# Cartan media: geometric continuum mechanics in homogeneous spaces

Lukas Kikuchi<EMAIL_ADDRESS>Department of Applied Mathematics and
Theoretical Physics, Centre for Mathematical Sciences, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom  Ronojoy
Adhikari  Department of Applied Mathematics and Theoretical Physics, Centre
for Mathematical Sciences, University of Cambridge, Wilberforce Road,
Cambridge CB3 0WA, United Kingdom

###### Abstract

We present a geometric formulation of the mechanics of a field that takes
values in a homogeneous space $\mathbb{X}$ on which a Lie group $G$ acts
transitively. This generalises the mechanics of Cosserat media where
$\mathbb{X}$ is the frame bundle of Euclidean space and $G$ is the special
Euclidean group. Kinematics is described by a map from a space-time manifold
to the homogeneous space. This map is characterised locally by generalised
strains (representing spatial deformations) and generalised velocities
(representing temporal motions). These are, respectively, the spatial and
temporal components of the Maurer-Cartan one-form in the Lie algebra of $G$.
Cartan’s equation of structure provides the fundamental kinematic relationship
between generalised strains and velocities. Dynamics is derived from a
Lagrange-d’Alembert principle in which generalised stresses and momenta,
taking values in the dual Lie algebra of $G$, are paired, respectively, with
generalised strains and velocities. For conservative systems, the dynamics can
be expressed completely through a generalised Euler-Poincare action principle.
The geometric formulation leads to accurate and efficient structure-preserving
integrators for numerical simulations. We provide an unified description of
the mechanics of Cosserat solids, surfaces and rods using our formulation. We
further show that, with suitable choices of $\mathbb{X}$ and $G$, a variety of
systems in soft condensed matter physics and beyond can be understood as
instances of a class of materials we provisionally call Cartan media.

##  I Introduction

Cosserat media are models of continuua in which the elementary constituent is
notionally a rigid body. The configuration of a Cosserat medium is determined
by the position and orientation of each constituent in Euclidean space. Pairs
of configurations are considered equivalent if a single rigid transformation,
consisting of a translation and a rotation, takes every constituent in the
first configuration to the corresponding constituent in the second
configuration. Pairs of configurations are deformed with respect to each other
if different translations and rotations are needed to bring the constituents
into correspondence. A measure of deformation can then be constructed by
examining how the translations and rotations differ from a single uniform
rigid transformation. The kinematic theory of strain in Cosserat media can be
constructed entirely on this basis. A dynamic theory of stress can be
constructed in duality with the kinematic theory of strain, in which stress
and strain are paired, in a precise mathematical sense, to yield a virtual
work. The mechanical equations of motions can then be obtained by imposing
suitable conditions on the virtual work.

At the heart of this elegant formulation of the mechanics of Cosserat media
(Schaefer, 1967a, b) is a symmetry, namely that of rigid transformations, or
isometries of Euclidean space. Mathematically, the configuration of the
Cosserat medium can be mapped to a space
$\mathbb{X}=\mathcal{F}(\mathbb{E}^{3})$, the frame bundle of Euclidean space
consisting of the union of pairs of points and rigid frames. A continuous
symmetry group $G=SE(3)$, the Euclidean group, acts on the frame bundle, such
that any two elements of the bundle are related by a rigid transformation.
Like Euclidean space itself, its frame bundle is a homogeneous space, in
which, informally speaking, there is no “distinguished point”. The homogeneity
of the frame bundle and the existence of a group of symmetries in terms of
which “displacements” in the configuration space can be characterised is the
basis of the geometric structure of Cosserat media.

The question we ask and address here is the following: what is the general
geometric structure of the mechanics of a continuum whose configuration takes
values in a homogeneous space $\mathbb{X}$ on which a Lie group $G$ acts
transitively? As we shall show below, this seemingly abstract question arises
repeatedly in concrete form in the modelling of many continuum systems but
seems not to have been consciously recognised. The Cosserat medium provokes
merely one instance of this question, namely when $\mathbb{X}$ is the frame
bundle of Euclidean space and $G$ is the special Euclidean group. In the
remainder of this paper, we show that the mechanics of such continuua can be
constructed by analogy with, and as a generalisation of, the geometric
mechanics of Cosserat media. The configuration in the homogeneous space
$\mathbb{X}$ can be “lifted” to the group $G$ and then the infinitesimal
structure of $G,$ encoded in its Lie algebra $\mathfrak{g}$ and its Lie-
algebra valued Maurer-Cartan one-form $\xi$, can be used to study
infinitesimal deformations of the medium. The dynamical theory can be
constructed analogously by pairing the infinitesimal deformations with
corresponding generalised momenta and generalised stresses to obtain the
virtual work from which the dynamical equations follow. In this manner, we
recognise a single geometric framework within which to study a multitude of
continuum systems that share the properties of homogeneity and symmetry.

The study of the geometry of space curves through the use of infinitesimal
rigid motions of orthogonal frames in Euclidean space was initiated by Bartels
and developed by Frenet and Serret (Clelland, 2017). Darboux extended it to
the study of surfaces embedded in Euclidean space (Darboux, 1887). Cartan
recognised the essential underlying principle as the action of a Lie group on
a homogeneous space and, in a remarkable generalisation, developed his method
of the _repère mobile_ to study the geometry of submanifolds of homogeneous
spaces (Cartan, 1935). Cartan made (Cosserat  _et al._ , 2009) a significant
reference to the symmetries and mechanics of Cosserat media when presenting
his method. Given this historical background, it does not seem inappropriate
to recognise the class of systems studied in this paper as “Cartan media”.

The remainder of the paper is organised as follows. In Section II we introduce
the mathematical notation used in the paper. In Section III we construct the
kinematics of Cartan media and derive the kinematic equations of motion using
the local properties of the Lie group $G$. The resulting equations of motion
are defined in terms of intrinsic quantities that are invariant to global
group action. In Section IV we derive the geometrised dynamics of Cartan
media, in both the conservative and non-conservative settings. Sections V
applies the formulation to unify the presentation of Cosserat solids, surfaces
and rods. Section VI shows how surfaces and filaments are also instances of
Cartan media, and how to derive their geometrised mechanics in terms of their
intrinsic and extrinsic curvatures. Section VII provides further examples in
soft matter physics and beyond. Finally, in Section VII we summarise and
conclude our work.

##  II Notation

The set of positive real numbers is denoted $\mathbb{R}_{+}$, and the set of
$m\times m$ matrices as $\mathbb{R}^{m\times m}$, and positive definite
matrices as $\mathbb{R}_{+}^{m\times m}$. Euclidean space $\mathbb{E}^{3}$ is
the vector space $\mathbb{R}^{3}$ equipped with the standard Euclidean inner
product. Elements of $\mathbb{E}^{3}$ are denoted as column vectors
$\mathbf{a}=(a_{1}\ a_{2}\ a_{3})^{T}\in\mathbb{E}^{3},\ a_{i}\in\mathbb{R}$,
such that the inner product is
$\mathbf{a}\cdot\mathbf{b}=\mathbf{a}^{T}\mathbf{b}$ for any
$\mathbf{a},\mathbf{b}\in\mathbb{E}^{3}$, and using the isomorphism
$\mathbb{E}^{3}\cong T\mathbb{E}^{3}$ the inner product extends to tangent
vectors as well. A vector in a _spatial frame of reference_ is denoted with a
superscript as $\mathbf{v}^{s}=(v_{1}^{s}\ v_{2}^{s}\ v_{3}^{s})^{T}\in
T\mathbb{E}^{3}$. That is, $\mathbf{v}^{s}=v_{i}^{s}\mathbf{d}_{i}$ where
$\mathbf{d}_{i}=(\delta_{i1}\ \delta_{i2}\ \delta_{i3})^{T},\ i=1,2,3$ is a
fixed basis for $T\mathbb{E}^{3}$. This is in contrast to the corresponding
vector in the _body frame of reference_ , a moving frame $E=(\mathbf{e}_{1}\
\mathbf{e}_{2}\ \mathbf{e}_{3})$ coincident with the material points of a
continuum body, which is written without the superscript $\mathbf{v}\in
T\mathbb{E}^{3}$. This notational rule is applied to all tangent vectors, with
the exception of basis vectors such as $\mathbf{e}_{i}$ and $\mathbf{d}_{i}$,
and derivatives of $\mathbb{E}^{3}$-valued fields. Corresponding vectors in
the two frames are related by a rotation $R\in SO(3)$ as
$\mathbf{v}^{s}=R\mathbf{v}$, where the Lie group $SO(3)$ is the special
orthogonal group. We will often make the identification $R=E$, such that
$\mathbf{v}^{s}=v_{i}^{s}\mathbf{d}_{i}=v_{i}\mathbf{e}_{i}$, where repeated
indices indicates a summation.. The distinction between the spatial and body
frames will reoccur throughout this text in relation to Euclidean, but also
more general, settings. We will often utilise the isomorphism between
$3$-vectors in $T\mathbb{E}^{3}$and $\mathfrak{so}(3)$, where the latter is
the Lie algebra of $SO(3)$. For $\mathbf{v},\mathbf{w}\in T\mathbb{E}^{3}$, we
can write $\mathbf{v}\times\mathbf{w}=\hat{v}\mathbf{w}$, where
$\hat{v}\in\mathfrak{so}(3)$ denotes the _hat map_ , given in Eq. 47. See the
appendix for further details on vector and matrix operations. We write general
Lie groups as $G$, and their corresponding Lie algebra as $\mathfrak{g}$. A
_homogeneous space_ is a space $\mathbb{X}$ that admits a transitive action of
a Lie group $G$. That is, given any two points $q_{1},q_{2}\in\mathbb{X}$
there exists a Lie group element $g\in G$ such that $g\cdot q_{1}=q_{2}$,
where the latter denotes the action of $G$ on $\mathbb{X}$. An example of a
homogeneous space is the $2$-sphere, on which $SO(3)$ acts transitively.

##  III Kinematics

We begin with a brief overview of kinematics in classical continuum mechanics
in the Lagrangian description (Batra, 2006), which will serve as a reference
point for the generalised kinematics we develop in this section. In classical
continuum mechanics the kinematics of a continuum body can be described by a
map $\mathbf{x}:[0,T]\times M\to\mathbb{E}^{3}$, where $\mathbf{u}\in M$ is a
material coordinate of a given reference configuration $M$, and
$\mathbf{x}(t,\mathbf{u})\in\mathbb{E}^{3}$ is the current vector of
displacement at time $t\in[0,T]$. The reference configuration $M$ shares the
same topology and dimensionality as that of the continuum body itself, but can
otherwise differ in shape. At time $t$, each material coordinate
$\mathbf{u}\in M$ is thus assigned a value in $\mathbb{E}^{3}$ via the map
$\mathbf{x}(t,\cdot):M\to\mathbb{E}^{3}$, such that the current configuration
is given by the image $\mathbf{x}(t,M)$. Consequently, $M$ can be seen as an
index set over the $\mathbb{E}^{3}$-valued point-continua of the continuum
body, and the material coordinate $\mathbf{u}$ a continuous multi-dimensional
index over the current configuration $\mathbf{x}$. We therefore call $M$ the
_material base space_ and $\mathbb{E}^{3}$ the _configuration space_ of the
system. In what follows we will generalise this to consider continuum bodies
with alternate configuration spaces, material dimensionalities and topologies.

