta\wedge du^{3}$

is a volume-form (non-degenerate). We must thus have
$\partial_{2}P=-\sqrt{g}B^{3}\neq 0$, which assumes that the magnetic field is
transverse to the level sets of $u^{3}$ acting as Poincaré sections. Often
omitted and quite tedious in practice, the final step to relate the magnetic
field to a Hamiltonian system is to express the function $H$ in terms of the
new coordinates, namely to define the Hamiltonian $\hat{H}$ so that
$\hat{H}(q,p,t)=H(u^{1},u^{2},u^{3})$ when
$(q,p,t)=(u^{1},P(u^{1},u^{2},u^{3}),t)$. Then, one reaches the desired
representation in the coordinates $(q,p,t)$

$\displaystyle\hat{\beta}=dp\wedge dq+dt\wedge d\bar{H}(q,p,t).$

###  II.3 Global coordinate-free procedure exploiting the Weyl gauge

A coordinate-free and global version of the construction described in the
previous section is when a smooth (strong) deformation retract
$F:M\times[0,e)\to M$ with $e>0$ of the manifold $M$ onto a sub-manifold $L$
can be manufactured as the semi-flow of the (autonomous) vector field $X$
fitting the description below. The maps $F_{s}:M\to M$, $x\mapsto F(x,s)$ is a
family of diffeomorphisms satisfying the ordinary differential equation

$\displaystyle\partial_{s}F_{s}=X\circ F_{s},$

with initial condition $F_{0}=\text{id}$. It is clear that the semi-flow of
$X$ enjoys the group property $F_{a}\circ F_{b}=F_{a+b}=F_{b}\circ F_{a}$
whenever $0\leq a+b<e$. This property implies in particular that the
generating vector field is invariant under its semi-flow, ${F_{s}}_{*}X=X$ for
all $s\in[0,e)$. We now compute the $s$-derivative of the pullback by $F_{s}$
of the magnetic two-form $\beta=i_{B}\mu$,

$\displaystyle\frac{d}{ds}F_{s}^{*}\beta=F_{s}^{*}\mathcal{L}_{X}\beta=d(F_{s}^{*}i_{X}\beta).$

Integrating with respect to $s$ and using the fundamental theorem of calculus,
we have

$\displaystyle\beta$ $\displaystyle=d\alpha+F^{*}_{e}\beta,$ where
$\displaystyle\alpha$ $\displaystyle=-\int_{0}^{e}F_{s}^{*}i_{X}\beta ds.$ (5)

If $L$ is a one-dimensional submanifold (e.g. $M$ retracts to a line, as is
the case for the solid torus), then $F_{e}^{*}\beta=0\in\Omega^{2}(L)$. If
$L\subset\partial M$ (e.g. $M$ retracts to a component of the boundary, as is
the case for the hollow torus) and the magnetic field is assumed to be
tangential to that boundary component, then $F_{e}^{*}\beta=0\in\Omega^{2}(L)$
because the two-form is Dirichlet, see Lemma IV.1. In either case, a unique
vector potential $\alpha$ is obtained to represent the (exact) magnetic two-
form, $\beta=d\alpha$. The direct evaluation of the vector potential of
Equation 5 corresponds to a choice of gauge; it is an instance of the _Weyl
gauge_ , also known as multi-polar gauge or Poincaré gauge. An interesting and
defining property of the potential one-form $\alpha$ is that its component
along $X$ is zero. Indeed,

$\displaystyle\alpha(X)=-\int_{0}^{e}(F_{s}^{*}i_{X}\beta)(X)ds=-\int_{0}^{e}F_{s}^{*}(\beta(X,{F_{s}}_{*}X))ds=-\int_{0}^{e}F_{s}^{*}(\beta(X,X))ds=0.$

Now, to arrive at the double-Clebsch representation of $\beta$, we further
assume that there exists a function $\rho$ and two linearly independent closed
one-form $\kappa_{1}$ and $\kappa_{2}$ on $M\setminus L$, such that

$\displaystyle i_{X}\mu=-\rho\kappa_{1}\wedge\kappa_{2}.$

There is freedom in the choices of $\rho$, $\kappa_{1}$ and $\kappa_{2}$
beyond the fact that they must produce a well-defined deformation retract $F$
of $M$ onto $L$, which can be difficult to assess in practice. The regularity
of $\kappa_{1}$ and $\kappa_{2}$ as closed one-forms on $M\setminus L$ may not
extend to the whole manifold $M$, in which case the function $\rho$ has to be
suitably tailored to compensate singularities in order for $X$ to be well-
defined, see Example II.1.

Since $X\in\ker\kappa_{1}\cap\ker\kappa_{2}$, we have
$\mathcal{L}_{X}\kappa_{j}=i_{X}\cancel{d\kappa_{j}}+d\cancel{i_{X}\kappa_{j}}=0$,
which means that the semi-flow preserves the closed one-forms, namely
$F_{s}^{*}\kappa_{j}=\kappa_{j}$ for all $s\in[0,e)$ and $j\in\\{1,2\\}$.

With the above prescription for $X$, an explicit expression for the potential
one-form $\alpha$ is obtained. First, note that

$\displaystyle-
i_{X}\beta=-i_{X}i_{B}\mu=i_{B}i_{X}\mu=-i_{B}(\rho\kappa_{1}\wedge\kappa_{2})=-\rho\kappa_{1}(B)\kappa_{2}+\rho\kappa_{2}(B)\kappa_{1}=\kappa_{2}(\rho
B)\kappa_{1}-\kappa_{1}(\rho B)\kappa_{2}.$

Because the flow preserves the one-forms $\kappa_{1}$ and $\kappa_{2}$, the
potential one-form is then given by

$\alpha=P\kappa_{1}-H\kappa_{2}$

where

$\displaystyle P$ $\displaystyle=\int_{0}^{e}F_{s}^{*}(\kappa_{2}(\rho B))ds,$
$\displaystyle H$ $\displaystyle=\int_{0}^{e}F_{s}^{*}(\kappa_{1}(\rho B))ds.$

Consequently, the two-form will be globally represented in the double-Clebsch
form

$\beta=dP\wedge\kappa_{1}-dH\wedge\kappa_{2}.$

To recover a genuine Hamiltonian representation from here, one needs to
identify canonical coordinates and a Hamiltonian function. For the same
reasons as in Section II.2, this (morally) works if the top-form

$\displaystyle dP\wedge\kappa_{1}\wedge\kappa_{2}=\beta\wedge\kappa_{2}$

is a volume-form. Then, $\kappa_{2}$ provides the _time_ -coordinate,
$dP\wedge\kappa_{1}$ defines a symplectic form on the leaves (Poincaré
sections) of the foliation integrated from $\ker\kappa_{2}$, and $H$ qualifies
as a non-autonomous Hamiltonian function. The frailty of the method is
highlighted in the following example.

######  Example II.1.

On a manifold diffeomorphic to a solid torus $M\cong S^{1}\times D^{2}$ with
volume-form $\mu=\sqrt{g}dr\wedge d\theta\wedge d\phi$, consider the vector
field $X=-r\partial_{r}$ which satisfies

$\displaystyle i_{X}\mu=-r\sqrt{g}d\theta\wedge d\phi$

where $\phi:M\to S^{1}$ and $(r,\theta)$ are polar coordinates on $D^{2}$.
Here, $\rho=r\sqrt{g}$, $\kappa_{2}=d\phi$ and $\kappa_{1}=d\theta$ which is a
well-defined one-form on $M\setminus L$, namely away from the coordinate axis
$L\cong S^{1}\times\\{0\\}$. The semi-flow of $X$ is given in coordinates by

$\displaystyle F_{s}(r,\theta,\phi)=(re^{-s},\theta,\phi)$

which adequately deformation retracts the solid torus to the coordinate axis
$L$ in the limit $s\to\infty$. Then, the functions
$P=\alpha(\partial_{\theta})$ and $H=-\alpha(\partial_{\phi})$ are the
covariant components of the vector potential, uniquely determined by the
contravariant components of the magnetic field $B^{\phi}=d\phi(B)$ and
$B^{\theta}=d\theta(B)$ through the following integrals

$\displaystyle P(r,\theta,\phi)$
$\displaystyle=r\int_{0}^{\infty}e^{-s}\sqrt{g}(re^{-s},\theta,\phi)B^{\phi}(re^{-s},\theta,\phi)ds=r\int_{0}^{1}\sqrt{g}(ur,\theta,\phi)B^{\phi}(ur,\theta,\phi)du$
$\displaystyle H(r,\theta,\phi)$
$\displaystyle=r\int_{0}^{\infty}e^{-s}\sqrt{g}(re^{-s},\theta,\phi)B^{\theta}(re^{-s},\theta,\phi)ds=r\int_{0}^{1}\sqrt{g}(ur,\theta,\phi)B^{\theta}(ur,\theta,\phi)du.$

This method extends to the hollow torus by using a similar radial vector field
$X\propto\partial_{r}$ under the additional assumption that the magnetic field
is tangential to the boundary. The representation of the vector potential
without a radial component is extremely common in plasma physics, in
particular in numerical codes such as VMEC [36] and SPEC [37].

This method is limited to manifolds admitting i) a deformation retract onto
the “right” submanifold, ii) closed but nowhere-vanishing one-forms. For
example, it does not seem possible to apply the Weyl gauge to the case of a
solid torus in which two toroidal loops are removed (yielding a cross-section
diffeomorphic to a disk with two holes). It is also not immediate that the
functions $P$ and $H$ defined above on $M\setminus L$ are smooth on $M$. While
$B^{\theta}=d\theta(B)$ is often singular at the coordinate axis, the product
$\sqrt{g}B^{\theta}$ is usually integrable. For example, consider the magnetic
field $B=\partial_{x}$ on a periodic cylinder with the flat metric
$g=dx^{2}+dy^{2}+d\phi^{2}=dr^{2}+r^{2}d\theta+d\phi^{2}$ where $(x,y)$ are
Cartesian coordinates of the disk, one finds that $B^{\theta}=-\sin\theta/r$,
but $\sqrt{g}B^{\theta}=-\sin\theta$ so that $H=-r\sin\theta=-y$ is smooth.
Overall, the method is still sensitive to the choice of coordinates and
magnetic fields.

In the remainder of the paper, we explore a vastly different and possibly more
robust method that circumvents some of the caveats discussed above. We first
start by reviewing in precise terms what non-autonomous Hamiltonian systems
means.

##  III Non-autonomous Hamiltonian systems

###  III.1 Classical non-autonomous Hamiltonian systems in canonical
coordinates

Classically, a non-autonomous Hamiltonian system is considered to be an
“$n+\tfrac{1}{2}$” degree of freedom dynamical system on the cotangent bundle
$T^{*}Q$ of a smooth $n$-manifold $Q$. It is defined by a _time-dependent_
Hamiltonian function $H:\mathbb{R}\times T^{*}Q\to\mathbb{R}$ or
$H:S^{1}\times T^{*}Q\to\mathbb{R}$ for periodicity in time. We will work with
the periodic case.

In canonical coordinates $(q^{i},p_{i})$ on $T^{*}Q$, Hamilton’s equations
read

$\displaystyle\dot{q}^{i}(t)$ $\displaystyle=\ \ \frac{\partial H}{\partial
p_{i}}(t,q(t),p(t))$ $\displaystyle\dot{p}_{i}(t)$
$\displaystyle=-\frac{\partial H}{\partial q^{i}}(t,q(t),p(t))$

which can be recorded as the dynamics of the non-autonomous vector field on
$T^{*}Q$,

$\displaystyle X_{t}=\frac{\partial H_{t}}{\partial
p_{i}}\partial_{q^{i}}-\frac{\partial H_{t}}{\partial q^{i}}\partial_{p_{i}},$

where $H_{t}:T^{*}Q\to\mathbb{R}$ is now a family of Hamiltonian functions on
$T^{*}Q$ defined by $H_{t}(q,p)=H(t,q,p)$ for $t\in S^{1}$.

It is natural to view _time_ as an independent coordinate and elevate the non-
autonomous system to an autonomous one on $S^{1}\times T^{*}Q$, given by the
differential equations

$\displaystyle\dot{t}(s)$ $\displaystyle=1$ $\displaystyle\dot{q}^{i}(s)$
$\displaystyle=\ \ \frac{\partial H}{\partial p_{i}}(t(s),q(s),p(s))$
$\displaystyle\dot{p}_{i}(s)$ $\displaystyle=-\frac{\partial H}{\partial
q^{i}}(t(s),q(s),p(s)).$

In turn, this can be thought of as the dynamics of the _evolution vector
field_ $E_{H}$ on $S^{1}\times T^{*}Q$, which can be expressed as the sum

$\displaystyle E_{H}=\partial_{t}+X_{H},$ (6)

where the vector field $X_{H}$ on $S^{1}\times T^{*}Q$ evaluates to
${X_{H}}(t,q,p)=X_{t}(q,p)$ for $t\in S^{1}$ and $(q,p)\in T^{*}Q$.

###  III.2 Some trivial bundle formalism

Before diving into a modern representation of non-autonomous Hamiltonian
dynamics, this section introduces some basic formalism around product
manifolds and trivial bundles. A review of the classification of fibre bundles
over the circle can be found in Appendix E.

A fibre bundle over the circle would be defined in terms of a projection
$\phi:M\to S^{1}$. A trivial bundle $M=S^{1}\times\Sigma$ comes with natural
projections onto each factor,

$\displaystyle\phi$ $\displaystyle=\text{proj}_{1}:S^{1}\times\Sigma\to
S^{1},\quad(t,x)\mapsto t$ (7) $\displaystyle\pi$
$\displaystyle=\text{proj}_{2}:S^{1}\times\Sigma\to\Sigma,\quad\ (t,x)\mapsto
x.$ (8)

For a given $t\in S^{1}$, consider the inclusion map of each fibre

$\displaystyle i_{t}$ $\displaystyle:\Sigma\subset S^{1}\times\Sigma,\quad
x\mapsto(t,x).$ (9)

It is useful to list the following identities about these projection and
inclusion maps. With $(t,x)\in S^{1}\times\Sigma$, we have

$\displaystyle\pi\circ i_{t}$ $\displaystyle=\text{id}_{\Sigma},$ (10)
$\displaystyle\phi\circ i_{t}$ $\displaystyle=t,$ (11) $\displaystyle
d\pi_{(t,x)}\circ(di_{t})_{x}$ $\displaystyle=\text{id}_{T_{x}\Sigma},$ (12)
$\displaystyle d\phi_{(t,x)}\circ(di_{t})_{x}$ $\displaystyle=0$ (13)

In words, the projection $\pi$ is a left-inverse of the inclusion $i_{t}$ for
each $t\in S^{1}$. Consequently, as pullback operators on $k$-forms, we have

$\displaystyle i_{t}^{*}\pi^{*}$
$\displaystyle=\text{id}_{\Omega^{k}(\Sigma)},$ $\displaystyle
i_{t}^{*}\phi^{*}$ $\displaystyle=0.$ (14)

It is immediately recognised that the pre-images of the projection $\phi$ onto
the circle are copies of the fibre $\Sigma$, namely
$i_{t}(\Sigma)=\phi^{-1}(t)$ for all $t\in S^{1}$.

######  Lemma III.1.

For any point $(t,x)\in S^{1}\times\Sigma$, $(di_{t})_{x}(T_{x}\Sigma)=\ker
d\phi_{(t,x)}$.

###### Proof.

That $(di_{t})_{x}(T_{x}\Sigma)\subseteq\ker d\phi_{x}$ is a consequence of
identity in Equation 13. The subspace $\ker d\phi_{(t,x)}$ has co-dimension
one since $\phi$ is nowhere critical. Since the inclusion map $i_{t}$ is
injective, the co-dimension of $(di_{t})_{x}(T_{x}\Sigma)$ is also one, thus
both subspaces coincide. ∎

As a closed one-form on $S^{1}\times\Sigma$, $d\phi$ represents a basis class
of first cohomology. We let $\int_{\Gamma_{T}}d\phi=2\pi$ where $\Gamma_{T}$
is a toroidal loop, representing the generator of first homology of the
$S^{1}$ factor.

###  III.3 Porting non-autonomous Hamiltonian systems into cosymplectic
manifolds

A modern treatment of Hamiltonian systems generalises the canonical cotangent
bundle to consider instead a $2n$-manifold $\Sigma$ together with a symplectic
form $\omega_{o}\in\Omega^{2}(\Sigma)$. An (autonomous) Hamiltonian system is
the dynamics of a Hamiltonian vector field $X_{H_{o}}$ on $\Sigma$ induced by
a (single) Hamiltonian function $H_{o}:\Sigma\to\mathbb{R}$ satisfying

$\displaystyle\iota_{X_{H_{o}}}\omega_{o}=-dH_{o}.$

A non-autonomous Hamiltonian system in this setting is a smooth family of
Hamiltonian functions $H_{t}:\Sigma\to\mathbb{R}$ inducing the non-autonomous
Hamiltonian vector field $X_{t}$ on $\Sigma$, via
$\iota_{X_{t}}\omega_{o}=-dH_{t}$. Here, the time-parameter takes values on
the circle, $t\in S^{1}$. The symplectic form
$\omega_{o}\in\Omega^{2}(\Sigma)$ is fixed (time-independent). A Dirichlet
boundary condition is imposed

$\displaystyle{\jmath}^{*}dH_{t}=0$

for all $t\in S^{1}$, where ${\jmath}:\partial\Sigma\subset\Sigma$. This
guarantees that the flow exists for all time (complete flow) when $\Sigma$ is
compact.

It is natural to view the dynamics of our non-autonomous Hamiltonian system as
taking place on the $2n+1$ dimensional trivial bundle $S^{1}\times\Sigma$. The
smooth family of Hamiltonian functions $H_{t}$ converts to the smooth function
$H:S^{1}\times\Sigma\to\mathbb{R}$, $(t,x)\mapsto H_{t}(x)$. The goal is to
construct an evolution vector field $E_{H}=\partial_{t}+X_{H}$ on
$S^{1}\times\Sigma$ analogously to Equation 6.

####  III.3.1 Time component of the evolution vector field

The time vector field $\partial_{t}\in\Gamma(T(S^{1}\times\Sigma))$ is
uniquely characterised by

$\displaystyle d\pi\circ\partial_{t}$ $\displaystyle=0,$ $\displaystyle
d\phi(\partial_{t})$ $\displaystyle=1.$

Defining the closed two-form
$\omega=\pi^{*}\omega_{o}\in\Omega^{2}(S^{1}\times\Sigma)$, the statement
$d\pi\circ\partial_{t}=0$ is equivalent to $\iota_{\partial_{t}}\omega=0$ by
the non-degeneracy of $\omega_{o}\in\Omega^{2}(\Sigma)$. The time vector field
$\partial_{t}\in\Gamma(T(S^{1}\times\Sigma))$ is thus uniquely determined by

$\displaystyle\partial_{t}$ $\displaystyle\in\ker\omega,$ $\displaystyle
d\phi(\partial_{t})$ $\displaystyle=1.$

Since $\omega^{n}\wedge d\phi\neq 0$, the tuple
$(S^{1}\times\Sigma,\omega,d\phi)$ forms what is known as a _cosymplectic
manifold_ [38, 39] whose _Reeb field_ is precisely $\partial_{t}$.

######  Definition III.2.

The triple $(M,\beta,\eta)$, where $M$ is a $2n+1$-manifold with boundary,
$\beta$ a closed two-form on $M$, and $\eta$ a closed one-form on $M$, is said
to be a cosymplectic manifold if $\beta^{n}\wedge\eta$ is a non-vanishing top
form and $\beta$ satisfies the Dirichlet boundary condition $j^{*}\beta=0$
with $j:\partial M\to M$. The unique vector field $R$ such that

$\displaystyle R$ $\displaystyle\in\ker\beta,$ $\displaystyle\eta(R)$
$\displaystyle=1$

is called the Reeb field of $(M,\beta,\eta)$.

The integral curves of $\partial_{t}$ are excessively simple in the sense that
they fix points on $\Sigma$ and translate them along the circle direction. In
particular, all integral curves of $\partial_{t}$ are recurrent with the
circle’s period, or in other words, its flow $\Phi_{s}^{\partial_{t}}$ has the
property that $\Phi^{\partial_{t}}_{2\pi}=\text{id}_{M}$. The vector field
$\partial_{t}$ gives rise to an Ehresmann connection on the trivial fibre
bundle $S^{1}\times\Sigma$. While viewing $\partial_{t}$ as an Ehresmann
connection is excessive in the present setting, we show the utility of this
perspective for non-trivial fibre bundles in Appendix E.

####  III.3.2 Hamiltonian component of the evolution vector field

A characterisation of the vector field $X_{H}$ is that it is tangent to every
copy of $\Sigma$ in the product $S^{1}\times\Sigma$, and that it agrees with
the vector field $X_{t}$ on the specific copy where the last coordinate is
equal to $t\in S^{1}$. This means that for a given time coordinate $t\in
S^{1}$, $X_{t}$ and $X_{H}$ are related by the push-forward of the inclusion
map $i_{t}:\Sigma\to S^{1}\times\Sigma$ of Equation 9,

$\displaystyle di_{t}\circ X_{t}=X_{H}\circ i_{t}.$ (15)

It is noted that $X_{H}$ automatically belongs to $\ker d\phi$. Rather than
through the family of Hamiltonian fields $\\{X_{t}\\}_{t\in S^{1}}$, we wish
to characterise $X_{H}$ through the function $H$. For this, the following
result applies.

######  Lemma III.3.

The vector field $X_{H}\in\ker d\phi$ in Equation 15 is determined by

$\displaystyle\iota_{X_{H}}\omega$ $\displaystyle=-dH+\partial_{t}Hd\phi.$

###### Proof.

Indeed, by naturality of the interior product, we have
$i_{t}^{*}\iota_{X_{H}}\omega=\iota_{X_{t}}\omega_{o}$ and $i_{t}^{*}H=H\circ
i_{t}=H_{t}$. This implies $i_{t}^{*}dH=di_{t}^{*}H=dH_{t}$ for all $t\in
S^{1}$, so both sides agree when restricted to copies of $\Sigma$, namely on
$\ker d\phi$. The expression also holds on
$\ker\omega=\text{span}\\{\partial_{t}\\}$; the left-hand side immediately
evaluates to zero, and the extra term on the right-hand side makes the latter
vanish too,

$\displaystyle-
dH(\partial_{t})+\partial_{t}H\cancelto{1}{d\phi(\partial_{t})}=-\partial_{t}H+\partial_{t}H=0.$

∎

####  III.3.3 The evolution vector field and the related cosymplectic manifold

The construction of the evolution vector field is summarised in the following
proposition.

######  Proposition III.4.

Let $(\Sigma,\omega_{o},\\{H_{t}\\}_{t\in S^{1}})$ be a non-autonomous
Hamiltonian system. The evolution vector field $E_{H}$ on $S^{1}\times\Sigma$
associated to the family of Hamiltonian fields $\\{X_{t}\\}_{t\in S^{1}}$ is
given by

$\displaystyle E_{H}=\partial_{t}+X_{H},$

where the time component $\partial_{t}$ and the Hamiltonian component $X_{H}$
are (uniquely) determined by

$\displaystyle\partial_{t}$ $\displaystyle\in\ker\omega,$ $\displaystyle
d\phi(\partial_{t})$ $\displaystyle=1$ $\displaystyle X_{H}$
$\displaystyle\in\ker d\phi,$ $\displaystyle\iota_{X_{H}}\omega$
$\displaystyle=-dH+\partial_{t}Hd\phi,$

where $H:S^{1}\times\Sigma\to\mathbb{R}$ is the smooth function $(t,x)\mapsto
H_{t}(x)$, $\omega=\pi^{*}\omega_{o}$, and $\phi:S^{1}\times\Sigma\to S^{1}$,
$\pi:S^{1}\times\Sigma\to\Sigma$ are the natural projections.

######  Remark III.5.

In Section III.3.1, we identified a cosymplectic manifold
$(S^{1}\times\Sigma,\omega,d\phi)$ such that $\partial_{t}$ was its Reeb
field. By modifying the cosymplectic system, the Reeb field may be adjusted to
become the evolution vector field $E_{H}$. Indeed, take $\beta_{o}=\omega-
dH\wedge d\phi$ and note that, as above, $\beta_{o}^{n}\wedge
d\phi=\omega^{n}\wedge d\phi\neq 0$ so that
$(S^{1}\times\Sigma,\beta_{o},d\phi)$ is a cosymplectic manifold. The Reeb
field $R_{o}$ of this cosymplectic structure satisfies $R_{o}\in\ker\beta_{o}$
and $d\phi(R_{o})=1$. A calculation reveals that $R_{o}=E_{H}$. Therefore, one
can always view the evolution field $E_{H}$ for a given $H$ as being the
(unique) Reeb field of $\beta_{o}$.

Propositions III.4 and III.5 show how to port the dynamics of a non-autonomous
Hamiltonian system from the symplectic manifold $(\Sigma,\omega_{o})$ into the
cosymplectic manifold $(\Sigma\times S^{1},\beta_{o},d\phi)$ whose Reeb field
is an evolution vector field. This motivates the following definition.

######  Definition III.6.

A cosymplectic manifold $(M,\beta,\eta)$ is said to be represented by the non-
autonomous Hamiltonian system $(\Sigma,\omega_{o},\\{H_{t}\\}_{t\in S^{1}})$
if there exists a diffeomorphism $\Psi:S^{1}\times\Sigma\to M$ such that
$\Psi^{*}\eta=d\phi$ and

$\Psi^{*}\beta=\omega-dH\wedge d\phi$

where $\omega=\pi^{*}\omega_{o}$ and $H\in C^{\infty}(S^{1}\times\Sigma)$ is
given by $H(t,x)=H_{t}(x)$.

######  Remark III.7.

The representation entails a decomposition of
$\beta=\Omega-d\mathcal{H}\wedge\eta$ where
$\Omega=\Upsilon^{*}\omega=\Upsilon^{*}\pi^{*}\omega_{o}$ and
$\mathcal{H}=\Upsilon^{*}H$ with $\Upsilon=\Psi^{-1}$. Consequently, the Reeb
field $R$ of $(M,\beta,\eta)$ also admits a decomposition
$R=\mathcal{T}+\mathcal{X}_{\mathcal{H}}$ where
$\mathcal{T}=\Psi_{*}\partial_{t}$ and
$\mathcal{X}_{\mathcal{H}}=\Psi_{*}X_{H}$ neatly characterised by

$\displaystyle\mathcal{T}$ $\displaystyle\in\ker\Omega,$
$\displaystyle\eta(\mathcal{T})$ $\displaystyle=1,$
$\displaystyle\mathcal{X}_{\mathcal{H}}$ $\displaystyle\in\ker\eta,$
$\displaystyle\iota_{\mathcal{X}_{\mathcal{H}}}\Omega$
$\displaystyle=-d\mathcal{H}+d\mathcal{H}(\mathcal{T})\eta.$

The flow of $\mathcal{T}$, denoted $\Phi^{\mathcal{T}}_{s}$, is $\Psi$-related
to the flow of $\partial_{t}$, in the sense that
$\Psi\circ\Phi_{s}^{\partial_{t}}=\Phi_{s}^{\mathcal{T}}\circ\Psi$. Hence,
$\Phi_{2\pi}^{\mathcal{T}}=\text{id}_{M}$ and the field-lines of $\mathcal{T}$
are consequently recurrent with the circle’s period. The vector field
$\mathcal{T}$ is called the _time-like part_ of $R$ and the vector field
$\mathcal{X}_{\mathcal{H}}$ is called the _Hamiltonian part_ of $R$. In
Appendix E, the time-like part is viewed as an Ehresmann connection producing
a monodromy map of the fibre bundle homotopic to the identity.

We will now consider an important class of examples that enter in the proof of
our main result, Theorem IV.5.

######  Example III.8.

An important class of examples fitting Definition III.6 is generated by
diffeomorphisms $\Psi:M\to M$ on the trivial bundle $M=S^{1}\times\Sigma$
satisfying $\phi\circ\Psi=\phi$. These are known as _bundle automorphisms_. We
begin with a non-autonomous Hamiltonian system on $(\Sigma,\omega_{o})$ with
time-dependent Hamiltonian $H\in C^{\infty}(S^{1}\times\Sigma)$ and use
Proposition III.4 to port this system into the cosymplectic manifold
$(S^{1}\times\Sigma,\beta_{o},d\phi)$ where, as in Remark III.5,
$\beta_{o}=\omega-dH\wedge d\phi$ with $\omega=\pi^{*}\omega_{o}$. Then,
setting $\Upsilon=\Psi^{-1}$, the cosymplectic manifold

$(S^{1}\times\Sigma,\beta,d\phi)=(S^{1}\times\Sigma,\Upsilon^{*}\beta_{o},d\phi)$

is represented by, and therefore equivalent to, the dynamics of the non-
autonomous system $(\Sigma,\omega_{o},H)$.

On the other hand, the Reeb field of $(S^{1}\times\Sigma,\beta,d\phi)$ induces
dynamics on $\Sigma$ via the family of functions
$\mathcal{H}_{t}=i_{t}^{*}\mathcal{H}$ where $\mathcal{H}=\Upsilon^{*}H$ and
the family of symplectic two-forms $\omega_{t}=i_{t}^{*}\beta$ yielding the
family of Hamiltonian vector fields $\mathcal{X}_{t}$ satisfying

$\iota_{\mathcal{X}_{t}}\omega_{t}=-d\mathcal{H}_{t},$

which we consider as a natural generalisation of a non-autonomous Hamiltonian
system on $\Sigma$. In local _non-canonical_ coordinates
$z=(z^{1},\ldots,z^{2n})\in\Sigma$, such a system would be represented by the
differential equations

$\displaystyle\frac{dz^{i}}{dt}(t)=\Pi^{ij}(t,z(t))\frac{\partial\mathcal{H}}{\partial
z^{j}}(t,z(t))$

where $\Pi^{ij}$ is the _Poisson tensor_ defined as the inverse of the
symplectic tensor [40]; if
$(\omega_{t})_{ij}:=\omega_{t}(\partial_{z_{i}},\partial_{z_{j}})$ then
$(\omega_{t})_{ij}(t,z)\Pi^{jk}(t,z)=\delta_{i}^{k}$.

In the next section, we consider the slightly more general notion of _a non-
autonomous locally-Hamiltonian system_.

###  III.4 Non-autonomous locally-Hamiltonian systems

The essential idea of a locally-Hamiltonian system is that the usual exact
one-forms $dH_{t}$ derived from the Hamiltonian function can be replaced by
closed one-forms $\nu_{t}$ on $\Sigma$. This generalisation is elementary from
the perspective that closed one-forms are locally exact.

The triple $(\Sigma,\omega_{o},\\{\nu_{t}\\}_{t\in S^{1}})$ is said to be a
non-autonomous locally-Hamiltonian system if $\Sigma$ is a manifold with
(possibly empty) boundary, $\omega_{o}$ is a symplectic two-form on $\Sigma$,
and $\\{\nu_{t}\\}_{t\in S^{1}}$ is a smooth family of closed one-forms on
$\Sigma$, namely $d\nu_{t}=0$ for all $t\in S^{1}$. The smooth adjective means
that the assignment $S^{1}\times\Sigma\to T^{*}\Sigma$,
$(t,x)\mapsto\nu_{t}\big{|}_{x}$ is smooth. A Dirichlet boundary condition is
imposed

${\jmath}^{*}\nu_{t}=0$

for all $t\in S^{1}$, where ${\jmath}:\partial\Sigma\subset\Sigma$ is the
inclusion. The dynamics of $(\Sigma,\omega_{o},\\{\nu_{t}\\}_{t\in S^{1}})$ on
$\Sigma$ are given by the flow of the non-autonomous vector field $X_{t}$
defined by

$i_{X_{t}}\omega_{o}=\nu_{t}$

which is necessarily tangent to $\partial\Sigma$ by the Dirichlet boundary
condition on $\nu_{t}$ and makes the flow complete when $\Sigma$ is compact.