Cosserat point-continua have configuration spaces that include an internal
micropolarity (Cosserat and Cosserat, 1909; Eremeyev  _et al._ , 2012;
Altenbach and Eremeyev, 2013a; Rubin, 2000; Non, 2005), as well as their
external configuration in $\mathbb{E}^{3}$. For instance, in addition to the
displacement, we may associate a unit vector $\mathbf{p}^{s}\in
T\mathbb{E}^{3}$, satisfying $\mathbf{p}^{s}\cdot\mathbf{p}^{s}=1$, to the
point-continua, representing its micropolarity. The configuration space of
such a system is the product $\mathbb{E}^{3}\times S^{2}$, where $S^{2}$ is
the 2-sphere, such that the configurations are
$q=(\mathbf{x},\mathbf{p}^{s})\in\mathbb{E}^{3}\times S^{2}$. If the
orientation around the polar vector is a degree of freedom, the internal
configuration can be represented with an orthonormal triad of vectors
$E=(\mathbf{e}_{1}\ \mathbf{e}_{2}\ \mathbf{e}_{3})$. The configuration space
for such a system can be identified with the _frame bundle_
$\mathcal{F}(\mathbb{E}^{3})$ of $\mathbb{E}^{3}$, where elements
$q=(\mathbf{x},E)\in\mathcal{F}(\mathbb{E}^{3})$ are called _trihedrons_. The
above are all examples of homogeneous spaces, on which $SE(3)$, the special
Euclidean group, acts transitively. We therefore call $SE(3)$ the _symmetry
group_ of $\mathbb{E}^{3}$, $\mathbb{E}^{3}\times S^{2}$ and
$\mathcal{F}(\mathbb{E}^{3})$.

There may in general be multiple Lie groups that act transitively on a given
homogeneous space. The choice of symmetry group $G$ determines how we treat
the symmetries of the configuration space. For example, we could equivalently
choose the product $G=T(3)\times SO(3)$ as the symmetry group of
$\mathbb{X}=\mathbb{E}^{3}\times S^{2}$, where $T(3)$ is the group of
translations on $\mathbb{E}^{3}$. The two terms of $G$ then act individually
on the two respective terms of $\mathbb{X}$, thus treating $\mathbb{E}^{3}$
and $S^{2}$ as two unrelated spaces. This separation does not reflect most
applications of Cosserat media, where the micropolarity is often considered,
either implicitly or explicitly, as tangent vectors on the ambient space. That
is, $S^{2}$ is often considered the subset of unit vectors in
$T\mathbb{E}^{3}$. The implication of this is that transformations on
$\mathbb{E}^{3}$ and $T\mathbb{E}^{3}$ should be done consistently; this
reflects the fact that the point-continua are _rigid_. This is accomodated by
choosing the symmetry group to be a semidirect product $G=T(3)\rtimes
SO(3)=SE(3)$, which in this case acts on $\mathbb{E}^{3}\times
T\mathbb{E}^{3}\supset\mathbb{E}^{3}\times S^{2}$, as will be shown below.
This example highlights why we bundle the external and internal (micropolar)
configuration spaces of a continuum system as one collective homogeneous space
$\mathbb{X}$, and the utility, and importance, of this will be borne out
through the results of this text.

Filaments (Nordgren, 1974; Goldstein and Langer, 1995; Soda, 1973) and
surfaces (Powers, 2010) are examples of continuum bodies that are embedded in
three-dimensional Euclidean space but with a lower intrinsic dimensionality.
That is, the configuration space of filaments and surfaces is
$\mathbb{E}^{3}$, but their material base spaces are one and two-dimensional
respectively. Similarly, _Cosserat rods_ and _surfaces_ (Rubin, 2000;
Altenbach and Eremeyev, 2013a) are continua with micropolar configuration
spaces, but with one- and two-dimensional material base spaces respectively.
For all continua, the topology of the continuum body must be reflected in the
topology of the material base space. For example, a closed surface will be
topologically equivalent to the unit sphere $S^{2}$, and we may thus let
$M=S^{2}$. Similarly, an adequate material base space for a closed rod is the
the periodic unit interval.

We will now generalise the kinematics of classical continuum mechanics, by
considering systems of general material base spaces and homogeneous
configuration spaces. Let the material base space $M$ be a topological
manifold of dimension $d$, and let the configuration space $\mathbb{X}$ be an
$n$-dimensional homogeneous space. At time $t\in[0,T]$, each _material point_
$p\in M$ is mapped to a point in $\mathbb{X}$ via the _spatio-temporal
configuration_ $q:W\to\mathbb{X}$, where we have defined the _kinematic base
space_ $W=[0,T]\times M$. The configuration of the system at time $t$ is thus
given by the image $q(t,M)$. Let $G$ be a _symmetry group_ of $\mathbb{X}$_,_
an $r$-dimensional Lie group that acts transitively on $\mathbb{X}$ under a
given group action $g\cdot q\in\mathbb{X}$, for $g\in G$ and $q\in\mathbb{X}$.
In general the choice of symmetry group is not unique, and may also be of
larger dimensionality than the configuration space $r\geq n$. In the special
case when $r=n$, the symmetry group is diffeomorphic to the configuration
space $\mathbb{X}$. 111Note that a Lie group $G$ may be considered a
homogeneous space, as it acts on itself transitively. For any Lie group $G$
there is an associated Lie algebra $\mathfrak{g}$, which is an $r$-dimensional
vector space equipped with a _Lie bracket_
$[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$.

Figure 1: The figure depicts an open Cosserat rod, mathematically described as
a mapping between a material base space $M=[0,L_{0}]$ and the Lie group
configuration space $\mathbb{X}=\mathcal{F}(\mathbb{E}^{3})$, where
$\mathcal{F}(\mathbb{E}^{3})$ is the frame bundle of $\mathbb{E}^{3}$, the set
of all orthonormal frames of Euclidean space. At any time $t\in[0,T]$, the
Cosserat rod is the image $q(t,M)\subset\mathcal{F}(\mathbb{E}^{3})$. In the
figure, two material points $u_{1},u_{2}\in M$ are shown mapped to
configurations
$(\mathbf{r}(t,u_{1}),E(t,u_{1})),(\mathbf{r}(t,u_{2}),E(t,u_{2}))\in\mathbb{X}$.
The temporal argument $t$ is suppressed in the figure.

Figure 2: The figure depicts a Cosserat surface, with material base space
$M=S^{2}$ and configuration space $\mathbb{X}=\mathbb{E}^{3}\times S^{2}$. The
material base space is the $2$-sphere $S^{2}$, which ensuresthat the surface
is closed. At any time $t\in[0,T]$, the Cosserat surface is the image
$q(t,M)\subset\mathbb{E}^{3}\times S^{2}$. In the figure, two material points
$p_{1},p_{2}\in M$ are shown mapped to
$(\mathbf{r}(p_{1}),\mathbf{p}^{s}(p_{2})),(\mathbf{r}(p_{2}),\mathbf{p}^{s}(p_{2}))\in\mathbb{X}$.
The temporal argument $t$ is suppressed in the figure.

We will refer to systems that can be kinematically described using the
formalism outlined above as _Cartan media_. Classical three-dimensional
continuum bodies, surfaces and rods, as well as their corresponding Cosserat
variants, all conform to this class of systems. See Fig. 2 and Fig. 2 for
illustrations of an open Cosserat rods and a closed Cosserat surface
respectively. In Sec. VIII we will give further examples of Cartan media,
including relativistic rods and field theories.

Let $W=[0,T]\times M$ be the _kinematic base space_. The transitive action of
$G$ on $\mathbb{X}$, guarantees the existence of a map $\Phi:W\to G$, which we
call a _structure field_ , that satisfies

$q(t,p)=\Phi(t,p)\cdot q_{r}$ (1)

for all $p\in M$ and $t\in[0,T]$, where $q_{r}\in\mathbb{X}$ is a fixed
reference configuration. Equation 1 shows that the spatio-temporal
configuration of the system is encapsulated in its entirety by the structure
field. The reference configuration $q_{r}$ will often carry the interpretation
of being the configuration of each point continua $p\in M$ as observed
_within_ its own body frame of reference.

As an example to illustrate Eq. 1, consider a Cosserat surface with
configuration space $\mathbb{X}=\mathbb{E}^{3}\times S^{2}$ and material base
space $M$. The spatial configuration at time $t$ will be a map
$q(t,\cdot):M\to\mathbb{E}^{3}\times S^{2}$, which we write as
$q=(\mathbf{r},\mathbf{p}^{s})$. Let $q_{r}=(\mathbf{0},\mathbf{p})$, where
$\mathbf{p}\in S^{2}$ is a fixed unit vector. Then the configuration
$q(t,\cdot)$ at each material point $p\in M$ can be constructed by rotation
and translation of $q_{r}$, which is precisely a group action of $SE(3)$ on
$\mathbb{E}^{3}\times S^{2}$. Now, let $R:W\to SO(3)$ be $R=(\mathbf{e}_{1}\
\mathbf{e}_{2}\ \mathbf{e}_{3})$, which is an orthonormal frame we define to
satisfy $\mathbf{p}^{s}=R\mathbf{p}=p_{i}\mathbf{e}_{i}$. Now, consider an
observer located at $\mathbf{r}(t,p)$ and with a frame of reference $R(t,p)$.
Then, the material point $p\in M$ will be located at $\mathbf{0}$, and
oriented as $\mathbf{p}$, relative to the observer. We thus see that $q_{r}$
can be interpreted as the observed configuration of each material point $p$
relative to an observer that is coincident and co-moving with $p$.

It should be noted that if $r>n$ there is no unique choice of structure field.
The gauge freedom in the choice of structure field will be discusssed in Sec.
III.5. A brief mathematical interlude: we note that the spatio-temporal
configuration of Cartan media can be be considered sections on the trivial
fibre bundle $M\times\mathbb{X}$. Similarly, $\Phi$ is a section on the
trivial principal bundle $M\times G$.