An important part of this paper is determining when a non-autonomous locally-
Hamiltonian system is in fact a (globally) non-autonomous Hamiltonian system.
The other direction is always true: if one begins with a (globally) non-
autonomous Hamiltonian system $(\Sigma,\omega_{o},\\{H_{t}\\}_{t\in S^{1}})$,
then the family of closed Dirichlet one-forms $\\{\nu_{t}=-dH_{t}\\}_{t\in
S^{1}}$ is smooth, and $(\Sigma,\omega_{o},\\{\nu_{t}=-dH_{t}\\}_{t\in
S^{1}})$ is a non-autonomous locally-Hamiltonian system. However, beginning
from a non-autonomous locally Hamiltonian system
$(\Sigma,\omega_{o},\\{\nu_{t}\\}_{t\in S^{1}})$, if there existed an
arbitrary family $\\{\tilde{H}_{t}\\}_{t\in S^{1}}$ of functions satisfying
$-d\tilde{H}_{t}=\nu_{t}$ for each $t\in S^{1}$, it is not immediate that it
may be replaced by a smooth family $\\{H_{t}\\}_{t\in S^{1}}$ satisfying
$-dH_{t}=\nu_{t}$. The following lemma, whose proof is contained in Appendix
B, resolves this. A similar result was shown in [41] for $t\in\mathbb{R}^{m}$.

######  Lemma III.9.

Let $(\Sigma,\omega_{o},\\{\nu_{t}\\}_{t\in S^{1}})$ be a non-autonomous
locally-Hamiltonian system, and suppose that $\nu_{t}$ is exact for each $t\in
S^{1}$. Then, there exists a smooth family $\\{H_{t}\\}_{t\in S^{1}}$ of
functions such that

$\nu_{t}=-dH_{t}.$

In particular, $(\Sigma,\omega_{o},\\{H_{t}\\}_{t\in S^{1}})$ is a non-
autonomous Hamiltonian system.

In order to deal more directly with non-autonomous locally-Hamiltonian
systems, we continue the analogy with the evolution vector field introduced
for non-autonomous (globally) Hamiltonian systems. In the local situation, we
can still port the dynamics into a cosymplectic manifold similarly to
Proposition III.4.

######  Proposition III.10.

Let $(\Sigma,\omega_{o},\\{\nu_{t}\\}_{t\in S^{1}})$ be a non-autonomous
locally-Hamiltonian system. The evolution vector field $E_{H}$ on
$S^{1}\times\Sigma$ associated to the family of locally-Hamiltonian fields
$\\{X_{t}\\}_{t\in S^{1}}$ is given by

$\displaystyle E_{\nu}=\partial_{t}+X_{\nu},$

where the time component $\partial_{t}$ and the locally-Hamiltonian component
$X_{\nu}$ are (uniquely) determined by

$\displaystyle\partial_{t}$ $\displaystyle\in\ker\omega,$ $\displaystyle
d\phi(\partial_{t})$ $\displaystyle=1$ $\displaystyle X_{\nu}$
$\displaystyle\in\ker d\phi,$ $\displaystyle\iota_{X_{\nu}}\omega$
$\displaystyle=\nu-\nu(\partial_{t})d\phi,$

where $\nu\in\Omega^{1}(S^{1}\times\Sigma)$ is such that
$i_{t}^{*}\nu=\nu_{t}$ for all $t\in S^{1}$, $\omega=\pi^{*}\omega_{o}$ and
$\phi:S^{1}\times\Sigma\to S^{1}$, $\pi:S^{1}\times\Sigma\to\Sigma$ are the
natural projections.

By virtue of Proposition III.10, we obtain a generalisation of Definition
III.6.

######  Definition III.11.

A cosymplectic manifold $(M,\beta,\eta)$ is said to be represented by the non-
autonomous locally-Hamiltonian system $(\Sigma,\omega_{o},\\{\nu_{t}\\}_{t\in
S^{1}})$ if there exists a diffeomorphism $\Psi:S^{1}\times\Sigma\to M$ such
that $\Psi^{*}\eta=d\phi$ and

$\displaystyle\Psi^{*}\beta=\omega+\nu\wedge d\phi$

where $\omega=\pi^{*}\omega_{o}$ and $\nu\in\Omega^{1}(S^{1}\times\Sigma)$
satisfies $i_{t}^{*}\nu=\nu_{t}$ for all $t\in S^{1}$.

######  Remark III.12.

The restriction of $\nu$ to a copy of $\Sigma$ agrees with $\nu_{t}$ at the
corresponding angle $t\in S^{1}$, which means that $\nu$ is determined by the
smooth family of one-forms $\\{\nu_{t}\\}_{t\in S^{1}}$ up to a term of the
form $fd\phi$ for some function $f$. An explicit representative is, for
example,

$\displaystyle\hat{\nu}|_{(t,x)}=(\hat{\nu}_{t})|_{x}\circ
d\pi_{(t,x)}=(\pi^{*}\hat{\nu}_{t})|_{(t,x)},$

which, incidentally, is Dirichlet on $S^{1}\times\Sigma$. Indeed, for each
$(t,x)\in\partial(S^{1}\times\Sigma)=S^{1}\times\partial\Sigma$,

$\displaystyle(j^{*}\hat{\nu})|_{(t,x)}$
$\displaystyle=\hat{\nu}|_{(t,x)}\circ dj_{(t,x)}=(\hat{\nu}_{t})|_{x}\circ
d\pi_{(t,x)}\circ dj_{(t,x)}=(\hat{\nu}_{t})|_{x}\circ d(\pi\circ
j)_{(t,x)}=(\hat{\nu}_{t})|_{x}\circ d({\jmath}\circ\pi_{\partial})_{(t,x)}$
$\displaystyle=(\hat{\nu}_{t})|_{x}\circ d{\jmath}_{x}\circ
d{\pi_{\partial}}_{(t,x)}=({\jmath}^{*}\hat{\nu}_{t})|_{x}\circ
d{\pi_{\partial}}_{(t,x)}=0,$

where $j:\partial(S^{1}\times\Sigma)\to S^{1}\times\Sigma$ and
${\jmath}:\partial\Sigma\to\Sigma$ are the inclusions and
$\pi_{\partial}:\partial(S^{1}\times\Sigma)\to\partial\Sigma$ is the
projection, so that $\pi\circ j={\jmath}\circ\pi_{\partial}$.

Representative one-forms $\nu$ in Proposition III.10 are closed on each copy
of $\Sigma$, i.e. $i_{t}^{*}d\nu=di_{t}\nu=d\nu_{t}=0$. Unfortunately, there
may not exist a choice that is closed on all of $S^{1}\times\Sigma$. This is a
serious obstacle to reconciling the notion of locally-Hamiltonian to its
global counter-part. The following lemma precisely describes the issue.

######  Lemma III.13.

Let $\\{\nu_{t}\\}_{t\in S^{1}}$ be a smooth family of closed one-forms on a
manifold $\Sigma$ with boundary. Then, the following are equivalent.

  1. 1.

There exists a closed one-form $\nu$ on $S^{1}\times\Sigma$ such that
$i_{t}^{*}\nu=\nu_{t}$ for each $t\in S^{1}$;

  2. 2.

The cohomology classes $[\nu_{t}]$ are constant.

###### Proof.

Assume that the one-form $\nu$ on $S^{1}\times\Sigma$ such that
$i_{t}^{*}\nu=i_{t}^{*}\nu=\nu_{t}$ for each $t\in S^{1}$ is closed. If one is
given a smooth closed curve $\gamma:[0,1]\to\Sigma$, then for any $s,t\in
S^{1}$, the curves $\gamma_{s}=i_{s}\circ\gamma$ and
$\gamma_{t}=i_{t}\circ\gamma$ are cohomologous on $S^{1}\times\Sigma$ and thus

$\int_{\gamma}\nu_{s}=\int_{\gamma_{s}}\nu=\int_{\gamma_{t}}\nu=\int_{\gamma}\nu_{t}$

which implies that the cohomology classes $[\nu_{s}]$ and $[\nu_{t}]$
coincide.

Now assume that the smooth family $\\{\nu_{t}\\}_{t\in S^{1}}$ of closed one-
forms has constant cohomology class. Then, fixing $o\in S^{1}$,
$\\{\nu_{t}-\nu_{o}\\}_{t\in S^{1}}$ is a smooth family of exact one-forms.
Therefore, by Lemma III.9, there exists a smooth family $\\{H_{t}\\}_{t\in
S^{1}}$ of functions such that $\nu_{t}-\nu_{o}=-dH_{t}$ for each $t\in
S^{1}$. That is, $\nu_{t}=-dH_{t}+\nu_{o}$. Defining the smooth function $H$
on $S^{1}\times\Sigma$ via $H(t,x)=H_{t}(x)$, the one-form

$\displaystyle\nu=-dH+\pi^{*}\nu_{o}$

is manifestly closed and, by virtue of Equation 14, such that
$i_{t}^{*}\nu=-di_{t}^{*}H+i_{t}^{*}\pi^{*}\nu_{o}=-dH_{t}+\nu_{o}=\nu_{t}$
for all $t\in S^{1}$ as required. ∎

The obstacle highlighted by Lemma III.13 is subtle. On a contractible
neighbourhood $U$ of any point in $\Sigma$, the closed family of
$\\{\nu_{t}\\}_{t\in S^{1}}$ are exact, which means that a global function
$h:S^{1}\times U\to\mathbb{R}$ exists such that $\nu_{t}|_{U}=-dh_{t}$ for
each $t\in S^{1}$. The temptation is to set $\tilde{\nu}=-dh$ on the extended
neighbourhood $S^{1}\times U$ as a candidate to close the gap between locally-
Hamiltonian and (globally) Hamiltonian. The strategy of covering $\Sigma$ with
contractible neighbourhoods in an attempt to glue $\tilde{\nu}$ into a
globally closed one-form on $S^{1}\times\Sigma$ will however fail unless the
conditions of Lemma III.13 are met.

It is easy to construct non-examples of Lemma III.13, including cases with
non-empty boundary. Indeed, if $(\Sigma,\omega_{o})$ is a symplectic manifold
with boundary admitting a closed Dirichlet one-form $\nu_{o}$ with non-trivial
de Rham cohomology class (see Appendix A), a family $\\{\nu_{t}\\}_{t\in
S^{1}}$ of Dirichlet closed one-forms with varying cohomology classes on
$\Sigma$ can be generated. For instance, taking $\nu_{t}=(2+\sin t)\nu_{o}$,
the cohomology class varies in $t$ and therefore there cannot be a closed
$\nu$ on $S^{1}\times\Sigma$ such that $i_{t}^{*}\nu=\nu_{t}$.

Remarkably though, when $\Sigma$ is a compact regular domain of
$\mathbb{R}^{m}$ where $m=2n$ (even with non-zero first de Rham cohomology
groups), such non-examples cannot be found. Briefly, closed Dirichlet one-
forms on such $\Sigma$ are always exact, in which case the notion of non-
autonomous Hamiltonian system automatically collapses to its global counter-
part.

######  Proposition III.14.

Let $\Sigma\subset\mathbb{R}^{m}$ be a compact regular domain. The boundary
inclusion map $j:\partial\Sigma\subset\Sigma$ is injective on the first
(absolute) cohomology as a pull-back map

$\displaystyle j^{*}:H_{\text{dR}}^{1}(\Sigma)\to
H_{\text{dR}}^{1}(\partial\Sigma).$

In particular, if $\nu_{o}$ is a closed one-form on $\Sigma$ such that
$j^{*}\nu_{o}=0$, then $\nu_{o}=-dH_{o}$ for some $H_{o}\in
C^{\infty}(\Sigma)$.

###### Proof.

We follow Cantarella’s sketch [42] for $m=3$ adapted to arbitrary $m$ with the
details filled-in for completeness. Specifically, we first work with the first
singular homology of $\Sigma$ via a Mayer-Vietoris sequence as follows. Recall
that $\Sigma$ has a normal collar, namely a smooth embedding
$\sigma:\partial\Sigma\times[0,1)\to\Sigma$. Let $C$ be the image of $\sigma$
and consider the open set

$N=\mathbb{R}^{m}\backslash(\Sigma\setminus
C)=(\mathbb{R}^{m}\setminus\Sigma)\cup C$

of $\mathbb{R}^{m}$. Then, the interiors of $\Sigma$ and $N$ in
$\mathbb{R}^{m}$ cover $\mathbb{R}^{m}$, so we may apply the Mayer-Vietoris
sequence to $\Sigma$ and $N$. Part of the sequence reads

$...\to H_{2}(\mathbb{R}^{m})\to H_{1}(\Sigma\cap N)\to H_{1}(\Sigma)\oplus
H_{1}(N)\to H_{1}(\mathbb{R}^{m})\to...$

which, because $C\cong\partial\Sigma\times[0,1)$ deformation retracts to
$\partial\Sigma\subset C$, reduces to

$0\to H_{1}(\partial\Sigma)\to H_{1}(\Sigma)\oplus H_{1}(N)\to 0.$

The map $H_{1}(\partial\Sigma)\to H_{1}(\Sigma)\oplus H_{1}(N)$ is
$(j_{*},k_{*})$ where $j:\partial\Sigma\subset\Sigma$ and
$k:\partial\Sigma\subset N$ are the inclusions. By exactness of the sequence,
$(j_{*},k_{*})$ is a surjection. In particular, the map
$j_{*}:H_{1}(\partial\Sigma)\to H_{1}(\Sigma)$ is a surjection. Dually, by de
Rham’s Theorem (for manifolds with boundary), the pull-back map
$\jmath^{*}:H_{\text{dR}}^{1}(\Sigma)\to H_{\text{dR}}^{1}(\partial\Sigma)$ is
injective.

In particular, if $\nu_{o}$ is a closed one-form on $\Sigma$ such that
$j^{*}\nu_{o}=0$, we have that $[\nu_{o}]\in H_{\text{dR}}^{1}(\Sigma)$ is in
the kernel of $j^{*}:H_{\text{dR}}^{1}(\Sigma)\to
H_{\text{dR}}^{1}(\partial\Sigma)$, and therefore, by injectivity of $j^{*}$,
$[\nu_{o}]=0$, and therefore, $\nu_{o}=-dH$ for some $H\in
C^{\infty}(\Sigma)$. ∎

We can now state and prove that non-autonomous locally-Hamiltonian systems
reduce to non-autonomous (globally) Hamiltonian systems on compact regular
domains in Euclidean space.

######  Corollary III.15.

Let $\Sigma$ be a compact regular domain of $\mathbb{R}^{2n}$. Then, if
$(\Sigma,\omega_{o},\\{\nu_{t}\\}_{t\in S^{1}})$ is a non-autonomous locally-
Hamiltonian system, there exists a smooth family $\\{H_{t}\\}_{t\in S^{1}}$ of
functions such that

$\nu_{t}=-dH_{t}.$

In particular, $(\Sigma,\omega_{o},\\{H_{t}\\}_{t\in S^{1}})$ is a non-
autonomous Hamiltonian system.

###### Proof.

For each $t\in S^{1}$, observe that, by definition of non-autonomous locally-
Hamiltonian system, the closed one-forms are Dirichlet,

${\jmath}^{*}\nu_{t}=0.$

In particular, by Proposition III.14, $\nu_{t}$ is exact. The result now
follows from Lemma III.9. ∎

##  IV Reduction of divergence-free vector fields to non-autonomous
1$\frac{1}{2}$D Hamiltonian systems

In this section, we will use the previously introduced technology to establish
the main result around a sufficient condition on magnetic fields with Poincaré
sections to admit a non-autonomous Hamiltonian representation. We start with a
vector field $B$ tangent to the boundary of a compact orientable three-
manifold $M$ equipped with a volume form $\mu$. We ask that $B$ is divergence-
free in the sense that $\beta=\iota_{B}\mu$ is a closed two-form. The boundary
condition on $\beta$ inherited from $B$ is the Dirichlet boundary condition.