There is flexibility in how we represent elements of the configuration space.
For instance, in the above we may have alternatively parameterised
configurations as $(\mathbf{r},\theta,\phi)$, where $\theta$ and $\phi$ are
angular coordinates on the sphere. Similarly, when
$\mathbb{X}=\mathcal{F}(\mathbb{E}^{3})$ we wrote configurations as pairs
$(\mathbf{r},E)\in\mathbb{X}$, though we could have also parameterised the
orthonormal triad in terms of Euler angles $E=E(\alpha,\beta,\gamma)$, such
that configurations are specified by six scalars
$(\mathbf{r},\alpha,\beta,\gamma)$. Strictly, these should be understood as
coordinates on $\mathbb{X}$, rather than elements of the configuration space
itself. However, we will not stress this distinction, and we will furthermore
make use of abuses of notations such as
$(\mathbf{r},\alpha,\beta,\gamma)\in\mathbb{X}$, identifying coordinates with
configurations. In full generality, we may express configurations as a vector
of real numbers $q\in\mathbb{R}^{m}$, where $m\geq n$. This in turn induces a
representation $\Pi:G\to GL(\mathbb{R}^{m})$ of the group action on the
configuration space, such that we can write $x(t,p)=\Pi(\Phi(t,p))x_{0}$. For
example, consider again a Cosserat surface $\mathbb{X}=\mathbb{E}^{3}\times
S^{2}$, with symmetry group $G=SE(3)$. We write group elements as pairs
$g=\left(\mathbf{a};A\right)\in G$, where $\mathbf{a}\in\mathbb{E}^{3}$ is a
translation and $A\in SO(3)$ a rotation, and let
$y=(\mathbf{s},\mathbf{q}^{s})\in\mathbb{X}$. A reasonable choice of group
action would then be $g\cdot
y=\Pi(g)y=(A\mathbf{s}+\mathbf{a},A\mathbf{q}^{s})$. We can thus construct the
structure field as $\Phi(t,p)=\left(\mathbf{r};R\right)$ which satisfies Eq.
1, where $R\in SO(3)$ satisfies $R\mathbf{p}=\mathbf{p}^{s}$. Different
choices of how the configuration space is expressed, whether as coordinates or
explicit elements, will lead to different representations of the Lie group.
The framework we present in this paper is agnostic with respect to this
choice, as long as the Lie group representation is constructed consistently.

We have shown that the spatio-temporal configuration of a continuum body in a
homogeneous space, which we refer to as Cartan media, can be parameterised in
terms of a Lie group-valued structure field. In the following subsection we
will make use of the Lie group-Lie algebra correspondence, to furthermore
express $\Phi$ in terms of Lie algebra-valued fields. The latter are
generalised strain and velocity fields, which are analogous to those of
classical continuum mechanics. We call this process _geometrisation_. In Sec.
III.1 we derive kinematic equations of motion for Cartan media, in terms of
generalised strain and velocity fields. The resulting kinematics is geometric
in the sense that the equations of motion are expressed in terms of the
differential geometry of the continuum body, and are constructed to respect
the symmetries of the configuration space. In Sec. III.5 we consider the case
of $\text{dim}(G)>\text{dim}(\mathbb{X})$, when there is a gauge freedom in
the choice of $\Phi$, and how the redundancy in the parameterisation can be
removed by _kinematic adaptation_.

###  III.1 Generalised strain and velocity fields

Let $u^{\alpha}:M\to\mathbb{R},\ \alpha=1,\dots,d$, be coordinates on $M$. For
the sake of simplicity, we have assumed that the coordinates cover $M$. This
is not true in general, as for example $M=S^{2}$ would require at least two
overlapping coordinate charts. We will consider the case of multiple charts in
Sec. III.4.

The exterior derivative of the structure field is
$d\Phi=\dot{\Phi}dt+\partial_{\alpha}\Phi du^{\alpha}$, where
$\dot{\Phi}=\partial_{t}\Phi$ and $\partial_{\alpha}=\frac{\partial}{\partial
u^{\alpha}}$, and where repeated indices denote an Einstein summation. Here
$\dot{\Phi}:M\to TG$ and $\partial_{\alpha}\Phi:M\to TG$ are the infinitesimal
generators of $\Phi$, and should be seen as vectors fields on the symmetry
group $G$, where $TG$ is the tangent bundle of $G$.

A unique property of Lie groups is the ability to relate vector fields to
corresponding Lie algebra-valued fields. That is, the tangent space $T_{g}G$
at $g\in G$ can be related to the tangent space at the identity
$\mathfrak{g}=T_{e}G$. In other words, we have the diffeomorphism $TG\cong
G\times\mathfrak{g}$, such that sections on $TG$ can be related to sections on
$G\times\mathfrak{g}$. Via left-translation to the identity
$T_{e}G\cong\mathfrak{g}$, we define the $\mathfrak{g}$-fields

$\displaystyle X_{\alpha}$ $\displaystyle=\Phi^{-1}\partial_{\alpha}\Phi,$
$\displaystyle\alpha=1,\dots,d$ (2a) $\displaystyle N$
$\displaystyle=\Phi^{-1}\dot{\Phi},$ (2b)

which we call respectively the _generalised strain_ and _velocity fields_.
Respectively, $X_{\alpha}$ and $N$ encode the spatial structure of $\Phi$ and
its temporal evolution. The generalised strain and velocity fields together
form a $\mathfrak{g}$-valued $1$-form

$\displaystyle\xi$ $\displaystyle=\Phi^{-1}d\Phi$ (3)
$\displaystyle=Ndt+X_{\alpha}du^{\alpha},$

which we call the _structure generator_. Equation 3 completes the process of
geometrisation, having gone from the spatio-temporal configuration, to the
structure field $\Phi$, and finally to $X_{\alpha}$ and $N$.

It is notable that $\xi$ is left-invariant under any global transformation in
$G$. That is if $\Phi^{\prime}=g\Phi$, for some $g\in G$, then
$\xi^{\prime}=\Phi^{\prime-1}d\Phi^{\prime}=\xi$. Conversely, if the
respective generators $\xi_{1}$ and $\xi_{2}$ of two structure fields
$\Phi_{1}$ and $\Phi_{2}$ satisfy $\xi_{1}=\xi_{2}$, then there exists a $g\in
G$ such that $\Phi_{1}=g\Phi_{2}$. The implication of this fact is that $\xi$
does not contain global information about the configuration of the system, and
it further indicates that the components of $\xi$ are _differential
invariants_ (Clelland, 2017), and is therefore the appropriate mathematical
object to define the kinematics of the system in terms of its intrinsic, and
extrinsic, geometry 222To provide further mathematical context, we also note
that within differential geometry $\Phi$ is called an _immersion_ of $G$ into
$M$, and we have $\xi=\Phi^{*}\omega$, where $\Phi^{*}$ denotes the pull-back,
and where $\omega=g^{-1}dg,\ g\in G$ is known as the _Maurer-Cartan form_. See
(Clelland, 2017) for a detailed proof, and for further mathematical
exposition.. The structure field can be reconstructed from the structure
generator, up to global transformation in $G$ by solving Eq. 3 for $\Phi$. The
numerical algorithm for the reconstruction is described in detail in Sec. V.1.

We should emphasise that, at each time $t$, the spatial component of $\xi$ is
a $\mathfrak{g}$-valued $1$-form on $M$ that generates the spatial structure
of $\Phi$. Let $d_{M}$ denote the exterior derivative on $M$, then for a given
time $t$, the spatial structure generator is
$\xi_{M}=\Phi(t,\cdot)^{-1}d_{M}\Phi(t,\cdot)$, which in local coordinates is
$X_{\alpha}(t,\cdot)du^{\alpha}$. As $\xi_{M}$ is a coordinate-independent
object, this emphasises that only the topological properties of $M$ are of
importance for the kinematics, whilst the the particular choice of coordinates
are not. The structure generator can thus be decomposed as $\xi=Ndt+\xi_{M}$,
which is made possible by the fact that the kinematic base space has a product
structure $W=[0,T]\times M$.

The generalised strain and velocity fields should be seen as rates-of-
deformation defined in the body frame of reference of the system. In contrast,
the corresponding fields in the spatial frame of reference are
$N^{s}=\dot{\Phi}\Phi^{-1}$ and
$X_{\alpha}^{s}=(\partial_{\alpha}\Phi)\Phi^{-1}$which are found by right-
translation to the identity. To see this, first let
$\rho:\mathfrak{g}\to\mathfrak{gl}(\mathbb{R}^{m})$ be the representation of
the action of the Lie algebra on the configuration $\mathbb{X},$ induced by
$\Pi$ as $\rho(N^{s})=\partial_{t}\Pi(\Phi)\Pi(\Phi)^{-1}$ and
$\rho(X_{\alpha}^{s})=\partial_{\alpha}\Pi(\Phi)\Pi(\Phi)^{-1}$. Then, from
Eq. 1, we then find that $\dot{q}=\rho(N^{s})x$ and
$\partial_{\alpha}q=\rho(X_{\alpha}^{s})x$. We can relate $N^{s}$ and
$X_{\alpha}^{s}$ to their corresponding kinematic fields in the body frame as
$N^{s}=\Phi N\Phi^{-1}=\text{Ad}_{\Phi}N$ and
$X_{\alpha}^{s}=\text{Ad}_{\Phi}X_{\alpha}$, where
$\text{Ad}:G\times\mathfrak{g}\to\mathfrak{g}$ is the adjoint action of $G$ on
$\mathfrak{g}$. Therefore, we will make us of the composition
$\rho\circ\text{Ad}_{\Phi}$, which is a representation of the action of the
body frame fields on the configuration. We have that
$\dot{q}=\rho(\text{Ad}_{\Phi}N)q$ and
$\partial_{\alpha}q=\rho(\text{Ad}_{\Phi}X_{\alpha})q$ or, more compactly

$\displaystyle dq$
$\displaystyle=\rho(\text{Ad}_{\Phi}N)qdt+\rho(\text{Ad}_{\Phi}X_{\alpha})qdu^{\alpha}$
(4) $\displaystyle=\rho(\text{Ad}_{\Phi}\xi)q$

which expresses the derivatives of the spatio-temporal configuration in terms
of the generalised strain and velocity fields. Though Eq. 4 are kinematic
equations of motion for the system, they will not be used as such, as a
geometric formulation of the kinematics will be developed in the following
subsection. Equation 4 is however useful in two main regards: 1) It relates
the often more conceptually intuitive expressions for $\dot{q}$ and
$\partial_{\alpha}q$, to the Lie algebraic $N$ and $X_{\alpha}$, so as to lend
this intuition in understanding the latter. 2) If the spatio-temporal
configuration and its derivatives are assigned physical units, we can infer
the units of the components of $N$ and $X_{\alpha}$ by dimensional analysis.
An example of the former use-case will be shown at the end of this subsection,
and further examples of both use-cases will be shown in Sec. VI-VIII.

It is worth reiterating the requisite steps required to reach Eq. 4. After
having identified the configuration space $\mathbb{X}$ of a given system, and
an appropriate symmetry group $G$. We must then specify the action of $G$ on
configurations $\mathbb{X}\subseteq\mathbb{R}^{m}$. That is, we must find an
appropriate representation $\Pi:G\to GL(\mathbb{R}^{m}$) that acts on
$\mathbb{X}$. Note that in practice, we often identify the group elements with
their matrix representations. Once $\Pi$ has been specified, we must identify
the corresponding matrix representation of the Lie algebra $\rho$. Finally, we
can relate the differential of the configuration $dx$ with the action of the
kinematic fields as Eq. 4. The reader should note that the choice of group
action, though in principle arbitrary as long as it satisfies Eq. 1,it is
often desirable, in practice, to choose it such that the generalised strain
and velocity have intuitive physical intepretations.

To illustrate the above, we briefly provide an example (see Sec. VI-VIII for
further examples. In particular Sec. VI.1 is a good example of the procedure).
Consider a filament lying on the surface of the unit-sphere. Its spatio-
temporal configuration is $\mathbf{p}^{s}:W\to\mathbb{X}$, where
$\mathbb{X}=S^{2}$ and $M=[0,L_{0}]$, and $L_{0}\in\mathbb{R}^{+}$. For fixed
$t$, the image $\mathbf{p}^{s}(t,[0,L_{0}])$ is then a curve on the sphere.
The spatio-temporal configuration can be written in terms of a structure field
$R:W\to SO(3)$, such that $\mathbf{p}^{s}=R\mathbf{p}$, where $\mathbf{p}\in
S^{2}$. We write the generalised strain and velocity fields as
$\hat{\Omega}=R^{-1}\dot{R}$ and $\hat{\pi}=R^{-1}\partial_{u}R$, where the
hat denotes the hat map, defined in Eq. 47, such that
$\boldsymbol{\Omega},\boldsymbol{\pi}:W\to\mathbb{R}^{3}$ are angular velocity
vectors along the time and material direction respectively. We then have that
$\dot{\mathbf{p}}^{s}=\dot{R}R^{-1}\mathbf{p}^{s}=\hat{\Omega}^{s}\mathbf{p}^{s}=\boldsymbol{\Omega}^{s}\times\mathbf{p}^{s}$
and $\partial_{u}\mathbf{p}^{s}=\boldsymbol{\pi}^{s}\times\mathbf{p}^{s}$,
where $\boldsymbol{\Omega}^{s}=R\boldsymbol{\Omega}$ and
$\boldsymbol{\pi}^{s}=R\boldsymbol{\pi}$. Alternatively, we can write these in
terms of the body frame velocity and strain as
$\dot{\mathbf{p}}^{s}=\text{Ad}_{R}\hat{\Omega}\mathbf{p}^{s}$ and
$\partial_{u}\mathbf{p}^{s}=\text{Ad}_{R}\hat{\pi}\mathbf{p}^{s}$. In the
framework we present here we will prioritise the body frame. Physically, this
is often the appropriate choice as, for instance, the moment of inertia of a
rotating rigid body is a constant matrix in its body frame. Therefore, the
relation Eq.4 will be useful in Sec. IV, where we derive the dynamical force
balance equations for Cartan media.

In the two-step procedure we call _geometrisation_ , we have gone from a
kinematic description in terms of a spatio-temporal configuration, and
replaced it with a structure field satisfying Eq. 1, and finally a structure
generator Eq. 3. As will be demonstrated through multiple examples, the
generalised strain provides a description of the system in terms of its
intrinsic geometry. Where the geometrisation does not lead to a fully
intrinsic description, this can be ameliorated through _kinematic adaptation_
, discussed in Sec. III.5. The geometric formulation lends itself to the study
of the constitutive mechanics of continuum systems, which is fundamentally the
dynamics of the differential properties of manifolds, where stress is the
dynamical response to strain. In the setting of Cartan media, which we have
defined as sub-manifolds of homogeneous spaces, we are thus developing a
generalised notion of strains (and their conjugate stresses, in Sec. IV). In
the following subsection, we show how the geometrised description, encoded in
$X_{\alpha}$, evolves over time in response to the generalised velocity $N$.

###  III.2 Kinematic equations of motion

Thus far we have explicitly considered the structure field as defined globally
over the material base space $M$ and time $[0,T]$. However, in applications,
we often want the temporal evolution of the frame given an initial
configuration at $t=0$. That is, we are interested in how the spatial
configuration deforms under an applied velocity field. Here we derive the
kinematic equations of motion of Cartan media in a geometric form, the
solution of which yields the structure generator $\xi$, from which $\Phi$ can
be reconstructed using Eq. 3.

As it turns out, the equations of motion can be found by deriving the
conditions under which a $\mathfrak{g}$-valued $1$-form is a structure
generator. That is, the condition under which a $\mathfrak{g}$-valued $1$-form
$\xi$ on $M$ is related to a $G$-valued function on $M$ via Eq. 3. To find
this condition, we take the exterior derivative of both sides of Eq. 3, to
find

$\displaystyle d\xi$ $\displaystyle=d(\Phi^{-1}d\Phi)$ (5)
$\displaystyle=-(\Phi^{-1}d\Phi\Phi^{-1})\wedge d\Phi+\Phi^{-1}d(d\Phi)$
$\displaystyle=-\xi\Phi^{-1}\wedge d\Phi$

where $\wedge$ is the wedge product and where we have used that $d\Phi$ must
be an exact differential $d(d\Phi)=0$. The matrix wedge product is defined
element-wise using the standard definition of the wedge product. For any two
matrices of forms $A$ and $B$, their wedge product $C=A\wedge B$ is given
element-wise as $C_{ij}=A_{ij}\wedge B_{ij}$. Using the fact that the wedge
product commutes with matrix multiplication, we arrive at

$d\xi+\xi\wedge\xi=0,$ (6)

which is an integrability condition on $\xi$. It should be seen as analogous
to the requirement of the equality of mixed partials of an exact differential
in multi-variate calculus. By substituting Eq. 3 into Eq. 6, we find

$\displaystyle\dot{X}_{\alpha}$ $\displaystyle=\mathcal{D}_{\alpha}N,$
$\displaystyle\alpha=1,\dots,d$ (7a) $\displaystyle\partial_{\beta}X_{\alpha}$
$\displaystyle=\mathcal{D}_{\alpha}X_{\beta},$
$\displaystyle\alpha=1,\dots,d-1,$ (7b) $\displaystyle\beta=\alpha+1,\dots,d.$

where $\mathcal{D}_{\alpha}=\partial_{\alpha}+\text{ad}_{X_{\alpha}}$ is a
covariant derivative with respect to $G$ along the $\alpha$th material
direction, and $\text{ad}_{Z}:\mathfrak{g}\to\mathfrak{g}$ is the adjoint
action of any $Z\in\mathfrak{g}$, given by $\text{ad}_{Z}Y=[Z,Y],\
Y\in\mathfrak{g}$. Equation 7a are the geometrised kinematic equations of
motion of Cartan continua, describing the temporal evolution of the
differential geometry of the system, as encoded in the generalised strain
fields $X_{\alpha}$, as a response to the generalised velocity field $N$.
Equation 7b are not equations of motion, but are rather spatial integrability
conditions. See the subsequent subsection for more details. The kinematics and
the spatial integrability conditions can also be expressed in a non-coordinate
and invariant manner. Inserting the decomposition $\xi=Ndt+\xi_{M}$ into Eq.
6, we find

$\displaystyle\dot{\xi}_{M}=$ $\displaystyle\ \mathcal{D}_{M}N,$ (8a)
$\displaystyle d_{M}\xi_{M}$ $\displaystyle+\xi_{M}\wedge\xi_{M}=0,$ (8b)

which are respectively the kinematic equation of motion of the system, and the
spatial integrability conditions on the generalised strain, and where
$\mathcal{D}_{M}=d_{M}+\text{ad}_{\xi_{M}}$. In local coordinates, the time
derivative of the material structure generator is to be understood as
$\dot{\xi}_{M}=\dot{X}_{\alpha}du^{\alpha}$. We see that the spatial
integrability condition Eq. 8b mirrors Eq. 6. This is to be expected as
integrability needs to be obeyed irrespective of the base, whether $M$ or $W$.
We can thus also interpret Eq. 8a as an integrability condition in the
temporal direction.

In simulations the coordinatised formulation of the kinematics is simpler to
implement. However, Eq. 8a may be particularly useful for more sophisiticated
discretisation techniques, that preserve the geometric properties of the
exterior product (Desbrun  _et al._ , 2005; Hirani, ).

Equation 7a, or Eq. 8a, can be used to derive the equations of motion of any
Cartan media. Consider again the example of the filament on the unit-sphere.
From Eq. 7a we have that
$\dot{\boldsymbol{\pi}}=\partial_{u}\boldsymbol{\Omega}+\boldsymbol{\pi}\times\boldsymbol{\Omega}$,
where we have used that
$\text{ad}_{\hat{a}}\hat{b}=[\hat{a},\hat{b}]=\widehat{\mathbf{a}\times\mathbf{b}}$,
for any $\hat{a},\hat{b}\in\mathfrak{so}(3)$. Further examples are presented
in Sec. VI-VIII.

From an initial boundary value of the structure field $\Phi_{0}:M\to G$, the
initial conditions of the kinematic equations of motion can be computed using
$X_{\alpha}|_{t=0}=\Phi_{0}^{-1}\partial_{\alpha}\Phi_{0}$. Then, given a
generalised velocity field $N:W\to\mathfrak{g}$, Eq. 7a can be solved to find
the structure generator $\xi$. It is important to recall that $\xi$ only
captures the kinematics of the differential geometry of the system. That is,
$\xi$ only specifies $\Phi$ at any given time $t$ up to a global
transformation $g\in G$. Therefore a separate equation must be solved to track
the global movement of the system. Let $p_{r}\in M$ be a given reference
material point, with material coordinate $\mathbf{u}_{r}$, and let
$\Phi_{r}(t)=\Phi(t,p_{r})$ be the structure field evaluated at $p_{r}$. The
equation of motion of $\Phi_{r}$ is then given by

$\dot{\Phi}_{r}(t)=\Phi_{r}(t)N(t,p_{r})$ (9)

which is a matrix ODE, that can be solved using standard methods (Hale, 2009).
The full global spatio-temporal configuration can thus be constructed by
solving Eq. 3 with boundary conditions $\Phi(t,p_{r})=\Phi_{r}(t),\
t\in[0,T]$. The reconstruction algorithm is described in detail in Sec. V.1.

###  III.3 Spatial integrability

Equation 7b are a set of $(d+1)d/2$ spatial integrability conditions that must
be simultaneously satisfied at all times $t\in[0,T]$. Note that though there
are $d$ strain fields $X_{\alpha}$, adding up to a seeming total of $rd$
degrees of freedom, Eq. 7b shows that these are not independent. As expected,
only $r$ independent degrees of freedom remain to determine the structure
generator once the spatial integrability conditions have been imposed.