######  Lemma IV.1.

Denoting the natural inclusion of the boundary $j:\partial M\subset M$, the
following statements are equivalent.

  1. 1.

The magnetic two-form $\beta$ satisfies the _Dirichlet_ boundary condition,
namely $j^{*}\beta=0$;

  2. 2.

The magnetic field $B$ is tangential to the boundary, namely $B_{j(x)}\in
dj_{x}(T_{x}\partial M)$ for all $x\in\partial M$.

Using any metric, the latter statement amounts to the fact that the magnetic
field on the boundary is orthogonal to the normal vector, $\bm{n}\cdot
B|_{\partial M}=0$.

###### Proof.

Let $x\in\partial M$ and $w_{1}=dj_{x}(v_{1}),w_{2}=dj_{x}(v_{2})\in
dj_{x}(T_{x}\partial M)$ be two linearly independent vectors spanning the
subspace of tangential vectors to the boundary at point $j(x)$. Then,
$(j^{*}\beta)_{x}(v_{1},v_{2})=\beta_{j(x)}(w_{1},w_{2})=\mu_{j(x)}(B_{j(x)},w_{1},w_{2})$.
The Dirichlet condition $j^{*}\beta=0$ implies that $B_{j(x)}$ is linearly
dependent with $w_{1},w_{2}$, namely $B_{j(x)}$ is tangential to the boundary,
and conversely. ∎

######  Remark IV.2.

Proposition III.14 implies that all tangential magnetic fields $B$ on a
compact regular domain $M\subset\mathbb{R}^{3}$ give rise to an exact two-form
$\iota_{B}\mu=\beta=d\alpha$. Indeed, let $\nu$ be a closed one-form with
Dirichlet boundary condition. Then, $j^{*}\nu=0$ so that on cohomology,
$j^{*}[\nu]=[j^{*}\nu]=0$. By injectivity of $j^{*}$, we have that $[\nu]=0$;
that is, $\nu=dH$ for some $H$. Thus

$\int_{M}\nu\wedge\beta=\int_{M}dH\wedge\beta=\int_{M}d(H\beta)=\int_{\partial
M}H\beta=0.$

By Poincaré-Lefschetz duality, this implies that $\beta$ must be exact in the
absolute sense. That is, $\beta=d\alpha$.

We restrict our attention to magnetic fields that admit a compact global
Poincaré section; namely a compact connected embedded submanifold $\Sigma$ of
$M$ with $\partial\Sigma\subset\partial M$, such that every field-line of $B$
transversely intersects $\Sigma$. This limiting but physically motivated
assumption is equivalent to the existence of an angle function $\phi:M\to
S^{1}$ such that the $d\phi(B)>0$ has no zeros (so that, in particular, $B$
has no zeros), as a result of the following lemma whose proof is deferred to
Appendix D.

######  Lemma IV.3.

If $M$ is a compact orientable manifold with boundary and $X$ is a vector
field tangent to $\partial M$ with a global Poincaré section $\Sigma$, then
there exists an angle function $\phi:M\to S^{1}$ such that $d\phi(X)>0$ and
$\Sigma$ is a level-set of $\phi$.

The angle function enables us to view $M$ as the fibre-bundle
$(M,S^{1},\phi,\Sigma)$ over the circle. This hugely limits its topology. For
example, removing a ball from a solid torus will prevent any tangential vector
field from admitting a Poincaré section (Hairy Ball Theorem).

For reference, the positive function $d\phi(B)>0$ is called the _toroidal
component_ of $B$. In this setting, $\tilde{\mu}:=\beta\wedge d\phi$ is a
volume form and the tuple $(M,\beta,d\phi)$ forms a cosymplectic manifold. It
is straightforward to verify that the Reeb field $R_{B}$ such that
$R_{B}\in\ker\beta$ and $d\phi(R_{B})=1$ is the magnetic field normalised to
its toroidal component,

$\displaystyle R_{B}=\frac{B}{d\phi(B)}.$

It is divergence-free with respect to the volume-form $\tilde{\mu}=\beta\wedge
d\phi$. The field-lines of the Reeb field and those of the magnetic field
coincide. The fact that $R_{B}$ is normalised to one in the circle direction,
$d\phi(R)=1$, makes the correspondence between the circle value of an integral
curve (field-line) and the time-parameter simple. It is in this sense that the
circle direction will constitute the “$\frac{1}{2}$” dimension of the
equivalent Hamiltonian system.

We restrict to the case where $M=S^{1}\times\Sigma$, where $\Sigma$ is an
orientable connected compact two-manifold possibly with boundary, and
$\phi:M\to S^{1}$ is the natural projection. The standard case of a solid
torus embedded in real three-space, $\hat{M}\subseteq\mathbb{R}^{3}$ and
$\hat{M}\cong S^{1}\times D^{2}$, is an important example covered by our
analysis. If one has a divergence-free vector field $\hat{B}$ on a solid torus
$\hat{M}\subset\mathbb{R}^{3}$, if $F:\hat{M}\to S^{1}\times D^{2}$ is a
diffeomorphism such that $F^{*}\phi=\hat{\phi}$ is an angle function with
$d\hat{\phi}(\hat{B})>0$, then the algebraic results will be based on the two-
form $\beta$ on $S^{1}\times D^{2}$ such that $F^{*}\beta=\hat{\beta}$, and
the conclusions carry over to the original two-form $\beta$ diffeomorphically.
Another application of our results is to the hollow torus $\tilde{M}\cong
S^{1}\times A_{1,2}$ embedded in real three-space, where $A_{1,2}$ is an
annulus (or cylinder), see Equation 2.

In fact, as soon as the Poincaré section $\Sigma$ is diffeomorphic to a disk
or an annulus, then $\hat{M}$ is necessarily a trivial bundle over the circle,
as concluded from Theorem E.15 presented in Appendix E.

######  Corollary IV.4.

Let $(\hat{M},S^{1},\hat{\phi},\Sigma)$ be a fibre-bundle over the circle with
typical fibre $\Sigma$. If $\Sigma$ is diffeomorphic to the disk or annulus,
then $(\hat{M},S^{1},\hat{\phi},\Sigma)$ is necessarily trivialisable. That
is, there exists a diffeomorphism $F:\hat{M}\to S^{1}\times\Sigma$ such that
$F^{*}\phi=\hat{\phi}$.

###### Proof.

By Theorem E.15, when $\Sigma$ is a disk or an annulus, the mapping class
group is trivial. Moreover, from Theorem E.9, the conjugacy classes of the
mapping class group are in one-to-one correspondence with isomorphism classes
of fibre bundles with fibre $\Sigma$ (Theorem E.9). It follows that all fibre
bundles with fibre $\Sigma$ an annulus or disk are bundle isomorphic to the
trivial bundle. ∎

Now, we ask whether the cosymplectic manifold
$(S^{1}\times\Sigma,\beta,d\phi)$ can be represented by some non-autonomous
Hamiltonian as in Definitions III.6 and III.7. Using a general Moser’s trick
(see Theorem IV.12, proven later in this section), and Corollary III.15, we
obtain the following result in the case where $\Sigma$ is planar; that is,
when $\Sigma$ is a compact regular domain in $\mathbb{R}^{2}$.

######  Theorem IV.5.

Consider the cosymplectic manifold $(S^{1}\times\Sigma,\beta,d\phi)$ where
$\Sigma$ is planar. Then, $(S^{1}\times\Sigma,\beta,d\phi)$ admits a non-
autonomous Hamiltonian representation. In fact, there exists a smooth family
of functions $\\{H_{t}\\}_{t\in S^{1}}$ such that
$(\Sigma,i_{o}^{*}\beta,\\{H_{t}\\}_{t\in S^{1}})$ is a non-autonomous
Hamiltonian system and there exists a bundle automorphism $\Psi$ such that

$\beta=\Omega-d\mathcal{H}\wedge d\phi$

where $(\Psi^{*}\mathcal{H})(t,x)=H_{t}(x)$ and
$\Psi^{*}\Omega=\pi^{*}i_{o}^{*}\beta$ for an arbitrary (fixed) reference
angle on the circle $o\in S^{1}$.

######  Remark IV.6.

We note that if $\Sigma$ is a given two-manifold with boundary such that the
product $M=S^{1}\times\Sigma$ can be embedded in $\mathbb{R}^{3}$, then
$\Sigma$ is automatically planar. This fact is proven in Appendix C.

We conclude this section by highlighting the following remarkable conclusion;
any volume-preserving vector field on a three-manifold with a global Poincaré
section diffeomorphic to a disk or an annulus admits a non-autonomous
Hamiltonian representation. This is a direct application of Theorems IV.5,
IV.3 and IV.4, which relies upon the classification of bundles over a circle
and the relation to mapping class groups, detailed in Appendix E.

The next section revolves around the proof of Theorem IV.5.

###  IV.1 Global procedure applying Moser’s trick

We consider the family of top-forms on $\Sigma$ labelled by $t\in S^{1}$

$\displaystyle\omega_{t}:=i_{t}^{*}\beta.$

Because $\beta\wedge d\phi\neq 0$, we have for all linearly independent vector
fields $Y_{1},Y_{2}\in\ker d\phi$ that $\iota_{Y_{2}}\iota_{Y_{1}}(\beta\wedge
d\phi)=\beta(Y_{1},Y_{2})d\phi\neq 0\iff\beta(Y_{1},Y_{2})\neq 0$. This means
that $\beta$ is non-degenerate on the kernel of $d\phi$, or equivalently that
each two-form $\omega_{t}$ is non-degenerate on $\Sigma$ so that they form a
family of symplectic forms. As top-forms, $\omega_{t}$ satisfy a Dirichlet
boundary condition, namely ${\jmath}^{*}\omega_{t}=0$, $\forall t\in S^{1}$
where ${\jmath}:\partial\Sigma\to\Sigma$.

It is well-known in plasma physics that the flux through any cross-section is
invariant, provided the magnetic field is tangential to the boundary of the
toroidal domain considered. We capture this physical property in the following
lemma.

######  Lemma IV.7.

The magnetic flux through the level sets of $\phi$ is constant, namely
$\forall a,b\in S^{1}$,

$\displaystyle\int\limits_{\phi^{-1}(a)}\beta=\int\limits_{\phi^{-1}(b)}\beta.$

###### Proof.

Assume $a\neq b$. Write $a=a_{0}+2\pi\mathbb{Z}$ and $b=b_{0}+2\pi\mathbb{Z}$
where $0<b_{0}-a_{0}<2\pi$. Consider the path $\gamma:[0,1]\to S^{1}$ where
$\gamma(t)=a_{0}+(b_{0}-a_{0})t+2\pi\mathbb{Z}$. Then, the map
$V:[0,1]\times\Sigma\to S^{1}\times\Sigma$ given by

$V(t,x)=(\gamma(t),x)$

is an embedding of the manifold with corners $[0,1]\times\Sigma$ into
$S^{1}\times\Sigma$. The image, $V([0,1]\times\Sigma)$, has boundary

$\partial
V([0,1]\times\Sigma)=V(\partial([0,1]\times\Sigma))=V(\partial[0,1]\times\Sigma)\cup
V(\\{0\\}\times\Sigma)\cup
V(\\{1\\}\times\Sigma)=C_{ab}\cup\phi^{-1}(a)\cup\phi^{-1}(b).$

The surface
$C_{ab}=V(\partial[0,1]\times\Sigma)\subseteq\partial(S^{1}\times\Sigma)=S^{1}\times\partial\Sigma$
belong to the boundary of the manifold, over which the magnetic two-form pulls
back to zero by the Dirichlet boundary condition. With this in mind and
because the magnetic two-form is closed, Stokes theorem yields

$\displaystyle
0=\int\limits_{V}d\beta=\int\limits_{\phi^{-1}(a)}\beta-\int\limits_{\phi^{-1}(b)}\beta$

where the minus sign comes from the opposite orientation of the level sets
$\phi^{-1}(a)$ and $\phi^{-1}(b)$ under $V$. ∎

######  Corollary IV.8.

The area of $\Sigma$ computed via the symplectic forms $\omega_{t}$ is
independent of $t\in S^{1}$. As closed Dirichlet two-forms, $\omega_{t}$ thus
belong to the same relative cohomology class.

###### Proof.

Indeed,

$\displaystyle\int\limits_{\Sigma}\omega_{t}=\int\limits_{\Sigma}i_{t}^{*}\beta=\int\limits_{i_{t}(\Sigma)}\beta=\int\limits_{\phi^{-1}(t)}\beta.$

∎

That all two-forms $\omega_{t}$ are relative cohomologous allows us to relate
them by a family of diffeomorphisms. This is essentially the content of
Moser’s stability theorem [27]. However, it is important to note that the
family of two-forms $\omega_{t}$ has a periodic parameter $t\in S^{1}$. If one
naively applies Moser’s stability theorem, then one would obtain a smooth
family $\\{\Psi_{t}:\Sigma\to\Sigma\\}_{t\in[0,1]}$ of diffeomorphisms such
that $\Psi_{t}^{*}\omega_{\gamma(t)}=\omega_{o}$ where $o\in S^{1}$ is fixed
and $\gamma:[0,1]\to S^{1}$ is the quotient projection. However, it is not
immediate that the family $\\{\Psi_{t}:\Sigma\to\Sigma\\}_{t\in[0,1]}$ can be
chosen with $\Psi_{0}=\Psi_{1}$. Therefore, some care is needed to apply
Moser’s result to ensure that one obtains a well-defined bundle automorphism
of $S^{1}\times\Sigma$. This leads to a _parameterised_ version of his
theorem. First, a definition to easily state it.

######  Definition IV.9.

Let $\nu_{1},\nu_{2}\in\Omega^{k}(S^{1}\times\Sigma)$ be $k$-forms on
$S^{1}\times\Sigma$. We write

$\displaystyle\nu_{1}=\nu_{2}\text{ mod }S^{1}\quad\text{ if }\quad
i_{t}^{*}\nu_{1}=i_{t}^{*}\nu_{2},\quad\forall t\in S^{1}.$

######  Lemma IV.10.

The statement $\nu_{1}=\nu_{2}\text{ mod }S^{1}$ is equivalent to

$\displaystyle\nu_{1}(Y_{1},\ldots,Y_{k})=\nu_{2}(Y_{1},\ldots
Y_{k}),\quad\forall Y_{1},\ldots,Y_{k}\in\ker d\phi.$

###### Proof.

Fix $t\in S^{1}$, $x\in\Sigma$ and let $(t,x)\in S^{1}\times\Sigma$. By Lemma
III.1, $(di_{t})_{x}$ forms a bijection between $T_{x}\Sigma$ and $\ker
d\phi_{(t,x)}$. So, evaluating the pullback $(i^{*}_{t}\nu_{m})_{x}$ over
$T_{x}\Sigma$ is the same as evaluating $(\nu_{m})_{(t,x)}$ on $\ker
d\phi_{(t,x)}$. Explicitly, let $v_{1},\ldots,v_{k}\in T_{x}\Sigma$ and the
corresponding $w_{1},\ldots,w_{k}\in\ker d\phi_{(t,x)}$ such that
$w_{n}=(di_{t})_{x}(v_{n})$, for $n=1,\ldots,k$. Then,

$\displaystyle(i_{t}^{*}\nu_{m})_{x}(v_{1},\ldots,v_{k})=(\nu_{m})_{(t,x)}((di_{t})_{x}(v_{1}),\ldots,(di_{t})_{x}(v_{k}))=(\nu_{m})_{(t,x)}(w_{1},\ldots,w_{n}),\quad
m=1,2.$

∎

######  Remark IV.11.

Applied to one-forms, $\alpha_{1}=\alpha_{2}$ mod $S^{1}$ is equivalent to
$\alpha_{1}=\alpha_{2}+fd\phi$ for some function $f$. In the case of two-
forms, $\omega_{1}=\omega_{2}$ mod $S^{1}$ is equivalent to
$\omega_{1}=\omega_{2}+\chi\wedge d\phi$ for some one-form $\chi$.