At first glance it may seem that Eq. 7 overdetermines the system, but Eq. 7a
and Eq. 7b are compatible. To see this, we compute the time-derivative of the
latter to get

$\displaystyle\partial_{t}\left(\partial_{\beta}X_{\alpha}-(\partial_{\alpha}+\text{ad}_{X^{\alpha}})X_{\beta}\right)$
(10)
$\displaystyle=\partial_{\beta}\dot{X}_{\alpha}-\partial_{\alpha}\dot{X}_{\beta}-[\dot{X}_{\alpha},X_{\beta}]-[X_{\alpha},\dot{X}_{\beta}]$
$\displaystyle=(\partial_{\beta}+\text{ad}_{X_{\beta}})\dot{X}_{\alpha}-(\partial_{\alpha}+\text{ad}_{X_{\alpha}})\dot{X}_{\beta}$
$\displaystyle=\partial_{\beta}([X_{\alpha},N])+[X_{\beta},\partial_{\alpha}N]+[X_{\beta},[X_{\alpha},N]]$
$\displaystyle-\partial_{\alpha}([X_{\beta},N])-[X_{\alpha},\partial_{\beta}N]-[X_{\alpha},[X_{\beta},N]]$
$\displaystyle=-[\partial_{\alpha}X_{\beta},N]+[X_{\beta},[X_{\alpha},N]]$
$\displaystyle+[\partial_{\beta}X_{\alpha},N]-[X_{\alpha},[X_{\beta},N]]$
$\displaystyle=[\partial_{\beta}X_{\alpha},N]-[\partial_{\alpha}X_{\beta},N]+[[X_{\beta},X_{\alpha}],N]$

where we used the Jacobi identity $[A,[B,C]]=-[C,[A,B]]-[B,[C,A]]$. Finally,
we get

$\partial_{t}\Delta_{\alpha\beta}^{\text{int}}=[\Delta_{\alpha\beta}^{\text{int}},N].$
(11)

where

$\Delta_{\alpha\beta}^{\text{int}}=\partial_{\beta}X_{\alpha}-\mathcal{D}_{\alpha}X_{\beta}$
(12)

is the residual error in spatial integrability
$\Delta_{\alpha\beta}^{\text{int}}:W\to\mathbb{R}$. If $X_{\alpha}$ satisfies
Eq. 7b at time $t=0$, then the right-hand side of Eq. 11 vanishes. Therefore
we see that the kinematic equations of motion Eq. 7a preserves the spatial
integrability conditions Eq. 7b at all future times. Equation 11 also serves
as a scaling law of the amplitude of the residual error in spatial
integrability, which can be studied by considering the eigenvalues of the
adjoint operator $-\text{ad}_{N}$.

###  III.4 Local charts on the material base space

In general, the material base space does not admit a global chart. Let
$\mathcal{A}_{M}$ be an atlas over $M$, with elements
$\mathcal{A}_{M}=\\{(U_{a},\mathbf{u}_{a})\ |\ a\in I\\}$, where $U_{a}\subset
M$ and $\mathbf{u}_{a}:M\to\mathbb{R}^{d}$ and $I$ is an index set. Due to the
product structure of the kinematic base space $W$, $\mathcal{A}_{M}$ extends
trivially to an atlas $\mathcal{A}_{W}=\\{([0,T]\times U_{a},\mathbf{w}_{a})\
|\ a\in I\\}$ over $W$, where $\mathbf{w}_{a}=(t,u_{a}^{1},\dots,u_{a}^{d})$.

Consider two local charts
$(U,\mathbf{u}),(U^{\prime},\mathbf{u}^{\prime})\in\mathcal{A}_{M}$ with
intersecting domains $U^{\prime}\cap U\neq\emptyset$, such that the spatial
structure generator can be written as
$\xi_{M}=X_{\alpha}du^{\alpha}=X^{\prime}_{\alpha}du^{\prime\alpha}$ on
$U^{\prime}\cap U$. Then we must have

$X^{\prime}_{\beta}=X_{\alpha}\frac{\partial u^{\alpha}}{\partial
u^{\prime\beta}}$ (13)

on $U^{\prime}\cap U$. If the material base space $M$ does not admit global
coordinates, then it is necessary to construct a patchwork of generalised
strain fields over coordinate charts that cover $M$, satisfying Eq. 13.

###  III.5 Adapted structure fields

If $r>n$, that is if the dimensionality of the symmetry group is larger than
that of the configuration space, then there is no unique structure field that
satisfies Eq. 1 for a given spatio-temporal configuration. This can be shown
as follows. For any homogeneous space $\mathbb{X}$ with symmetry group $G$,
there exists a $(r-n)$-dimensional subgroup $H\subseteq G$ such that
$G/H\cong\mathbb{X}$, where $G/H=\left\\{gH:g\in G\right\\}$ is the left coset
space. The subgroup $H$ is called a _stabiliser_ , and for any
$q\in\mathbb{X}$ we have that $h\cdot q=q$, for all $h\in H$. Therefore if
$\Phi(t,p)$ satisfies Eq. 1 at a point $(t,p)\in W$, then so does $\Phi(t,p)h$
for all $h\in H$.

There are therefore $r-n$ redundant degrees of freedom in the kinematic
configuration, once the system configuration has been expressed in terms of
the generalised strain fields. This is in principle not of consequence, as the
redundancies will have no effect on the reconstruction of the true spatio-
temporal configuration, via Eq. 1. However, it is possible to eliminate these
redundant degrees of freedom by choosing a ’gauge’ in $H$. Formally, a gauge
is a prescription that selects a unique structure field $\Phi$ that solves Eq.
1, given a spatio-temporal configuration $q:W\to\mathbb{X}$. We call this a
_kinematic adaption_ , which is a concept closely related to the notion of an
_adapted frame_ of space curves and surfaces (Frenet, 1852; Darboux, 1887) and
the _theory of moving frames_(Clelland, 2017; Cartan, 1951; Fels and Olver,
1998, 1999; Olver, ; Olver, 2005). For space curves and surfaces, the
adaptation leads to the desirable result of a parameterisation in terms of the
intrinsic and extrinsic curvatures of the systems. In the following we outline
the required steps to construct a kinematic adaptation, which we will then
briefly illustrate using an example. The reader may also find the detailed
derivations of some examples systems in Sec. VII helpful to understand the
procedure.

The construction of an adapted structure field is equivalent to choosing an
element $h\in H$ at each time and material point $(t,p)\in W$, which is
formally a function $h=h(q,\partial_{t}q,\partial_{\alpha}q,\dots)$, leading
to a one-to-one “map-between-maps” from spatio-temporal configurations
$q:W\to\mathbb{X}$ to structure fields $\Phi:W\to\mathbb{X}$. The reduction of
the degrees of freedom of the structure field results in the elimination of
$n-r$ constraints on the components of the generalised strain and velocity
fields each, which are to be inferred from Eq. 2 and Eq. 7.

In most applications, as in the example we will give below, the adapted
structure field is adapted to the _spatial_ kinematics of the system. That is,
the gauge is chosen as $h=h(q,\partial_{\alpha}q,\dots)$, without
$\partial_{t}x$. However, this is not always the case, see Sec. VIII.3 for an
example. To simplify the discussion, we will presume we are working with a
system in $d=1$ material dimensions, with a spatially adapted structure field
(see Sec. VII for higher-dimensional examples). In this case, from Eq. 2 we
find constraints on the strain, such that it takes value in a vector subset
$X\in V\subseteq\mathfrak{g}$, where $\text{dim}(V)=n$. Let $V^{\perp}$ be the
orthogonal complement of $V$ respectively, where $\text{dim}(V^{\perp})=r-n$,
and let $b_{i}\in V,\ i=1,\dots,n,$ and $b_{j}^{\perp}\in V,\ j=1,\dots,r-n$
be basis vectors for $V$ and $V^{\perp}$ respectively. A given Lie algebra
element $C\in\mathfrak{g}$ can be expanded as
$C=C_{i}b_{i}+C_{j}^{\perp}b_{j}^{\perp}$, where $C_{i}$ and $C_{i}^{\perp}$
are the components of $C$ in the respective bases. Then Eq. 7 decomposes into

$\displaystyle\dot{X}_{i}$ $\displaystyle=(\mathcal{D}_{\alpha}N)_{i},$
$\displaystyle i=1,\dots,n,$ (14a) $\displaystyle 0$
$\displaystyle=(\mathcal{D}_{\alpha}N)_{j}^{\perp},$ $\displaystyle
j=1,\dots,r-n,$ (14b)

which is the kinematic equation of motion for the strain, and a set of
constraints on $N$, respectively. In effect, Eq. 14b constrains $N$ onto the
subsurface of $\mathfrak{g}$ that ensures that the structure field remains
adapted in time. Note that the kinematic equations of motions in its original
unconstrained form, Eq. 7a, hold as before, under the condition that Eq. 14b
is satisfied. The remaining components of the generalised strain will in
general comprise the extrinsic, in addition to the intrinsic, geometry of the
system.

To illustrate the above, let us again consider a filament on the unit sphere.
The configuration space of the filament is $\mathbb{X}=S^{2}$, and its
symmetry group $G=SO(3)$; that is, $n=2$ and $r=3$. This will lead to an
$SO(2)$ gauge freedom in the kinematic description. To see this, we write the
structure field as $R:W\to SO(3)$ as an orthonormal frame $R=(\mathbf{e}_{1}\
\mathbf{e}_{2}\ \mathbf{e}_{3})$ that satisfies $R\mathbf{p}=\mathbf{p}^{s}$,
where $\mathbf{p}\in S^{2}$ and $\mathbf{p}^{s}:W\to S^{2}$. We furthermore
let $\mathbf{p}=(0\ 0\ 1)^{T}$, such that
$\mathbf{p}^{s}=R\mathbf{p}=\mathbf{e}_{3}$. The intrinsic geometry of the
filament corresponds to the longitudinal deformations along its length, and
the extrinsic geometry its curvature. We see Eq. 1 remains invariant under
rotations of $R$ around $\mathbf{e}_{3}$, at any point $u\in M$. The
corresponding generalised strain $\hat{\pi}:W\to\mathfrak{so}(3)$ will thus
have a superfluous degree of freedom. To rid the geometric kinematic
description of the system of this gauge freedom, we can construct a kinematic
adaptation by constraining $\mathbf{e}_{1}$ to be tangent to the filament at
all times; that is, we set $\mathbf{e}_{1}=h^{-1}\partial_{u}\mathbf{p}^{s}$,
where $h=|\partial_{u}\mathbf{p}^{s}|$. This effectively chooses a unique
element $R=R(\mathbf{p}^{s},\partial_{u}\mathbf{p}^{s})$ for a given filament
$\mathbf{p}^{s}(t,\cdot)$. From Eq. 2 we then find that $\pi_{1}=0$, and we
write $\boldsymbol{\pi}=(0\ h\ \kappa)^{T}$, where $\kappa$ is the scalar
curvature of the filament, and $h^{2}$ the metric along the filament induced
from ambient Euclidean space. The components $\kappa$ and $h$ correspond to
the extrinsic and intrinsic geometry of the filament respectively. To see the
latter, note that the length of the filament is given by
$L=\int_{0}^{L_{0}}|\partial_{u}\mathbf{p}^{s}|du=\int_{0}^{L_{0}}|\boldsymbol{\theta}|du=\int_{0}^{L_{0}}hdu$.
Now, let us assume that the structure field is adapted at $t=0$, such that
$\mathbf{e}_{1}(0,u)=(h^{-1}\partial_{u}\mathbf{e}_{3})\big{|}_{t=0}$ and
$\boldsymbol{\pi}(0,u)=(R^{-1}\partial_{u}R)_{t=0}=(0\ h(0,u)\
\kappa(0,u))^{T}$. The generalised velocity must be constrained so as the
maintain the adaption of the system in time, which we can find from Eq. 7,
leading to $\partial_{u}\Omega_{1}-\Omega_{2}\kappa+\Omega_{3}h=0$. Solving
the constraint for $\Omega_{2},$we have that
$\boldsymbol{\Omega}=\boldsymbol{\Omega}(h,\kappa,\Omega_{3})$; that is, the
generalised velocity is a function the generalised strain and $\Omega_{3}$.