We are now ready to state and prove an important result.

######  Theorem IV.12.

There exists a bundle isomorphism $\Psi:S^{1}\times\Sigma\to
S^{1}\times\Sigma$ with $\Psi^{*}\phi=\phi$ such that
$\Psi^{*}\beta=\omega\mod S^{1}$, where $\omega=\pi^{*}i_{o}^{*}\beta$ for an
arbitrary (fixed) reference angle on the circle $o\in S^{1}$.

###### Proof.

The procedure is known as Moser’s trick. Consider the family of closed two-
forms $\beta_{s}=s\beta+(1-s)\omega$ for $s\in[0,1]$, linearly interpolating
between the closed two-forms $\omega$ and $\beta$. We wish to construct a
family of diffeomorphisms $\Psi_{s}$ so that $\Psi_{s}^{*}\phi=\phi$ and
$\Psi_{s}^{*}\beta_{s}=\omega$ mod $S^{1}$ for each $s\in[0,1]$. To achieve
this, we manufacture a non-autonomous vector field $X_{s}$ on
$S^{1}\times\Sigma$ and solve the linear differential equation with variable
coefficients, $\frac{d}{ds}\Psi_{s}=X_{s}\circ\Psi_{s}$, supplemented with
initial condition $\Psi_{0}=\text{id}_{S^{1}\times\Sigma}$. The non-autonomous
vector field $X_{s}$ uniquely determines $\Psi_{s}$, and vice-versa.

The property that $\Psi_{s}^{*}\phi=\phi$ is equivalent to the generating
vector field $X_{s}$ belonging to $\ker d\phi$. This can be seen by
differentiating the condition with respect to $s$,

$\displaystyle 0$
$\displaystyle=\frac{d}{ds}(\Psi_{s}^{*}\phi-\phi)=\Psi_{s}^{*}\mathcal{L}_{X_{s}}\phi\iff
d\phi(X_{s})=0\iff X_{s}\in\ker d\phi.$

The second property $\Psi_{s}^{*}\beta_{s}=\omega$ mod $S^{1}$ is equivalent
to the non-autonomous vector field $X_{s}\in\ker d\phi$ additionally
satisfying

$\displaystyle 0$
$\displaystyle=\frac{d}{ds}(\Psi_{s}^{*}\beta_{s}-\omega)=\Psi_{s}^{*}(\partial_{s}\beta_{s}+\mathcal{L}_{X_{s}}\beta_{s})=0\text{
mod }S^{1}\iff\beta-\omega+di_{X_{s}}\beta_{s}=0\text{ mod }S^{1},$

where, when $\Psi_{s}$ preserves $\ker d\phi$ as a result of the first
condition, the statement $\Psi_{s}^{*}\nu=0$ mod $S^{1}$ is equivalent to
$\nu=0$ mod $S^{1}$, for $\nu\in\Omega^{k}(S^{1}\times\Sigma)$. Now by
Corollary IV.8, $i_{t}^{*}(\beta-\omega)$ is relative cohomologous to zero for
all $t\in S^{1}$, so $\beta-\omega=d\alpha$ mod $S^{1}$ for some Dirichlet-mod
$S^{1}$ one-form $\alpha$ on $S^{1}\times\Sigma$. The requirements on the
generating vector field are thus $X_{s}\in\ker d\phi$ and
$d(\iota_{X_{s}}\beta_{s}+\alpha)=0$ mod $S^{1}$. Conveniently, it is
sufficient that $X_{s}\in\ker d\phi$ satisfies

$\displaystyle\iota_{X_{s}}\beta_{s}(Y)$
$\displaystyle=-\alpha(Y),\quad\forall Y\in\ker d\phi,$

which determines $X_{s}$ uniquely because $\beta_{s}$ is non-degenerate on
$\ker d\phi$, for all $s\in[0,1]$.

Now that the non-autonomous vector field $X_{s}$ has been prescribed, we
verify that the initial-value problem
$\frac{d}{ds}\Psi_{s}=X_{s}\circ\Psi_{s}$ with $\Psi_{0}=\text{id}$ is well-
posed. Since $\alpha$ is Dirichlet-mod $S^{1}$, the vector fields $X_{s}$ are
tangential to the boundary of $S^{1}\times\Sigma$. Indeed, at any point
$x\in\partial(S^{1}\times\Sigma)\subset S^{1}\times\Sigma$, let $v\in
dj_{x}(T_{x}\partial(S^{1}\times\Sigma))\cap\ker d\phi_{x}$. Then,
$0=-\alpha_{x}(v)=(\beta_{s})_{x}((X_{s})_{x},v)$, which means that
$(X_{s})_{x}$ are co-linear to $v$, namely tangential to the boundary. The
manifold $S^{1}\times\Sigma$ is compact and $X_{s}$ is tangential to the
boundary, which implies that $X_{s}$ has global non-autonomous flow,
$\Psi_{s}:S^{1}\times\Sigma\to S^{1}\times\Sigma$ for all times $t\in[0,1]$.
Hence, the desired diffeomorphism $\Psi=\Psi_{1}$ exists.

∎

######  Remark IV.13.

Theorem IV.12 implies an important decomposition of the magnetic two-form
$\beta$. Consider the closed two-form $\Omega$ on $S^{1}\times\Sigma$ such
that $\Psi^{*}\Omega=\omega=\pi^{*}i_{o}^{*}\beta$ where $\Psi$ is the
diffeomorphism of Theorem IV.12. Because $\Psi$ preserves the level sets of
$\phi$, the two-form $\Omega$ is Dirichlet, $j^{*}\Omega=0$. By virtue of
Theorem IV.12, the magnetic two-form $\beta$ matches $\Omega$ on the level
sets of $\phi$, namely $\beta=\Omega$ mod $S^{1}$ (or
$i_{t}^{*}\beta=i_{t}^{*}\Omega=\omega_{t}$, $\forall t\in S^{1}$) which
means, by Remark IV.11,

$\displaystyle\beta=\Omega+\chi\wedge d\phi$

for some one-form $\chi$.

The one-form $\chi$ necessarily satisfies additional properties. Because
$\beta$, $\Omega$ and $d\phi$ are all closed, the one-form $\chi$ satisfies

$\displaystyle d\chi\wedge d\phi=0.$

That is $d\chi=0$ mod $S^{1}$, namely $\chi$ is closed mod $S^{1}$. We also
have that $\chi$ is Dirichlet mod $S^{1}$. Indeed, because $\beta$ and
$\Omega$ are Dirichlet but $j^{*}d\phi\neq 0$, we have that

$0=j^{*}\beta=\cancel{j^{*}\Omega}+j^{*}\chi\wedge j^{*}d\phi\iff
j^{*}\chi=0.$

Now, letting $t\in S^{1}$, $x\in\Sigma$ and $v\in T_{x}\partial\Sigma$, we
consider the vector $V=d(\partial i_{t})_{x}v$ where $\partial
i_{t}:\partial\Sigma\to\partial(S^{1}\times\Sigma)$ is inclusion inherited
from $i_{t}:\Sigma\to S^{1}\times\Sigma$ and the inclusions
$j:\partial(S^{1}\times\Sigma)\subset S^{1}\times\Sigma$ and
${\jmath}:\partial\Sigma\subset\Sigma$. Then, we find that

$0=\iota_{V}(j^{*}\chi\wedge
j^{*}d\phi)|_{x}=j^{*}\chi|_{x}(V)j^{*}d\phi|_{x}-\cancel{j^{*}d\phi|_{x}(V)j^{*}\chi|_{x}}$

so that $\chi|_{x}(v)=0$. Hence, the one-form $i_{t}^{*}\chi$ is Dirichlet for
each $t\in S^{1}$, as claimed. We therefore have that

$\Psi^{*}\beta=\omega+\nu\wedge d\phi$

whereby $\nu=\Psi^{*}\chi$ is both Dirichlet- and closed- mod $S^{1}$.
Considering the smooth family $\\{\nu_{t}=i_{t}^{*}\nu\\}_{t\in S^{1}}$, we
now summarise our immediate findings for the cosymplectic manifold
$(S^{1}\times\Sigma,\beta,d\phi)$ from Theorem IV.12.

######  Corollary IV.14.

The cosymplectic manifold $(S^{1}\times\Sigma,\beta,d\phi)$ is represented by
the non-autonomous locally-Hamiltonian system
$(\Sigma,\omega,\\{\nu_{t}\\}_{t\in S^{1}})$.

By virtue of Corollary III.15, more can be said in the case where $\Sigma$ is
planar, that is, a compact regular domain in $\mathbb{R}^{2}$.

######  Corollary IV.15.

Assume that $\Sigma$ is planar. Then, the cosymplectic manifold
$(S^{1}\times\Sigma,\beta,d\phi)$ is represented by a non-autonomous
Hamiltonian system.

By collecting and expanding the meaning of these results, we summarise our
findings in the proof of the main result, Theorem IV.5.

###### Proof of Theorem IV.5..

Recalling the assumptions, let $(S^{1}\times\Sigma,\beta,d\phi)$ be a
cosymplectic manifold where $\Sigma$ is planar, $\phi:S^{1}\times\Sigma\to
S^{1}$ is the trivial projection, and $\beta$ satisfies the Dirichlet boundary
condition, $j^{*}\beta=0$. Fix a reference angle $o\in S^{1}$. Implicit in
Corollary IV.15 from the planarity of $\Sigma$ is the result that the closed
one-form $\nu$ on $S^{1}\times\Sigma$ in Remark IV.13 is actually exact. That
is, $\nu=-dH$ for some function $H$ in $S^{1}\times\Sigma$. Hence,
$(\Sigma,i_{o}^{*}\beta,\\{i_{t}^{*}H\\}_{t\in S^{1}})$ is a non-autonomous
Hamiltonian system. The diffeomorphism of Theorem IV.12, which preserves the
projection $\Psi^{*}\phi=\phi$, allows us to write

$\displaystyle\Psi^{*}\beta=\omega-dH\wedge
d\phi\iff\beta=\Omega-d\mathcal{H}\wedge d\phi$

where $\Psi^{*}\Omega=\omega=\pi^{*}i_{o}\beta$ and $\Psi^{*}\mathcal{H}=H$. ∎

##  V Conclusion

In summary, this paper carefully investigated the relation between the field-
line dynamics of divergence-free tangential vector fields on a fibre bundle
over the circle and non-autonomous Hamiltonian systems.

First, various flavours of non-autonomous Hamiltonian systems (periodic in
time) were presented. Essentially, the evolution vector field $E_{H}$ of a
non-autonomous Hamiltonian system on a symplectic manifold
$(\Sigma,\omega_{o})$ induced by the family of functions $\\{H_{t}\\}_{t\in
S^{1}}$ is the Reeb field $E_{H}=\partial_{t}+X_{H}=R_{o}$ of the cosymplectic
manifold $(S^{1}\times\Sigma,\pi^{*}\omega_{o}-dH\wedge d\phi,d\phi)$, see
Proposition III.4 and Remark III.5. Of course, the non-autonomous Hamiltonian
structure and the corresponding dynamics on $S^{1}\times\Sigma$ should be
insensitive to diffeomorphisms; if a cosymplectic manifold $(M,\beta,\eta)$
can be converted to the one of the above type via a diffeomorphism, then we
say that it admits a non-autonomous Hamiltonian representation, see Definition
III.6. Paraphrasing Remark III.7, the closed two-form then necessarily
decomposes as $\beta=\Omega-d\mathcal{H}\wedge\eta$ and the Reeb field as
$R=\mathcal{T}+\mathcal{X}_{\mathcal{H}}$ where the time-like and Hamiltonian
components are characterised by

$\displaystyle\mathcal{T}$ $\displaystyle\in\ker\Omega,$
$\displaystyle\eta(\mathcal{T})$ $\displaystyle=1,$
$\displaystyle\mathcal{X}_{\mathcal{H}}$ $\displaystyle\in\ker\eta,$
$\displaystyle\iota_{\mathcal{X}_{\mathcal{H}}}\Omega$
$\displaystyle=-d\mathcal{H}+d\mathcal{H}(T)\eta.$

In this formalism, bundle automorphisms of $S^{1}\times\Sigma$ induce, in some
sense, non-canonical transformations; the family of functions
$\mathcal{H}_{t}=i^{*}_{t}\mathcal{H}$ and the family of symplectic two-forms
$\omega_{t}=i^{*}_{t}\beta$ produce dynamics on $\Sigma$ given by the flow of
the vector field $\mathcal{X}_{t}$ satisfying

$\displaystyle\iota_{\mathcal{X}_{t}}\omega_{t}=-d\mathcal{H}_{t},$

which can be viewed as a generalisation of the dynamics of a non-autonomous
Hamiltonian system, see Example III.8.

We then defined the notion of non-autonomous locally-Hamiltonian system, for
which the dynamics is induced by the family of Hamiltonian vector fields
$\\{X_{t}\\}_{t\in S^{1}}$ satisfying

$\displaystyle\iota_{X_{t}}\omega_{o}=\nu_{t},$

where $\\{\nu_{t}\\}_{t\in S^{1}}$ is a family of closed Dirichlet one-forms
on $\Sigma$, see Definition III.11. By extension, a cosymplectic manifold
$(M,\beta,\eta)$ admits a non-autonomous locally-Hamiltonian representation if
there exists a diffeomorphism back to a cosymplectic manifold of the form
$(S^{1}\times\Sigma,\pi^{*}\omega_{o}+\nu\wedge d\phi,d\phi)$, where $\nu$ is
a one-form on $S^{1}\times\Sigma$ such that $i^{*}_{t}\nu=\nu_{t}$, and the
diffeomorphism is required to respect the cosymplectic structure. It was
highlighted that $\nu$ could be chosen closed if and only if the cohomology
classes of $\\{\nu_{t}\\}_{t\in S^{1}}$ agree. If $\Sigma$ is a regular
compact domain of $\mathbb{R}^{2n}$, each closed one-form $\nu_{t}$ is
actually exact, as a consequence of the Dirichlet boundary condition and
injectivity of the boundary inclusion map on the first cohomology. Then, a
(global) function $H$ on $S^{1}\times\Sigma$ can be found so that $\nu=-dH$
and the notions of non-autonomous locally- and globally- Hamiltonian coincide,
see Corollary III.15.

In the second part of the paper, we attempted to reverse the argument by
considering a divergence-free tangential vector field $B$ on a compact
connected Riemannian three-manifold $M$. The manifold must at least be a fibre
bundle over the circle. This fibre-bundle structure comes about if the field
is transverse to a global Poincaré section $\Sigma$. In this case, the
projection (angle function) $\phi:M\to S^{1}$ can be arranged so that
$d\phi(B)>0$, see Lemma IV.3. Viewing $(M,\iota_{B}\mu,d\phi)$ as a
cosymplectic manifold, we asked whether it can be represented by some flavour
of non-autonomous Hamiltonian system. We focused on the situation in which $M$
is a trivial bundle over the circle, where a “time” direction can be defined.
By Corollary IV.4, this is the case if the sections $\Sigma$ are diffeomorphic
to the disk or the annulus. An adaptation of Moser’s trick allowed us to
conclude that $(S^{1}\times\Sigma,\iota_{B}\mu,d\phi)$ is always represented
by a non-autonomous locally-Hamiltonian system, see Theorem IV.12. In the case
$\Sigma$ is planar, Corollary III.15 allowed for the full identification.

On non-trivial fibre bundles, there is no natural choice of “time” direction;
there are as many candidates as there are conjugacy classes of the mapping
class group $\mathcal{M}(\Sigma)$, see Appendix E. The latter need not even be
finite, see Corollary E.14. Future work will investigate the idea of recording
the monodromy map induced by the vector field $B$, so that the time-like
component arises as an instance of a purely topological feature of the
manifold akin to a geometric (Berry) phase.

The work has potential for an extension to the equivalence between volume-
preserving vector fields on a $2n+1$ manifold and non-autonomous volume-
preserving vector fields on a $2n$-manifold, with possible implications for
Nambu mechanics.

The work has immediate implications for the design and optimisation of
magnetic confinement fusion devices. For instance, it must be recognised that
the dynamics of divergence-free vector fields on toroidal domains is
considerably richer than that of periodic non-autonomous Hamiltonian systems,
and that the tools of Hamiltonian mechanics have limited scope in the
comprehensive characterisation of magnetic configurations. In different terms,
non-autonomous Hamiltonian systems only represent a subset of all possible
magnetic field-line dynamics, and there is no reason to believe that the
former offers better confinement properties than the latter. It is clear that
the parameter space of three-dimensional stellarator magnetic fields is
significantly greater than that of the axisymmetric tokamak concept.
Exploration and optimisation of stellarator designs is guided by the intuition
offered by the Hamiltonian interpretation, for example in the
prediction/control of the onset of magnetic islands and stochastic fields, or
of the persistence of invariant surfaces under perturbations. Perhaps though,
configurations without classical symmetry exhibit outstanding confinement
properties and may be less sensitive to small adjustments. In probing the
design space for this eventuality, the theoretical framework and numerical
tools needs to be extended to general classes of magnetic fields and less
elementary domain topologies.

## Acknowledgements

The authors are indebted to H. Dullin, J.D. Meiss, P.J. Morrison, S.R. Hudson,
A.H. Boozer, A. Bhattacharjee, P. Helander and many other members of the
Simons Collaboration on Hidden Symmetries and Fusion Energy for insightful
discussions. The latter collaboration is acknowledged for nurturing strong
synergies and providing multiple opportunities for the authors to meet. A
special thought goes to the late R.L. Dewar, whom we believe would have
enjoyed the exposition.

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##  Appendix A Dirichlet closed one-forms with non-zero cohomology classes

######  Example A.1.

Given a closed $n$-manifold $\Sigma_{0}$ with
$H^{1}_{\text{dR}}(\Sigma_{0})\neq\\{0\\}$, we remove a ball from $\Sigma_{0}$
to construct a manifold $\Sigma_{1}$ with boundary which admits a closed one-
form $\nu$ on $\Sigma_{1}$ with non-zero cohomology class and satisfying
$j^{*}\nu=0$.

On the closed manifold $\Sigma_{0}$, there exists a smooth closed curve
$\gamma:[0,1]\to\Sigma_{0}$ and a closed one-form $\eta$ such that
$\int_{\gamma}\eta\neq 0$. Now, taking a point
$x\in\Sigma_{0}\setminus\gamma([0,1])$, consider a chart $(U,\varphi)$ about
$x$ with $U\subset\Sigma_{0}\setminus\gamma([0,1])$, $\varphi(x)=0$, and
$\varphi(U)=B_{\delta}$ for some $\delta>0$ (where $B_{r}$ denotes an open
ball of radius $r>0$ in $\mathbb{R}^{n}$ centred at $0$). By the Poincaré
Lemma, we have that $i^{*}\eta=dh$ for some $h\in C^{\infty}(U)$ where
$i:U\subset\Sigma_{0}$ denotes the inclusion. Now, let $\rho$ be a smooth
function in $B_{\delta}$ which is $1$ on $B_{\delta/3}$ and zero outside
$B_{\delta/2}$. Now define $f:\Sigma_{0}\to\mathbb{R}$ by

$f(x)=\begin{cases}\rho(\varphi(x))h(x),&x\in U\\\ 0,&x\in\Sigma_{0}\setminus
U\end{cases}$

Then, $f\in C^{\infty}(\Sigma_{0})$ and $\eta|_{V}=df|_{V}$ where
$V=\varphi^{-1}(B_{\delta/3})$. Let $\nu_{0}=\eta-df$. By construction,
$\nu_{0}$ is closed on $\Sigma_{0}$ and $\nu_{0}|_{V}=0$ and hence
$\nu_{0}|_{\overline{V}}=0$.

Consider the manifold $\Sigma_{1}=\Sigma_{0}\setminus V$ with boundary
$\partial\Sigma_{1}=\overline{V}\setminus V$. Letting
$i_{10}:\Sigma_{1}\subset\Sigma_{0}$ denote the inclusion, we obtain a closed
one-form $\nu=i_{10}^{*}\nu_{0}$ on $\Sigma_{1}$. We have that

$\int_{\gamma}\nu_{0}=\int_{\gamma}\eta\neq 0$

and because $\gamma([0,1])\subset\Sigma_{1}$, $\nu$ has non-trivial cohomology
class. Moreover, because $\nu_{0}|_{\partial\Sigma_{1}}=0$, it follows that
$\nu$ satisfies the Dirichlet boundary condition on $\partial\Sigma_{1}$.

In particular, this construction applies to closed manifolds which are
symplectic, and therefore one trivially obtains symplectic manifodls with
boundary admitting Dirichlet closed one-forms with non-trivial cohomology
class.

##  Appendix B Proof of Lemma III.9

To prove Lemma III.9, we first introduce a technical lemma about smoothness.

######  Lemma B.1.

Consider a star-shaped open subset $U$ with respect to the origin $0$ of
either $\mathbb{R}^{n}$ or the upper half-plane $\mathbb{H}^{n}$ and a smooth
family $\\{\nu_{t}\\}_{t\in S^{1}}$ of closed one-forms on $U$. Let
$h:S^{1}\to\mathbb{R}$ be a smooth function. Then, there exists a unique
smooth family $H_{t}$ of functions on $U$ such that

$-dH_{t}=\nu_{t},\qquad H_{t}(0)=h(t).$

###### Proof.

Indeed, for each $t\in S^{1}$, by Poincaré Lemma, we consider the function
$H_{t}:U\to\mathbb{R}$ given by

$H_{t}(x)=h(t)-\int_{\gamma_{x}}\nu_{t}$ (16)

where $\gamma_{x}:[0,1]\to U$ is given by $\gamma_{x}(t)=tx$ for $x\in U$. We
note that $H_{t}$ is the unique function such that

$-dH_{t}=\nu_{t},\qquad H_{t}(0)=h(t).$

However, by Equation 16, the function $H:S^{1}\times U\to\mathbb{R}$ given by

$H(t,x)=H_{t}(x)$

is smooth. ∎

We are now ready to prove Lemma III.9.

###### Proof of Lemma III.9.

First, fix a point $x_{0}\in\Sigma$. Then, for each $t\in S^{1}$, by exactness
of $\nu_{t}$, there exists a unique $H_{t}\in C^{\infty}(\Sigma)$ such that

$dH_{t}=\nu_{t},\qquad H_{t}(x_{0})=0.$

Consider the function $H:S^{1}\times\Sigma\to\mathbb{R}$ such that

$H(t,x)=H_{t}(x).$

Now, let $x\in\Sigma$ and let $\gamma:[0,1]\to\Sigma$ be a smooth path from
$x_{0}$ to $x$. Then, we have that

$H(t,x)=\int_{\gamma}\nu_{t}$

and therefore the function $h_{x}:S^{1}\to\mathbb{R}$ given by

$h_{x}(t)=H_{t}(x)=H(t,x)$

is smooth because the map $\nu:S^{1}\times\Sigma\to T^{*}\Sigma$ given by
$\nu(t,x)=\nu_{t}|_{x}$ is smooth. Thus, we may use a chart about $x$ to apply
Lemma B.1 to the setting of our proposition to establish smoothness of $H$.
That is, $\\{H_{t}\\}_{t\in S^{1}}$ is a smooth family. ∎

##  Appendix C Necessary planarity of a factor in an embedded product

######  Proposition C.1.

Let $\Sigma$ be a compact two-manifold with boundary. If $S^{1}\times\Sigma$
can be embedded in $\mathbb{R}^{3}$, then $\Sigma$ can be embedded in
$\mathbb{R}^{2}$ (planar).

###### Proof.

First note that it suffices to consider the case where $\Sigma$ is connected.
Indeed, assuming the result holds for this case, if $\Sigma$ is a compact two-
manifold with boundary such that $S^{1}\times\Sigma$ can be embedded in
$\mathbb{R}^{3}$, then, by compactness, there are only finitely many connected
components of $\Sigma$. Let $\Sigma_{1},...,\Sigma_{k}$ denote the components.
Note that each connected component $\Sigma_{i}$ of $\Sigma$ is such that
$S^{1}\times\Sigma_{i}$ can be embedded in $\mathbb{R}^{3}$. Therefore
$\Sigma_{i}$ can be embedded in $\mathbb{R}^{2}$. Since the $\Sigma_{i}$ are
compact and mutually disjoint, it follows that
$\Sigma=\cup_{i=1}^{k}\Sigma_{i}$ can also be embedded in $\mathbb{R}^{2}$.

From here, the proof is divided into two steps. Step 1 is showing that the
inclusion map $j:\partial\Sigma\to\Sigma$ is injective on first cohomology.
Step 2 is to use the classification of surfaces (together with the necessary
orientability of $\Sigma$) to conclude that $\Sigma$ must be planar.

_Step 1._ The image $M\subset\mathbb{R}^{3}$ of an embedding is a compact
regular domain. By Proposition III.14, the boundary inclusion $\partial M\to
M$ is injective on first cohomology. Since
$\partial(S^{1}\times\Sigma)=S^{1}\times\partial\Sigma$, we also have that
$l:S^{1}\times\partial\Sigma\to S^{1}\times\Sigma$ is injective on first
cohomology.

We now show that $j:\partial\Sigma\to\Sigma$ is injective on first cohomology.
First, denote $\pi_{\partial}:S^{1}\times\partial\Sigma\to\partial\Sigma$.
Note that $j\circ\pi_{\partial}=\pi\circ l$, i.e.
$\pi_{\partial}^{*}j^{*}=l^{*}\pi^{*}$ as pullback maps on one-forms. Also
recall that $\pi\circ i_{o}=\text{id}$, i.e. $i_{o}^{*}\pi^{*}=\text{id}$ for
any $o\in S^{1}$.

Let $\nu\in\Omega^{1}(\Sigma)$ be a closed one-form such that $j^{*}\nu$ is
exact. Consider the closed one-form
$\tau=\pi^{*}\nu\in\Omega^{1}(S^{1}\times\Sigma)$. The one-form

$\displaystyle l^{*}\tau=l^{*}\pi^{*}\nu=\pi_{\partial}^{*}j^{*}\nu$

is thus exact. By injectivity of $l^{*}:H^{1}_{dR}(S^{1}\times\Sigma)\to
H^{1}_{dR}(S^{1}\times\partial\Sigma)$, $\pi^{*}\nu=\tau$ is exact. Hence,
$\nu=i_{o}^{*}\pi^{*}\nu$ is exact too. This proves injectivity of
$j:\partial\Sigma\subset\Sigma$ on cohomology.

_Step 2._ We first note that $\Sigma$ is orientable and has non-empty boundary
because $M\subset\mathbb{R}^{3}$ is orientable and has non-empty boundary.
From Step 1, we know that $j:\partial\Sigma\subset\Sigma$ is injective on
cohomology. Now, each connected component of $\partial\Sigma$, as a closed
connected one-manifold, is diffeomorphic to $S^{1}$. We form a two-manifold
$S$ by patching each hole, detected by the connected components of
$\partial\Sigma$, with a disk. More precisely, if there are $m$-connected
components of $\partial\Sigma$, then we take a diffeomorphism
$f:\partial\Sigma\to\sqcup_{i=1}^{m}\partial D_{i}$ where each $D_{i}$ is a
disk, and form the adjunction $S=\Sigma\cup_{f}\sqcup_{i=1}^{m}D_{i}$, see
[43, Theorem 9.29].

It follows that $S$ is a compact orientable two-manifold and is therefore a
surface of genus-$g$. Because the components of $\partial\Sigma$ that are
embedded in $S$ are contractible in $S$ and $j$ is injective on first
cohomology (surjective on first homology), it follows that the first homology
of $S$ is trivial. Therefore, $g=0$ and $S$ is a sphere. We have by
construction that $\Sigma$ is a proper subset of $S$. That is, $\Sigma\subset
S\setminus\\{x\\}$ for some $x\in S$. Thus, by stereographic projection,
$\Sigma$ can be embedded in $\mathbb{R}^{2}$. ∎

##  Appendix D Proof of Lemma IV.3

In this section, a proof of Lemma IV.3 is given. Different formulations of the
lemma with varying hypothesis have been given in several previous works, with
the first instance of the result stated by Birkhoff [44, Ch. V.10]. See, for
example, the work of [45, 46, 47]. Here we state and prove it for compact
manifolds with boundary.

Specifically, it shown that, if $M$ is a compact orientable manifold with
boundary $\partial M$ and $X$ is a vector field on $M$ tangent to $\partial M$
that admits a global Poincaré section $\Sigma$, then there exists an angle
valued function $\phi:M\to S^{1}$ such that $d\phi(X)>0$ and $\Sigma$ is a
level-set of $\phi$. By a global Poincaré section of $X$ on a manifold with
boundary $M$, we mean a compact connected embedded submanifold $\Sigma$ of $M$
such that $\partial\Sigma\subset\partial M$ and every orbit of $X$ intersects
$\Sigma$.

The proof proceeds as follows. First, a Flowout Theorem for manifolds with
boundary is proved in Lemma D.2. The proof relies upon Lemma D.1 which shows
that $\Sigma$ can be extended to a submanifold $\tilde{\Sigma}$ without
boundary of the boundaryless double $D(M)$, and $X$ can be extended to a
$\tilde{X}$ on $D(M)$ such that $\tilde{X}$ is everywhere transverse to
$\tilde{\Sigma}$. Using this extension, it is shown how the Flowout Theorem
for manifolds without boundary [43, Theorem 9.20] implies the Flowout Theorem
for manifolds with boundary. The Flowout Theorem is then used to prove Lemma
D.3 which states that the classical notion of first return time
$\tau(x):=\min\\{t>0\,:\,\psi_{t}^{X}(x)\in\Sigma\\}$ is well-defined and
smooth for manifolds with boundary. In turn, Lemma D.3 is used to show that
the Poincaré map $f(x):=\psi^{X}_{\tau(x)}(x)$ is a diffeomorphism of $\Sigma$
in Proposition D.4. Finally, the smoothness of $\tau(x),f(x)$ together with
the concept of a mapping torus and special flow (see Appendix E) are used to
prove the existence of the desired circle-valued function $\phi:M\to S^{1}$.

######  Lemma D.1.

Let $M$ be a manifold (without boundary) and let $\Sigma$ be a compact
embedded manifold of codimension $1$ that has boundary in $M$. Let $X$ be a
vector field on $M$ which is transverse to $\Sigma$. Then, there exists an
embedded manifold $\tilde{\Sigma}$ without boundary of $M$ and of codimension
$1$ such that $\Sigma\subset\tilde{\Sigma}$ and $X$ is transverse to
$\tilde{\Sigma}$.

###### Proof.

Let the dimension of $M$ be $n+1$ so that $\Sigma$ has dimension $n$. As shown
in [43, Theorem 5.51], for each $x\in\Sigma$, there exists a so called _slice
chart_ $(U,\varphi)$ with $x\in U$ such that, if
$x\in\operatorname{int}\Sigma$,

$\Sigma\cap U=\\{x\in U:\varphi^{n}(x)=0\\},$

and if $x\in\partial\Sigma$,

$\Sigma\cap U=\\{x\in U:\varphi^{n}(x)=0,\varphi^{n+1}(x)\geq 0\\}.$

For any function $f:U\to\mathbb{R}$ where $U$ is an open subset of $M$, denote
by $d_{*}f$ the local 1-form $U\to T^{*}M$ such that
$(d_{*}f)|_{x}=(di|_{x}^{-1})^{*}df|_{x}$ where $i:U\subset M$ is the
inclusion and $di|_{x}:T_{x}U\to T_{x}M$ is the associated tangent map (which
is an isomorphism for slice charts).