The above example shows that we can see kinematic adaptation as a way to
“emulate” $\mathbb{X}$-configured Cartan media, with symmetry group $G$, using
a kinematically constrained Cartan system in a larger configuration space
$\tilde{\mathbb{X}}\supset\mathbb{X}$ with the same symmetry group, for which
$\text{dim}(\tilde{\mathbb{X}})=\text{dim}(G)$. The homogeneous space
$\tilde{\mathbb{X}}$ is essentially $G$ itself, and can formally be written as
a coset-space $\tilde{\mathbb{X}}=G/\\{e\\}$, where $e\in G$ is the identity
element. In the above example, we had that $\mathbb{X}=S^{2}$ and
$\tilde{\mathbb{X}}=SO(3)$, where the latter can be seen as the configuration
space of a “Cosserat rod” on the sphere (which is the subject of Sec. VIII.1).
From this point-of-view, the adapted structure field amounts to a
“kinematically constrained” Cosserat rod on the sphere, where it is
constrained such that $\mathbf{e}_{1}$ is tangent to the center-line.

##  IV Dynamics

In the previous section we found kinematic equations of motion of Cartan media
in terms of their generalised strain and velocity fields. Here we derive
geometrised second-order equations of motion, in terms of _generalised
momentum_ and _stress fields_ and _generalised body force densities_.
Consequently, we obtain momentum balance equations analogous to those of
classical continuum mechanics. As will be seen here, and with more detail in
Sec. VI-VIII, the form of the balance equations reflect the symmetry group of
the configuration space. The Cauchy momentum equation of classical continuum
mechanics, which is the balance of linear momentum, derives from the
translational symmetry of Euclidean space. Analogously, if $\mathbb{X}=S^{2}$,
for which the appropriate symmetry group is $G=SO(3)$, then the rotational
symmetry of the configuration space will lead to angular momentum balance
equations.

The derivation will begin programmatically by way of Hamilton’s principle.
That is, given a configuration-dependent Lagrangian density, most commonly
constructed from kinetic and potential energy densities, the dynamical
trajectory of the system must then be a stationary point of the corresponding
action functional. The equations of motion of such trajectories can then be
found by variation of the action in the configurational degrees of freedom,
leading to the Euler-Lagrange equations. However, this will not naturally lead
to mechanics expressed in terms of the differential geometry of the system, in
the spirit of the geometrised kinematics developed in the previous section.
Instead, we relate the variation to the infinitesimal action of the
generalised strain and velocity fields. In so doing we formulate the mechanics
in terms of _dual Lie algebra_ -valued fields, that are conjugate to the Lie
algebraic kinematic variables: generalised momentum and stress fields, and the
generalised body force density.

This approach was first pioneered in (Poincaré, 1901) for point particles,
from which a large literature on geometric particle mechanics (Marle, 2013;
Marsden and Ratiu, 2013; Holm, 2008) has sprung. In those contexts, the
resulting equation of motion is known as the _Euler-Poincaré_ _equation_. The
derivation we present here can thus be seen as a generalisation of this
procedure to continuum systems in homogeneous spaces.

The conservative dynamics of a system is fully specified given a Lagrangian
$L(q,dq)$, which, given a spatio-temporal configuration $q:W\to\mathbb{X}$, is
a volume form over the material base space. In a local chart
$\mathbf{u}:U\to\mathbb{R}^{d},\ U\subseteq M$, the Lagrangian can be written
as $L(q,dq)=\mathcal{L}(q,\dot{q},\partial_{\alpha}q)\ dt\wedge dV$, where we
have defined the volume element $dV=du^{1}\wedge\dots\wedge du^{d}$, and where
$\mathcal{L}$ is the Lagrangian density, which is a scalar density
$\mathcal{L}(q(\cdot,\cdot),\dot{q}(\cdot,\cdot),\partial_{\alpha}q(\cdot,\cdot)):W\to\mathbb{R}$
given a spatio-temporal configuration. Under a change-of-coordinates we have
that volume element transforms as a tensor density of weight $-1$,

$dV^{\prime}=du^{\prime 1}\wedge\dots\wedge du^{\prime d}=|J|^{-1}dV$, and the
Lagrangian density is a scalar density of weight $1$,

$\mathcal{L}^{\prime}(q,\dot{q},\partial^{\prime}_{\alpha}q)=|J|\mathcal{L}(q,\dot{q},\partial_{\alpha}q).$
(15)

where $\partial^{\prime}_{\alpha}=J_{\alpha}^{\beta}\partial_{\beta}$,
$J_{\alpha}^{\beta}=\frac{\partial u^{\beta}}{\partial u^{\prime\alpha}}$ is
the Jacobian matrix of the coordinate transformation and
$|J|=\text{det}\left[J\right]$. Therefore we have
$\mathcal{L}(q,\dot{q},\partial_{\alpha}q)\ dt\wedge
dV=\mathcal{L}^{\prime}(q,\dot{q},\partial^{\prime}_{\alpha}q)\ dt\wedge
dV^{\prime}$ .

The _action_ is a functional of the spatio-temporal configuration, formed by
integrating $L$ over the kinematic base space $W=[0,T]\times M$. A spatio-
temporal configuration is called _physical_ if it is a stationary trajectory
of the action functional, that is

$\delta\int_{W}L(q,dq)=\delta\int_{W}\mathcal{L}(q,\dot{q},\partial_{\alpha}q)\
dt\wedge dV=0$ (16)

under variations $q\to q+\delta q$, where $\delta q:W\to T\mathbb{X}$ is a
variational test function, which must vanish at the temporal boundaries
$\delta q(0,p)=\delta q(T,p)=0$ for all $p\in M$. Note that Eq. 16 is an abuse
of notation, as the Lagrangian density is defined with respect to a chart,
which may only cover a subset of the material base space $U\subseteq M$.
Strictly, the Lagrangian must be varied over a set of charts
$\\{(U_{a},\mathbf{u}_{a})\ |\ a\in K\\}=\mathcal{B}\subset\mathcal{A}_{M}$,
where $K\subset I$ is an index set, such that the domains $U_{\alpha}$ form a
disjoint partition of $M$. That is, $U_{a}\cap U_{b}=\emptyset$ for $a\neq b$
and $\bigcup_{a\in K}\bar{U}_{a}=M$, where $\bar{U}_{k}$ denotes the closure
of the open set $U_{k}$. Hamilton’s principle is then $\delta\sum_{a\in
J}\int_{[0,T]\times
U_{a}}\mathcal{L}^{(a)}(q,\dot{q},\partial_{\alpha}^{(a)}q)\ dt\wedge
dV_{(a)}=0$. In the following, we will continue to use the notation in Eq. 16,
and address chart-related issues as they arise.

In most physical applications, the dynamics is defined via a kinetic
$\mathcal{K}(\dot{q})$ and potential $\mathcal{U}(q,\partial_{\alpha}q$)
energy density over the system, such that the Lagrangian is constructed as
$\mathcal{L}(q,\dot{q},\partial_{\alpha}q)=\mathcal{K}(\dot{q})-\mathcal{U}(q,\partial_{\alpha}q)$.
Equation 16 can be recognised as identical in form to the variational
principles found in classical field theory, where the first term would
correspond to local field effects, and the latter two correspond to temporal
and spatial deformations of the field. For example, in scalar field theory a
typical Lagrangian may be of the form
$\mathcal{L}=\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}|\partial_{\alpha}\phi|^{2}-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}$,
where $\phi\in\mathbb{X}=\mathbb{R}$. In Sec. VIII.2 we apply the geometrised
mechanical framework presented here to non-linear $\sigma$ field theories.

Recall Eq. 1 and Eq. 4, which gives us the spatio-temporal configuration in
terms of the structure field $q=\Phi\cdot q_{r}$, and its derivatives in terms
of the generalised strain and velocity $\dot{q}=\rho(\text{Ad}_{\Phi}N)q$ and
$\partial_{\alpha}q=\rho(\text{Ad}_{\Phi}X_{\alpha})q$. We can rewrite these
as $\dot{x}=\Pi(\Phi)\rho(N)q_{0}$ and
$\partial_{\alpha}q=\Pi(\Phi)\rho(X_{\alpha})q_{r}$, or combined as
$dq=\Pi(\Phi)\rho(\xi)q_{r}$. We can therefore define the _left-trivialised
Lagrangian_ (Marle, 2013; Marsden and Ratiu, 2013; Holm, 2008)

$\check{L}(\Phi,\xi)=L(\Pi(\Phi)q_{r},\Pi(\Phi)\rho(\xi)q_{r}),$ (17)

which is, in local coordinates,
$\check{\mathcal{L}}(\Phi,N,X_{\alpha})=\mathcal{L}(\Pi(\Phi)q_{0},\Pi(\Phi)\rho(N)q_{r},\Pi(\Phi)\rho(X_{\alpha})q_{r})$,
such that $\check{L}(\Phi,\xi)=\check{\mathcal{L}}(\Phi,N,X_{\alpha})\
dt\wedge dV$. The Lagrangian Eq. 17 is now expressed entirely in terms of Lie
group and Lie algebraic fields. The left-trivialised Lagrangian transforms as
$\check{\mathcal{L}}^{\prime}(\Phi,N,X^{\prime}_{\alpha})=|J|\check{\mathcal{L}}(\Phi,N,X_{\alpha})$
under a change-of-coordinates, where $X^{\prime}_{\alpha}$ is given by Eq. 13.
Note that the generalised strain and velocity fields are expressed in the body
frame in $\check{\mathcal{L}}^{\prime}(\Phi,N,X^{\prime}_{\alpha})$, therefore
the left-trivialised Lagrangian neatly separates non-constitutive and
constitutive parts of the dynamics, corresponding to the first and the latter
two arguments respectively.

Variations of Eq. 17 must preserve both the spatial and temporal integrability
of the system, Eq. 7. We must therefore derive the space of permissible
variations $\delta N$ and $\delta X_{\alpha}$. We begin by considering the
variation of the configurational Lagrangian, Eq. 16. Applying the variational
operator $\delta$ on both sides of Eq. 1 we find that $\delta
q=\delta\Pi(\Phi)q_{0}=\delta\Pi(\Phi)\Pi(\Phi^{-1})q=\rho(\text{Ad}_{\Phi}\eta)q$,
where $\eta=\Phi^{-1}\delta\Phi$, which must satisfy $\eta(0,p)=\eta(T,p)=0$
for all $p\in M$. We see that the configurational variation can be fully
expressed in terms of the Lie algebraic variational test function $\eta$. We
now vary the structure generator $\xi$ to get

$\displaystyle\delta\xi$
$\displaystyle=-\Phi^{-1}\delta\Phi\Phi^{-1}d\Phi+\Phi^{-1}\delta(d\Phi)$ (18)
$\displaystyle=d\eta-\eta\xi+\xi\eta$
$\displaystyle=d\eta+\text{ad}_{\xi}\eta$

where we used $d\eta=-\xi\eta+\Phi^{-1}d(\delta\Phi)$ and $d\delta=\delta d$.
In local coordiantes, we thus find that the variations of the generalised
strain and velocity fields are

$\displaystyle\delta N$
$\displaystyle=\mathcal{D}_{t}=\dot{\eta}+\text{ad}_{N}\eta,$ (19a)
$\displaystyle\delta X_{\alpha}$
$\displaystyle=\mathcal{D}_{\alpha}=\partial_{\alpha}\eta+\text{ad}_{X}\eta$
(19b)

for $\alpha=1,\dots,d$, and where we used Eq. 3. In the subsequent
subsections, we will perform this Lie algebraic variation to derive the
geometrised dynamical equations of motion of Cartan media.

In Sec. IV.1 and Sec. IV.2 we will derive generalised momentum balance
equations for Cartan media, using the left-trivialised Lagrangian and
Hamilton’s principle. Section IV.1 treats the purely constitutive case, where
the generalised stress fields are derived from strain-dependent potential
energy densities. Section IV.2 introduces generalised body force densities,
derived from configuration-dependent potential energy densities. In Sec. IV.3
we relax the assumption of conservative dynamics, formulating a generalised
Lagrange-D’Alembert principle for Cartan media, from which general non-
conservative dynamical equations of motion can be derived. Finally, in Sec.
III.5 we continue the discussion of Sec. III.5 and consider dynamics under
kinematic adaptation.

###  IV.1 Generalised stresses

We will now apply Hamilton’s principle assuming purely constitutive dynamics.
That is, we make the following two assumptions: Firstly, the configurational
Lagrangian is not dependent on its first argument $L(q,dq)=L(q_{0},dq)$.
Secondly, the latter argument satisfies _left-invariance_
$L(q,\Pi(g)\rho(\xi)q_{r})=L(q,\rho(\xi)q_{r})$ for any $g\in G$, which in
local coordinates corresponds to
$\mathcal{L}(q,\Pi(g)\rho(N)q_{r},\Pi(g)\rho(X_{\alpha})q_{r})=\mathcal{L}(q,\rho(N)q_{r},\rho(X_{\alpha})q_{r})$.
Both of these are equivalent to the condition that the left-trivialised
Lagrangian satisfies $\check{L}(\Phi,\xi)=\check{L}(e,\xi)$, where $e\in G$ is
the identity element. If these hold, we can construct a _reduced
Lagrangian_(Marle, 2013; Marsden and Ratiu, 2013; Holm, 2008)

$l(\xi)=L(q_{r},\rho(\xi)q_{r})$ (20)

and $\ell(N,X_{\alpha})=\mathcal{L}(q_{r},\rho(N)q_{r},\rho(X_{\alpha})q_{r})$
in local coordinates, such that $l(\xi)=\ell(N,X_{\alpha})\ dt\wedge dV$, with
which we can derive the dynamics using Hamilton’s principle. The resulting
second-order equations of motion will be purely constitutive, expressed in
terms of generalised momentum and stress fields.

An example of a reducible Lagrangian for the filament on the unit sphere is
$\mathcal{L}=\frac{m}{2}|\dot{\mathbf{p}}^{s}|^{2}-\frac{k}{2}|\partial_{u}\mathbf{p}^{s}|^{2}$,
which is only dependent on the derivatives of the configuration. This
Lagrangian is reducible, as
$|\dot{\mathbf{p}}^{s}|^{2}=|\dot{R}\mathbf{p}|^{2}=|R\hat{\Omega}\mathbf{p}|^{2}=|\hat{\Omega}\mathbf{p}|^{2}=|\boldsymbol{\Omega}\times\mathbf{p}|^{2}=\boldsymbol{\Omega}^{T}P_{0}\boldsymbol{\Omega}$
and
$|\partial_{u}\mathbf{p}^{s}|^{2}=|R\hat{\pi}\mathbf{p}|^{2}=\boldsymbol{\pi}^{T}P_{0}\boldsymbol{\pi}$,
where $\hat{\pi}=\Phi^{-1}\partial_{u}\Phi$ is the spatial angular rate-of-
change along the material coordinate $u$ and $P_{0}=\hat{p}^{T}\hat{p}$. We
can write the reduced Lagrangian as
$\ell=\frac{m}{2}\boldsymbol{\Omega}^{T}P_{0}\boldsymbol{\Omega}-\frac{k}{2}\boldsymbol{\pi}^{T}P_{0}\boldsymbol{\pi}$.
Note that $P_{0}$ is a rank-$2$ matrix, reflecting the fact that
$\text{dim}(S^{2})=n=2$, whilst $\text{dim}(SO(3))=r=3$. Such cases will not
affect the discussion that follows, but we will consider a worked out example
in Sec. VI.2. The Lagrangian would be irreducible if it contains a term like
$\mathbf{p}^{T}\mathbf{p}^{s}$, which breaks the first of the two assumptions
outlined above. More esoterically, the second assumption does not hold if the
strain couples with quantities in the spatial frame directly, such as if there
is a term like $\mathbf{p}\cdot\partial_{u}\mathbf{p}^{s}$. Finally, a term
like $\mathbf{p}^{s}\cdot\partial_{u}\mathbf{p}^{s}$ would violate both
assumptions simulatenously.

We now use Eq. 19 to vary the reduced Lagrangian. Let $b_{i}\in\mathfrak{g},\
i=1,\dots,r$, be a basis for the Lie algebra, such that we can write
$C=C_{i}b_{i}$ for any $C\in\mathfrak{g}$. The variation of the Lagrangian is
then $\delta\ell=\frac{\partial\ell}{\partial N_{i}}\delta
N_{i}+\frac{\partial\ell}{\partial(X_{\alpha})_{i}}\delta(X_{\alpha})_{i}$,
where we can identify $\frac{\partial\ell}{\partial N_{i}}$ and
$\frac{\partial\ell}{\partial(X_{\alpha})_{i}}$ as components of elements of
the dual Lie algebra $\mathfrak{g}^{*}$, which are contracting with the
variations $\delta N$ and $\delta X_{\alpha}$ respectively. We rewrite the
contractions as an inner product
$\langle\cdot,\cdot\rangle:\mathfrak{g}\times\mathfrak{g}^{*}\to\mathbb{R}$,
which we define as

$\langle C,Y\rangle=C_{i}Y_{i}$

for any $C\in\mathfrak{g}$ and $Y=Y_{i}B_{i}\in\mathfrak{g}^{*}$, where
$B_{i}\in\mathfrak{g}^{*},\ i=1,\dots,r$ is a basis for the dual Lie algebra
satisfying $\langle b_{i},B_{j}\rangle=\delta_{ij}$. We can thus rewrite the
variation as $\delta\ell=\left\langle\delta
N,S\right\rangle-\left\langle\delta X_{\alpha},Q^{\alpha}\right\rangle$, where
we have defined the _generalised momentum_ $S=\frac{\partial\ell}{\partial
N}=\frac{\partial\mathcal{K}}{\partial N}$ and _generalised stress_
$Q^{\alpha}=-\frac{\partial\ell}{\partial
X_{\alpha}}=\frac{\partial\mathcal{U}}{\partial X_{\alpha}}$ _fields_.

$\displaystyle S$ $\displaystyle=\frac{\partial\ell}{\partial
N}=\frac{\partial\mathcal{K}}{\partial N}\in\mathfrak{g}^{*},$ (21a)
$\displaystyle Q^{\alpha}$ $\displaystyle=-\frac{\partial\ell}{\partial
X_{\alpha}}=\frac{\partial\mathcal{U}}{\partial
X_{\alpha}}\in\mathfrak{g}^{*}.$ (21b)

Note that these derivatives are a short-hand for $\frac{\partial\ell}{\partial
N}=\frac{\partial\ell}{\partial N_{i}}B_{i}$. However, if we consider Lie
algebra elements in a matrix representation, $\frac{\partial\ell}{\partial N}$
can be interpreted as a matrix derivative. See App. A for more details. The
inner product also allows us to dualise linear operators on the Lie algebra.
Given a linear operator $\mathrm{A}:\mathfrak{g}\to\mathfrak{g}$, we define
its _dual_ as the operator
$\mathrm{A}^{*}:\mathfrak{g}^{*}\to\mathfrak{g}^{*}$ defined by the relation
$\left\langle\mathrm{A}C,Y\right\rangle=-\left\langle
C,\mathrm{A}^{*}Y\right\rangle$, for any $C\in\mathfrak{g}$ and
$Y\in\mathfrak{g}^{*}$.

The generalised momentum field transforms as a scalar density of weight $1$
under change of charts, and the generalised stress field as a vector density
of weight $1$. We have

$\displaystyle S^{\prime}$ $\displaystyle=|J|S,$ (22a) $\displaystyle
Q^{\prime\alpha}$ $\displaystyle=|J|\frac{\partial u^{\prime\alpha}}{\partial
u^{\beta}}Q{}^{\beta}$ (22b)

where we used Eq. 16 and Eq. 13.

We can now proceed to vary the action. We have

$\displaystyle\delta\int_{W}l(\xi)=\delta\int_{W}\ell(N,X_{\alpha})\ dt\wedge
dV$ (23) $\displaystyle=\int_{W}\left\\{\left\langle\delta
N,S\right\rangle-\left\langle\delta
X_{\alpha},Q^{\alpha}\right\rangle\right\\}dt\wedge dV$
$\displaystyle=\int_{W}\left\\{\left\langle\mathcal{D}_{t}\eta,S\right\rangle-\left\langle\mathcal{D}_{\alpha}\eta,Q^{\alpha}\right\rangle\right\\}dt\wedge
dV$
$\displaystyle=\int_{W}\left\\{-\left\langle\eta,\mathcal{D}_{t}^{*}S\right\rangle+\left\langle\eta,\mathcal{D}_{\alpha}^{*}Q^{\alpha}\right\rangle\right\\}dt\wedge
dV$ $\displaystyle\
+\int_{W}\left\\{\partial_{t}\langle\eta,S\rangle-\partial_{\alpha}\langle\eta,Q^{\alpha}\rangle\right\\}dt\wedge
dV$

where we used integration-by-parts, and where the dualised covariant
derivatives are $\mathcal{D}_{t}^{*}=\partial_{t}+\text{ad}_{N}^{*}$ and
$\mathcal{D}_{\alpha}^{*}=\partial_{t}+\text{ad}_{X_{\alpha}}^{*}$, and where
$\text{ad}_{C}^{*}:\mathfrak{g}^{*}\to\mathfrak{g}^{*},\ C\in\mathfrak{g}$ is
the dual of the adjoint action. See App. A for further details on the dual
adjoint, and an example of its matrix representation for $\mathfrak{se}(3)$.
We note here again that we are abusing the notation, as the integrands are
expressed in local charts. This is only of consequence for the final line of
Eq. 23, which will lead to boundary terms, and will be massaged into an
invariant form below.