We will say a slice chart $(U,\varphi)$ at $x\in\Sigma$ is _proper_ if
$\overline{U}$ is a compact subset of $M$ and $d_{*}{\varphi}^{n}(X|_{U})>0$.
We now show that a proper slice chart exists for all $x\in\Sigma$. Indeed, let
$(U,\varphi)$ be any slice chart at $x$. Consider the local 1-form
$d_{*}\varphi^{n}:U\to T^{*}M$. By transversality of $X$, we have that
$d_{*}\varphi^{n}(X|_{U})$ has no zeros. In particular, for any open ball
$B\subset\mathbb{R}^{n+1}$ such that $\bar{B}\subset\varphi(U)$ and
$\varphi(x)\in B$ it holds that either $d\varphi^{n}(X|_{\varphi^{-1}(B)})>0$
or $d_{*}\varphi^{n}(X|_{\varphi^{-1}(B)})<0$. That is,
$d_{*}\sigma\varphi^{n}(X|_{V})>0$ for some $\sigma\in\\{\pm 1\\}$. Then,
consider the chart $(U^{\prime}=\varphi^{-1}(B),\varphi^{\prime})$ where
${\varphi^{\prime}}^{i}=\varphi^{i}|_{U^{\prime}}$ for
$i\in\\{1,...,n+1\\}\setminus\\{n\\}$ and
${\varphi^{\prime}}^{n}=\sigma\varphi^{n}|_{U^{\prime}}$. We have that
$\overline{U^{\prime}}$ is a compact subset of $M$ and that the local 1-form
$d_{*}{\varphi^{\prime}}^{n}:U^{\prime}\to T^{*}M$ is such that
$d_{*}{\varphi^{\prime}}^{n}(X|_{U^{\prime}})>0$, that is,
$(U^{\prime},\varphi^{\prime})$ is a proper slice chart at $x$.

The compactness of $\Sigma$ guarantees that there exists $N\in\mathbb{N}$ and
proper slice-charts $\\{(U_{i},\varphi_{i})\\}_{i=1}^{N}$ such that
$\\{U_{1},...,U_{N}\\}$ forms an open cover of $\Sigma$. Now,
$\\{U_{1},...,U_{N}\\}$ trivially forms an open cover for the open submanifold
$S=\cup_{i=1}^{N}U_{i}\subset M$ and so there exists a partition of unity
$\\{\rho_{i}:S\to\mathbb{R}\\}_{i=1}^{N}$ subordinate to this open cover. Now,
for $i\in\\{1,...,N\\}$, we have that $\operatorname{supp}\rho_{i}\subset
U_{i}$. Therefore, the function $F_{i}:M\to\mathbb{R}$ given by

$F_{i}(y)=\begin{cases}\rho_{i}(y)\varphi_{i}^{n}(y)\text{ if }y\in U_{i}\\\
0\text{ otherwise},\end{cases}$

is smooth and compactly supported.

Now let $F=\sum_{i}F_{i}$ and observe that, if $y\in\Sigma$ then $F(y)=0$.
Moreover, because $dF_{j}|_{y}=0$ for all $j\in\\{1,...,N\\}$ such that
$y\notin\operatorname{supp}\rho_{j}$, we can write

$dF|_{y}(X|_{y})=\sum_{\\{i:y\in\operatorname{supp}\rho_{i}\\}}\varphi^{n}_{i}(y)(d_{*}\rho_{i})|_{y}(X|_{y})+\rho_{i}(y)d_{*}\varphi_{i}^{n}(X|_{y})=\sum_{\\{i:y\in\operatorname{supp}\rho_{i}\\}}\rho_{i}(y)d_{*}\varphi_{i}^{n}(X|_{y})>0,$

because for all $i\in\\{1,...,N\\}$, $\varphi_{i}^{n}(y)=0$,
$d_{*}\varphi_{i}^{n}(X|_{y})>0$ by definition of a proper slice chart, and
$\rho_{i}(y)>0$ if $y\in\operatorname{supp}\rho_{i}$.

Now, let $V=\\{x\in M:dF(X)|_{x}>0\\}$. Then, $V$ is an open subset of $M$.
The function $f=F|_{V}$ has no critical points. Therefore $f^{-1}(0)$ is a
regular level set of $f$. However, we have from the fact that
$dF|_{\Sigma}(X|_{\Sigma})>0$ that $\Sigma\subset V$. Therefore,
$\Sigma\subset f^{-1}(0)$. The result now follows from the Preimage Theorem by
setting $\tilde{\Sigma}=f^{-1}(0)$. ∎

Using Lemma D.1, we will now prove a flowout theorem for manifolds with
boundary.

######  Lemma D.2.

Let $M$ be a compact manifold with boundary. Let $\Sigma$ be a compact
codimension 1 submanifold with boundary of $M$ with
$\partial\Sigma\subset\partial M$. Let $X$ be a vector field on $M$ that is
tangent to $\partial M$, transverse to $\Sigma$, and let the global flow of
$X$ be denoted by $\psi^{X}$. Then, there exists $\delta>0$ such that, for any
$t\in\mathbb{R}$, the flow $\psi^{X}$ restricted to
$(t-\delta,t+\delta)\times\Sigma$ is a diffeomorphism onto its open image.

In particular, this makes $\Sigma$ transverse to the flow
$\psi^{X}:\mathbb{R}\times M\to M$ in the topological sense of [45].

###### Proof.

First, consider the boundaryless double $D(M)$ of $M$ (for an introduction to
the boundaryless double, see [43, Example 9.32]. We have that $D(M)$ is a
manifold without boundary and that there is an embedding $i_{1}:M\to D(M)$
whereby $i_{1}(M)$ is a compact regular domain of $D(M)$. By the Extension
Lemma for Vector Fields (see [43, Lemma 8.6]), there exists a vector field
$\tilde{X}$ extending $X$ to $D(M)$, in the sense that $\tilde{X}\circ
i_{1}=di_{1}\circ X$.

Now, $i_{1}(\Sigma)$ is a compact embedded submanifold with boundary in $M$ of
codimension 1 and $\tilde{X}$ is a vector field on $D(M)$ that is transverse
to $i_{1}(\Sigma)$. Hence, by Lemma D.1, there exists an embedded submanifold
$\tilde{\Sigma}$ of dimension $n$ in $D(M)$ whereby
$i_{1}(\Sigma)\subset\tilde{\Sigma}$ and $\tilde{X}$ is transverse to
$\tilde{\Sigma}$. Then, by the (classical) Flowout Theorem [43, Theorem 9.20],
there exists a smooth positive function
$\tilde{\delta}:\tilde{\Sigma}\to\mathbb{R}$ such that the restriction of the
global flow $\psi^{\tilde{X}}:\mathbb{R}\times D(M)\to D(M)$ to

$\tilde{U}=\\{(t,x)\in\mathbb{R}\times D(M):|t|<\tilde{\delta}(x)\\}$

is a diffeomorphism onto its open image. Now, consider the smooth positive
function $\delta=\tilde{\delta}\circ i_{1}$ and the open set

$U=\\{(t,x)\in\mathbb{R}\times M:|t|<\delta(x)\\}.$

Observe that the flows $\psi^{X}$ and $\psi^{\tilde{X}}$ are related by the
fact that, for $x\in M$ and $t\in\mathbb{R}$,

$\psi^{\tilde{X}}(i_{1}(x),t)=\psi^{X}(x,t).$

In particular,

$i_{1}(\psi^{X}(U))=\psi^{\tilde{X}}(\tilde{U})\cap i_{1}(M),$

so that $\psi^{X}(U)$ is an open subset of $M$. It follows that $\psi^{X}$
restricted to $U$ is a diffeomorphism onto its open image. Moreover, because
$\Sigma$ is compact, we have for some $\delta_{0}>0$ that
$(-\delta_{0},\delta_{0})\times M\subset U$. Thus, $\psi^{X}$ restricted to
$(-\delta_{0},\delta_{0})\times M$ is a diffeomorphism onto its open image.
Lastly, for any $t\in\mathbb{R}$, we have

$\psi^{X}|_{(t-\delta_{0},t+\delta_{0})\times
M}=\psi_{t}^{X}\circ\psi^{X}|_{(-\delta_{0},\delta_{0})\times M}\circ T_{t},$

where $T_{-t}:(t-\delta_{0},t+\delta_{0})\times
M\to(-\delta_{0},\delta_{0})\times M$ is the map given by
$(s,x)\mapsto(s-t,x)$. Therefore,
$\psi^{X}|_{(t-\delta_{0},t+\delta_{0})\times M}$ is also a diffeomorphism
onto its open image. ∎

Using the flowout theorem for manifolds with boundary, we can establish the
classical notion of the first return time for manifolds with boundary.

######  Lemma D.3.

Let $M$ be a compact manifold with boundary. Suppose that $X$ is a vector
field tangent to $M$ and that $X$ admits a Poincaré section $\Sigma$. Then,
the _first return time_ $\tau:\Sigma\to\Sigma$ given by

$\tau(x):=\min\left\\{t>0\,:\,\psi^{X}_{t}(x)\in\Sigma\right\\}$

exists for all $x\in\Sigma$ and is smooth.

###### Proof.

Throughout, we will use the fact proved in Lemma D.2, that there exists
$\delta>0$ such that, for every $t\in\mathbb{R}$, $\psi^{X}$ restricted to
$(t-\delta,t+\delta)\times\Sigma$ is a diffeomorphism onto its open image. Fix
such a $\delta>0$.

We begin by showing that for all $x\in\Sigma$ the first-return time $\tau(x)$
is well-defined. This will follow provided that the set of positive return
times

$T_{x}^{+}:=\left\\{t>0\,:\,\psi^{X}_{t}(x)\in\Sigma\right\\}$

for $x\in\Sigma$ is non-empty and satisfies a uniform discreteness condition
(see Equation (17)). In fact, $T_{x}^{+}$ is non-empty for all $x\in M$. To
see this, observe, as in [45], that the compactness of $M$ guarantees that the
$\omega$-limit set of any point $x\in M$ is non-empty. In particular, there
exists an increasing sequence $\\{t_{n}\\}_{n\in\mathbb{N}}$ of times with
$t_{n}\to\infty$ such that $\psi^{X}_{t_{n}}(x)\to y$ for some $y\in M$. Now,
we have that $\psi^{X}_{t}(y)\in\Sigma$ for some $t\in\mathbb{R}$ and so
$\psi^{X}_{t_{n}+t}(x)\to\psi^{X}_{t}(y)$ by continuity of $\psi^{X}_{t}$. On
the other hand, $U=\psi^{X}((-\delta,\delta)\times\Sigma)$ is a neighbourhood
of $\psi^{X}_{t}(y)\in\Sigma$. Hence, there exists $N\in\mathbb{N}$ such that
$t_{N}+t-\delta>0$ and $\psi^{X}_{t_{N}+t}(x)\in U$. However, because
$\psi^{X}|_{(-\delta,\delta)\times\Sigma}$ is a diffeomorphism onto its open
image $U$, there exists $s\in(-\delta,\delta)$ such that
$\psi^{X}_{s}(\psi^{X}_{t_{N}+t}(x))\in\Sigma$. By choice of $N$,
$s+t_{N}+t>0$. That is, for all $x\in M$, the set $T_{x}^{+}$ is non-empty.

We now show for $x\in\Sigma$ that $T_{x}^{+}$ satisfies a uniform discreteness
condition. In fact, this is true for the set of all return times

$T_{x}=\\{t\in\mathbb{R}:\psi^{X}_{t}(x)\in\Sigma\\}.$

Indeed, let $x\in\Sigma$ and $t\in T_{x}$. Then, $\psi^{X}_{t}(x)\in\Sigma$.
On the other hand, $\psi^{X}$ restricted to $(-\delta,\delta)\times\Sigma$ is
a diffeomorphism onto its open image. In particular, because
$\psi^{X}(\\{0\\}\times\Sigma)=\Sigma$ and
$\psi^{X}|_{(-\delta,\delta)\times\Sigma}$ is injective, for any
$s\in(-\delta,\delta)\setminus\\{0\\}$ and $y\in\Sigma$, we have that
$\psi^{X}_{s}(y)\notin\Sigma$. Hence, for any
$s\in(-\delta,\delta)\setminus\\{0\\}$, it follows that

$\psi^{X}_{s+t}(x)=\psi^{X}_{s}(\psi^{X}_{t}(x))\notin\Sigma,$

and therefore $s+t\notin T_{x}$. It follows that, for all $x\in\Sigma$ and
$t\in T_{x}$,

$T_{x}\cap(t-\delta,t+\delta)=\\{t\\}.$ (17)

From the fact that $T_{x}^{+}$ is non-empty and the fact that Equation (17)
holds for all $x\in\Sigma$, we can conclude that the minimum of $T_{x}^{+}$
exists and hence $\tau(x)$ is well-defined. Moreover, an immediate consequence
of Equation (17) that $\tau(x)$ satisfies the inequality

$\tau(x)\geq\delta.$ (18)

We will now show that $\tau$ is smooth in an indirect manner. First, it is
shown that for each $x\in\Sigma$ and $t\in T_{x}^{+}$ that there exists a
neighbourhood $V$ of $x$ in $\Sigma$ and a smooth function
$T:V\to(t-\delta,t+\delta)$ such that $T(x)=t$ and $T(y)\in T_{y}^{+}$ for all
$y\in V$. The function $T$ gives a ‘local’ return time function for all $y\in
V$. Using this local $T$, it is shown that $\tau$ satisfies a continuity
condition. Finally, the minimality $\tau$ is then used to show that, for all
$x\in\Sigma$, $\tau$ and the local return time function $T$ agree on a
neighbourhood $W$ of $x$ in $\Sigma$. Consequently, $\tau$ is smooth.

Claim 1. For all $x\in\Sigma$ and $t\in T_{x}^{+}$ there exists a
neighbourhood $V$ of $x$ in $\Sigma$ and a smooth function
$T:V\to(t-\delta,t+\delta)$ such that $T(x)=t$ and $T(y)\in T_{y}^{+}$ for all
$y\in V$.

Consider

$U=\psi^{X}((-\delta,\delta)\times\Sigma),$

and observe that there is a smooth function $f:U\to\mathbb{R}$ such that, for
all $x^{\prime}\in U$,

$\psi^{X}_{f(x^{\prime})}(x^{\prime})\in\Sigma.$ (19)

Indeed, because $\psi^{X}|_{(-\delta,\delta)\times\Sigma}$ is a diffeomorphism
onto its open image $U$, for any $x^{\prime}\in U$ there is a unique point
$(t_{x^{\prime}},y_{x^{\prime}})\in(-\delta,\delta)\times\Sigma$ such that
$\Psi^{X}_{t_{x^{\prime}}}(y_{x^{\prime}})=x^{\prime}$. Then, we can define a
smooth function $f:U\to\mathbb{R}$ by $f(x^{\prime})=-t_{x^{\prime}}$. It
follows that

$\psi^{X}_{f(x^{\prime})}(x^{\prime})=\psi^{X}_{-t_{x^{\prime}}}(\psi^{X}_{t_{x^{\prime}}}(y_{x^{\prime}}))=y_{x^{\prime}}\in\Sigma.$

Now, let $x\in\Sigma$ and $t\in T_{x}^{+}$. We will show that
$V=\Sigma\cap\psi^{X}_{-t}(U)$ is the desired neighbourhood of $x$ and

$T(y)=f(\psi^{X}_{t}(y))+t$

gives the desired smooth function $T:V\to(t-\delta,t+\delta)$ satisfying
$T(x)=t$ and $T(y)\in T^{+}_{y}$ for all $y\in V$. First, as
$\psi^{X}_{t}(x)\in\Sigma$, then $\psi^{X}_{t}(x)\in U$ and in turn
$x\in\psi^{X}_{-t}(U)$. Furthermore, $x\in\Sigma$ and so
$x\in\Sigma\cap\psi_{-t}^{X}(U)$. It follows that $\Sigma\cap\psi_{-t}^{X}(U)$
is a neighbourhood of $x$ in $\Sigma$. Now, $\psi_{t}^{X}(x)\in\Sigma$ and
therefore $f(\psi_{t}^{X}(x))=0$. Thus, $T(x)=t$. Moreover, for each
$y\in\Sigma\cap\psi_{-t}^{X}(U)$, we have that $y\in\Sigma$ and
$\psi_{t}^{X}(y)\in W$. Hence, by Equation (19),

$\psi^{X}_{T(y)}(y)=\psi^{X}_{f(\psi_{t}^{X}(y))+t}(y)=\psi^{X}_{f(\psi_{t}^{X}(y))}(\psi_{t}^{X}(y))\in\Sigma.$

Thus $T(y)\in T_{y}$. Finally, $T(y)\in T_{y}^{+}$ because $t\in T^{+}_{x}$
and Equation (18) guarantees that $t\geq\delta$, so that

$T(y)>t-\delta\geq\delta-\delta=0.$

In conclusion, $V=\Sigma\cap\psi^{X}_{-t}(W)$ and $T(y)=f(\psi_{t}^{X}(y))+t$
have the desired properties so that claim 1 holds.

Claim 2. The first-return time $\tau$ satisfies the continuity condition that,
for all $x\in\Sigma$ and for any sequence $\\{y_{n}\\}_{n\in\mathbb{N}}$ with
$y_{n}\to x$, there exists a subsequence $\\{y_{n_{k}}\\}_{k\in\mathbb{N}}$ of
$\\{y_{n}\\}_{n\in\mathbb{N}}$ such that $\tau(y_{n_{k}})\to\tau(x)$.

Indeed, for $x\in\Sigma$, we have that $\tau(x)\in T_{x}^{+}$ and so we may
take the neighbourhood $V$ of $x$ in $\Sigma$ and function
$T:V\to(\tau(x)-\delta,\tau(x)+\delta)$ as given by Claim 1. Without loss of
generality, let $\\{y_{n}\\}_{n\in\mathbb{N}}$ be a sequence in $V$ with
$y_{n}\to x$. For all $y\in V$, because $T(y),\tau(y)\in T_{y}^{+}$, it
follows from Equation (18) that

$\delta\leq\tau(y)\leq T(y)<\tau(x)+\delta.$ (20)

In particular, the sequence $\\{\tau(y_{n})\\}_{n\in\mathbb{N}}$ in
$[\delta,\tau(x)+\delta]$ has a convergent subsequence in
$[\delta,\tau(x)+\delta]$ (by compactness of $[\delta,\tau(x)+\delta]$).
Hence, there exists a subsequence $\\{y_{n_{k}}\\}_{k\in\mathbb{N}}$ of
$\\{y_{n}\\}_{n\in\mathbb{N}}$ in $\Sigma$ and a
$t^{*}\in[\delta,\tau(x)+\delta]$ such that

$\tau(y_{n_{k}})\to t^{*}.$

Moreover, because $y_{n}\to x$ by assumption, we have that $y_{n_{k}}\to x$.
Therefore, the sequence $\\{(\tau(y_{n_{k}}),y_{n_{k}})\\}_{k\in\mathbb{N}}$
in $\mathbb{R}\times\Sigma$ is such that

$(\tau(y_{n_{k}}),y_{n_{k}})\to(t^{*},x).$

Observe that for $n\in\mathbb{N}$, we have $y_{n}\in\Sigma$ and
$\tau(y_{n})\in T_{y_{n}}$ so that

$\psi^{X}(\tau(y_{n}),y_{n})\in\Sigma.$

However, the preimage $(\psi^{X}|_{\mathbb{R}\times\Sigma})^{-1}(\Sigma)$ is a
closed subset of $\mathbb{R}\times\Sigma$ because $\Sigma$ is a closed subset
of $M$ and $\psi^{X}|_{\mathbb{R}\times\Sigma}:\mathbb{R}\times\Sigma\to M$ is
continuous. Therefore, the limit $(t^{*},x)$ of
$\\{(\tau(y_{n_{k}}),y_{n_{k}})\\}_{k\in\mathbb{N}}$ is such that

$\psi^{X}(t^{*},x)\in\Sigma.$

Hence, $t^{*}\in T_{x}$. Moreover, $t^{*}\in[\delta,\tau(x)+\delta]$ and so
$t^{*}\in T_{x}^{+}$. Therefore, $\tau(x)\leq t^{*}$. On the other hand,
because $T$ is continuous and $y_{n_{k}}\to x$, we have

$T(y_{n_{k}})\to T(x)=\tau(x).$

However, for all $k\in\mathbb{N}$, from Equation (20), we have

$\tau(y_{n_{k}})\leq T(y_{n_{k}}).$

Therefore $t^{*}\leq\tau(x)$. Hence, $t^{*}=\tau(x)$ and

$\tau(y_{n_{k}})\to\tau(x).$

This completes the proof of Claim 2.

Finally, Claim 1 and Claim 2 guarantee the smoothness of $\tau$ as follows.
Let $x\in\Sigma$ and take the neighbourhood $V$ of $x$ in $\Sigma$ and smooth
function $T:V\to(\tau(x)-\delta,\tau(x)+\delta)$ whereby $T(x)=\tau(x)$ and
$T(y)\in T_{y}^{+}$ for $y\in V$. Now, suppose that for any neighbourhood $W$
of $x$ in $V$, there exists $y\in W$ such that $T(y)\neq\tau(y)$. Then, there
exists a sequence $\\{y_{n}\\}_{n\in\mathbb{N}}$ in $\Sigma$ with $y_{n}\to x$
such that $T(y_{n})\neq\tau(y_{n})$ for all $n\in\mathbb{N}$. For
$n\in\mathbb{N}$, we have that $T(y_{n})>\tau(y_{n})$ and
$T(y_{n}),\tau(y_{n})\in T_{y_{n}}^{+}$ so that by Equation (17), we have

$T(y_{n})\geq\delta+\tau(y_{n}).$

However, from Claim 2, there exists a subsequence
$\\{y_{n_{k}}\\}_{k\in\mathbb{N}}$ of $\\{y_{n}\\}_{n\in\mathbb{N}}$ such that
$\tau(y_{n_{k}})\to\tau(x)$. On the other hand, because $y_{n_{k}}\to x$, we
have that $T(y_{n})\to T(x)=\tau(x)$. This implies
$\tau(x)\geq\delta+\tau(x)$, which is a contradiction. Therefore, there exists
a neighbourhood $W$ of $x$ in $V$ such that

$\tau|_{W}=T|_{W}.$

It follows that $\tau$ is smooth. ∎

From Lemma D.3, we can introduce the notion of first return map and show that
it is a diffeomorphism.

######  Proposition D.4.

Let $M$ be a compact manifold with boundary. Suppose that $X$ is a vector
field tangent to $M$ and that $X$ admits a Poincaré section $\Sigma$. Then,
the function $f:\Sigma\to\Sigma$ given by

$f(x)=\psi^{X}_{\tau(x)}(x)$

is a diffeomorphism.

###### Proof.

Note that the function $F:\Sigma\to M$ given by $F(x)=\psi^{X}_{\tau(x)}(x)$
is smooth because $\tau$ is smooth and $\psi^{X}:\mathbb{R}\times M\to M$ is
smooth. Moreover, $F(\Sigma)\subset\Sigma$ and $\Sigma$ is an embedded
submanifold of $M$ with boundary. Therefore, the restricted map
$f:\Sigma\to\Sigma$ is also smooth (see [43, Theorem 5.53 and Problem 9-13]).

We show now that $f$ has a smooth inverse. Consider the vector field $Y=-X$
and note that $\Sigma$ is a Poincaré section for $Y$. Let
$\upsilon:\Sigma\to\mathbb{R}$ denote the smooth first-return time as in Lemma
D.3. As per the above paragraph, the first return map $g:\Sigma\to\Sigma$ of
$Y$, given by

$g(x)=\psi^{Y}_{\upsilon(x)}(x),$

is smooth. We now show that

$f\circ g=\operatorname{id}=g\circ f.$

It is enough to show the first equality because the second is proven
identically. To this end, for $x\in\Sigma$, set

$\displaystyle T^{X}_{x}$
$\displaystyle=\\{t\in\mathbb{R}:\psi^{X}_{t}(x)\in\Sigma\\},$ $\displaystyle
T^{Y}_{x}$ $\displaystyle=\\{t\in\mathbb{R}:\psi^{Y}_{t}(x)\in\Sigma\\}.$

Now, let $x\in\Sigma$ and observe that

$\psi^{Y}_{t}(f(x))=\psi^{X}_{-t}(f(x))=\psi^{X}_{-t}(\psi^{X}_{\tau(x)}(x))=\psi^{X}_{\tau(x)-t}(x).$

Therefore,

$t\in T^{Y}_{f(x)}\text{ and }t>0\iff\tau(x)-t\in T^{X}_{x}\text{ and
}\tau(x)-t<\tau(x)\iff\tau(x)-t\in T^{X}_{x}\text{ and }\tau(x)-t\leq 0.$

Hence, $\upsilon(f(x))=\tau(x)$ and

$\psi^{Y}_{\upsilon(f(x))}(f(x))=\psi^{X}_{\tau(x)-\tau(x)}(x)=x.$

That is,

$g(f(x))=\psi^{Y}_{\upsilon(f(x))}(f(x))=x.$

∎

With Lemma D.3 showing the first-return time $\tau$ is smooth and Proposition
D.4 showing the first-return map is a diffeomorphism for manifold with
boundary, we are ready to prove Lemma IV.3.

###### Proof of Lemma IV.3.

Let $f:\Sigma\to\Sigma$ be the first-return map, $\tau:\Sigma\to\mathbb{R}$
the first-return time of $X$, and $\Phi=\psi^{X}$ be the global flow of $X$.
Let $\Sigma^{\tau}_{f}$ be the manifold with boundary defined by the quotient
space

$\Sigma^{\tau}_{f}:=\frac{\\{(t,x)\in\mathbb{R}\times\Sigma\,|\,0\leq
t\leq\tau(x)\\}}{(0,f(x))\sim(\tau(x),x)}.$

Observe that $\Sigma_{f}^{\tau}$ is diffeomorphic to $M$. In fact, a
diffeomorphism is explicitly given by $\tilde{\Phi}:\Sigma_{f}^{\tau}\to M$,
$[t,x]\mapsto\Phi(t,x)$. Indeed, $\tilde{\Phi}$ is well-defined as

$\tilde{\Phi}([\tau(x),x])=\Phi(\tau(x),x)=f(x)=\Phi(0,f(x))=\tilde{\Phi}([0,f(x)]).$

Moreover, $\tilde{\Phi}$ is a local diffeomorphism by the Flowout Theorem for
manifolds with boundary Lemma D.2. Lastly, we note that $\tilde{\Phi}$ is
injective by construction of $\tau$ and $f$ and $\tilde{\Phi}$ is surjective
because $\Phi$ is surjective (as $\Sigma$ is a Poincaré section). Hence, as
claimed, $\tilde{\Phi}$ is a diffeomorphism.

Moreover, the so-called vertical vector field $\partial_{t}$ on
$\Sigma_{f}^{\tau}$ defined by the flow $(s,[t,x])\mapsto[t+s,x]$ is
$C^{\infty}$-orbitally equivalent to $X$;

$\tilde{\Phi}_{*}\partial_{t}=\left.\frac{d}{ds}\right|_{s=0}\tilde{\Phi}([t+s,x])=\left.\frac{d}{ds}\right|_{s=0}\Phi(t+s,x)=X.$

Hence, $\Sigma_{f}^{\tau}$ gives the flow of $X$ as the _flow under a
function_ or _special flow_ (see [48]).

Now, consider the mapping torus $\Sigma_{f}$ defined here as

$\Sigma_{f}:=\frac{\\{(t,x)\in\mathbb{R}\times\Sigma\,|\,0\leq t\leq
2\pi\\}}{(0,f(x))\sim(2\pi,x)}.$

The manifold $\Sigma_{f}^{\tau}$ and the mapping torus $\Sigma_{f}$ are
diffeomorphic [48, Prop. 2.2.5]. Indeed, following [48], let $k=\min\tau$,
$K=\max\tau$, $\tilde{k}=\min(\\{k,2\pi\\})$ and consider any smooth function
$g:[0,2\pi]\times[k,K]\to[0,K]$ such that

  1. 1.

$g(t,s)=t$ for $t\in[0,\tfrac{1}{4}\tilde{k}]$,

  2. 2.

$g(t,s)=t+s-2\pi$ for $t\in[2\pi-\tfrac{1}{4}\tilde{k},2\pi]$,

  3. 3.

$\frac{\partial}{\partial t}g(t,s)>0$.

Then, the map $\Psi:\Sigma_{f}\to\Sigma_{f}^{\tau}$ given by
$[t,x]\mapsto[g(t,\tau(x)),x]$ is a diffeomorphism.

Furthermore, the vector field $\tilde{\partial}_{t}$ on $\Sigma_{f}$ with flow
$(s,[t,x])\mapsto[t+s,x]$ is $C^{\infty}$-orbitally equivalent to the flow of
$\partial_{t}$ on $\Sigma_{f}^{\tau}$. Indeed, we have that

$\Psi_{*}\tilde{\partial}_{t}=\left.\frac{d}{ds}\right|_{s=0}\Psi([t+s,x])=\left.\frac{d}{ds}\right|_{s=0}[g(t+s,\tau(x)),x]=u\partial_{t}$

for some function $u:\Sigma_{f}^{\tau}\to\mathbb{R}$ that must be strictly
positive in consequence of the fact that $\frac{\partial g}{\partial t}>0$.

Finally, define $\tilde{\phi}:\Sigma_{f}\to S^{1}$ by
$\tilde{\phi}([t,x])=[t]$ and let $\phi:M\to S^{1}$ be the circle-valued
function given from the pushforward of $\tilde{\phi}$ by the diffeomorphism
$\tilde{\Phi}\circ\Psi:\Sigma_{f}\to M$. Then,
$d\tilde{\phi}(\tilde{\partial}_{t})=1\implies
d\phi(X)=(\tilde{\Phi}_{*}u)^{-1}>0$ and $\phi^{-1}(0)=\Sigma$. Therefore,
$\phi$ is the desired circle-valued function on $M$.

∎

##  Appendix E Mapping tori, mapping class groups, monodromy, and a
classification of fibre bundles

Let $\phi:M\to S^{1}$ and $\phi^{\prime}:M^{\prime}\to S^{1}$ be two fibre
bundles over a circle with respective fibres $\Sigma,\Sigma^{\prime}$ that are
both connected, compact, smooth manifolds (possibly with boundary). The fibre
bundles are said to be _bundle isomorphic_ if there exists a diffeomorphism
$\Psi:M\to M^{\prime}$ such that $\phi=\phi^{\prime}\circ\Psi$, that is, the
projections $\phi,\phi^{\prime}$ commute with $\Psi$. A natural question is
whether there exists a classification of all fibre bundles over $S^{1}$ up to
bundle isomorphism. In particular, it is crucial to know whether there exist
_non-trivial_ fibre bundles, namely fibre bundles that are not isomorphic to
the trivial fibre bundle $M=S^{1}\times\Sigma$.

In this section, we detail a representation of fibre bundles known as the
_monodromy representation_. For some classes of fibre bundles, the monodromy
representation is a complete invariant. This means that any two fibre bundles
are isomorphic if and only if they have the same monodromy representation. In
particular, the monodromy representation is a complete invariant for the class
of fibre bundles considered here; that of a fibre bundle over $S^{1}$. A
recent exposition on the monodromy representation is given in [49], where
surface bundles with arbitrary base manifold are treated. The more general
case of flat fibre bundles is treated in [50]. In this section we consider
only fibre bundles with base manifold $S^{1}$ and consequently provide
simplifications and an alternative formulation of the monodromy representation
to that given in the aforementioned references.

The section begins by establishing the monodromy representation and its
relation to mapping tori and the mapping class group of $\Sigma$. Then, the
monodromy representation is used to show that there are no non-trivial fibre
bundles only if $\Sigma$ is a disk or annulus.

###  E.1 The Monodromy Representation

Let $\phi:M\to S^{1}$ be a fibre bundle over $S^{1}$ with fibre $\Sigma$ that
is a connected, compact manifold (possibly with boundary). Henceforth, bundles