By imposing that the integral of the penultimate line of Eq. 23 must vanish
for all $\eta$, we find the momentum balance equations for constitutive
dynamics in local coordinates

$\mathcal{D}_{t}^{*}S=\mathcal{D}_{\alpha}^{*}Q^{\alpha},$ (24)

which, together with the kinematics equations of motion Eq. 7a, defines the
constitutive mechanics of Cartan media. It is of notethat $\Phi$ does not
appear explicitly in the constitutive momentum balance equations derived here,
as for the geometrised kinematic equations of motion Eq. 7a. Therefore, if the
dynamics is constitutive, the entire set of kinematic and dynamic equations of
motion can be simulated purely in terms of the generalised strain and velocity
fields $X_{\alpha}$and $N$. The balance equation can be expressed as equations
of motion expressed in terms of $N$ using Eq. 21. Combined, the kinematic and
dynamic equations of motion form a second-order system for constitutive Cartan
media mechanics.

Eq. 24 can now be used to derive the constitutive momentum balance equations
for any Cartan media. Let us again consider the filament on the sphere. Let
the reduced kinetic and potential energy densities be of the form
$\mathcal{K}=\frac{1}{2}\boldsymbol{\Omega}^{T}\mathbb{I}\boldsymbol{\Omega}$
and $\mathcal{U}=\frac{1}{2}\boldsymbol{\pi}^{T}\mathbb{N}\boldsymbol{\pi}$
respectively, where $\mathbb{I},\mathbb{N}\in\mathbb{R}_{+}^{3\times 3}$ are
symmetric positive-definite matrices. We write the generalised momentum and
stress as $\mathbf{L}=\mathbb{I}\boldsymbol{\Omega}$ and
$\mathbf{M}=\mathbb{K}\boldsymbol{\pi}$ respectively, where we have used bold-
faced notation to signify that these are vectors in $T\mathbb{E}^{3}$. In this
context we should see $\mathbf{L}(t,u)$ and
$\mathbf{M}(\boldsymbol{\pi}(t,u))$ as the angular momentum of the material
point $u$ at time $t$, and the moment it is experiencing, respectively. These
quantities are all in the body frame of the system. As
$\text{ad}^{*}=\text{ad}$ for $\mathfrak{so}(3)$ (see App. A) we have that
$\dot{\mathbf{L}}+\boldsymbol{\Omega}\times\mathbf{L}=\partial_{u}\mathbf{M}+\boldsymbol{\pi}\times\mathbf{M}$,
which are angular momentum balance equations. Further examples are presented
in Sec. VI-VIII.

We now consider the final line of Eq. 23. The first term in the integrand
vanishes, as $\int_{W}\partial_{t}\langle\eta,S\rangle dt\wedge
dV=\int_{0}^{T}dt\int_{M}\partial_{t}\langle\eta,S\rangle
dV=\int_{M}[\langle\eta,S\rangle]_{0}^{T}dV=0$, where we have used that $\eta$
vanishes at the temporal boundaries. To simplify the treatment of the second
term, we will introduce an arbitrary volume form $\omega$ over $M$, which we
write in local coordinates as $\omega=wdV$, where $w$ is a scalar density. Let
$A^{\alpha}=w^{-1}\langle\eta,Q^{\alpha}\rangle$, which are components of a
vector field $A=A^{\alpha}\frac{\partial}{\partial u^{\alpha}}$ on $M$, then
$\int_{M}\partial_{\alpha}\langle\eta,Q^{\alpha}\rangle
dV=\int_{M}g^{-1}\partial_{\alpha}(gA^{\alpha})\ \omega=\int_{M}\text{div}A\
\omega$, where we have used the definition of the divergence. From the
divergence theorem, we then have that $\int_{M}\text{div}A\
\omega=\int_{\partial M}(n,A)\ \omega_{S}$, where $\omega_{S}$ is the induced
volume form on the material surface $\partial M$, and where $n:\partial M\to
T^{*}M$ is the unique (up to scalar multiplication) covector-field normal to
$\partial M$, and the contraction is $(n,A)=n_{\alpha}A^{\alpha}$ in local
coordinates. We now impose that this term must vanish for all $\eta$, which
gives us the boundary condition

$n_{\alpha}Q^{\alpha}=0$ (25)

in local coordinates. This is a coordinate-independent condition, as
$n_{\alpha}=\frac{\partial u^{\prime\beta}}{\partial
u^{\alpha}}n^{\prime}_{\beta}$ under a change of charts, such that
$n^{\prime}_{\alpha}Q^{\prime\alpha}=|J|n_{\alpha}Q^{\alpha}=0$. As the stress
is a function of the strain $Q^{\alpha}=Q^{\alpha}(X_{1},X_{2},\dots,X_{d})$,
then these boundary conditions will in turn induce conditions on $X_{\alpha}$
at the material boundaries.

###  IV.2 Generalised body force densities

If the assumptions of the previous subsection do not hold, that is, if the
Lagrangian has explicit configurational dependence, or if the generalised
strain couples with the spatial frame, _generalised body force densities_ will
appear in the balance equations as a result. Furthermore, as the dynamics is
no longer purely constitutive, the equations of motion will now require
explicit knowledge of the structure field $\Phi$. Fundamentally, this is due
to the explicit involvement of the spatial frame in the dynamics, which
couples with intrinsic geometric formuation of the kinematics. Within the
context of the geometric mechanics of point-particles (Marle, 2013; Marsden
and Ratiu, 2013; Holm, 2008), such symmetry-breaking terms in the Lagrangian
are called _advected terms_.

Varying the left-trivialised Lagrangian Eq. 17, we find

$\displaystyle\delta\int_{W}\check{\mathcal{L}}\ dt\wedge d^{d}u$ (26)
$\displaystyle=\int_{W}\left\\{\mathcal{T}_{\mu}\delta\Phi^{\mu}+\left\langle\delta
N,S\right\rangle-\left\langle\delta
X_{\alpha},Q^{\alpha}\right\rangle\right\\}\ dt\wedge d^{d}u$

where $S=\frac{\partial\check{\mathcal{L}}}{\partial N}$ and
$Q=-\frac{\partial\check{\mathcal{L}}}{\partial X_{\alpha}}$, and we have
defined
$\mathcal{T}_{\mu}=\frac{\partial\check{\mathcal{L}}}{\partial\Phi^{\mu}}$. We
can identify $\mathcal{T}$ as a covector-field on the Lie group
$\mathcal{T}\in T^{*}G$, as it contracts with the variational test function
$\delta\Phi\in TG$. Depending on how $G$ is parameterised, $\Phi^{\mu}$ can
either be considered coordinates on $G$, or the components of a matrix
representation $G$. For example, if $G=SO(3)$, we may represent Lie group
elements in terms of Euler angles $(\alpha,\beta,\gamma)\mapsto
R(\alpha,\beta,\gamma)\in SO(3)$, or as rotation matrices $R=(\mathbf{e}_{1}\
\mathbf{e}_{2}\ \mathbf{e}_{3})\in\mathbb{R}^{3\times 3}$, where
$\mathbf{e}_{i},\ i=1,2,3$ are an orthonormal set of basis vectors in
$\mathbb{R}^{3}$. In the latter case, the contraction in the first term of Eq.
26 may be written as $\mathcal{T}_{\mu\nu}^{T}\delta\Phi_{\mu\nu}$, where
$\mathcal{T}_{\mu\nu}=\frac{\partial\check{\mathcal{L}}}{\partial\Phi_{\nu\mu}}$,
following the numerator-layout matrix derivative convention as defined in App.
A.

Now, recall that the variational test function can be written as
$\delta\Phi=\Phi\eta$. We thus have that
$\mathcal{T}_{\mu\nu}^{T}\delta\Phi_{\mu\nu}=\mathcal{T}_{\mu\nu}^{T}\Phi_{\mu\sigma}\eta_{\sigma\nu}=\text{Tr}\left[\mathcal{T}\Phi\eta\right]=\eta_{i}\text{Tr}\left[\mathcal{T}\Phi
b_{i}\right]=\langle\eta,T\rangle$, where we have defined the dual Lia
algebra-valued _generalised body force density_

$\displaystyle T$ $\displaystyle=T_{i}B_{i}\in\mathfrak{g}^{*}$ (27)
$\displaystyle T_{i}$ $\displaystyle=\mathcal{\text{Tr}}\left[\mathcal{T}\Phi
b_{i}\right].$

This equation can be seen as transforming the generalised force in the spatial
frame $\mathcal{T},$ into a generalised force in the body frame $T$, via a
pull-back from $T^{*}G$ to $\mathfrak{g}^{*}$. That is, $\mathcal{T}\in
T^{*}G$ is trivialised to a dual Lie algebra-valued vector
$T\in\mathfrak{g}^{*}$. The generalised body force density transforms as a
density under a change of charts, it thus follows follows the same
transformation law as the Lagrangian Eq. 15.

By imposing that the full variation must vanish for all $\eta$, we find the
momentum balance equations
$\mathcal{D}_{t}^{*}S=\mathcal{D}_{\alpha}^{*}Q^{\alpha}+T$. Note that in Eq.
27 we see explicitly that the the equations of motion are no longer expressed
purely in terms of the intrinsic geometry of the system, as $T$ is a function
of $\Phi$. In numerical simulations, this entails that we need to numerically
propagate the structure field in time using Eq. 2b, in addition to the
generalised strain $X_{\alpha}$ and velocity $N$ fields.

Let us now illustrate the usage of Eq. 27 by again considering the filament on
the sphere. We write a basis of the Lie algebra as
$\hat{b}_{i}\in\mathfrak{so}(3)$ Consider a configurational potential
$\mathcal{U}(\mathbf{p})=a\mathbf{p}^{s}\cdot\mathbf{p}$, which we can rewrite
as $\hat{\mathcal{U}}(R)=a\mathbf{p}^{T}R\mathbf{p}$. We write the structure
field as an orthonormal frame $R=\left(\mathbf{e}_{1}\ \mathbf{e}_{2}\
\mathbf{e}_{3}\right)$. Then, we have that
$\mathcal{T}=-g\mathbf{p}\mathbf{p}^{T}$, and
$T_{i}=\text{Tr}\left[\mathcal{T}R\hat{b}_{i}\right]=-a\text{Tr}\left[\mathbf{p}\mathbf{p}^{T}R\hat{b}_{i}\right]=-a\mathbf{p}^{T}R\hat{b}_{i}\mathbf{p}=-a\mathbf{p}^{T}R(\mathbf{d}_{i}\times\mathbf{p})=a\mathbf{p}^{T}(\mathbf{p}^{s}\times\mathbf{e}_{i})$.
We can write $\mathbf{T}=aR^{T}(\mathbf{p}\times\mathbf{p}^{s})$. Using the
results from the previous subsection, the angular momentum balance equations
are now
$\dot{\mathbf{L}}+\boldsymbol{\Omega}\times\mathbf{L}=\partial_{u}\mathbf{M}+\boldsymbol{\pi}\times\mathbf{M}+aR^{T}(\mathbf{p}\times\mathbf{p}^{s})$.

The generalised body force may be derived equivalently in terms of the
configurational degree of freedom, as $\check{\mathcal{L}}$ may be considered
a function of $\Phi$ and $x$ interchangeably. We can therefore write the
variation as $\delta\check{\mathcal{L}}=\mathcal{T}_{\mu}\delta
x^{\mu}+\dots$, where
$\mathcal{T}_{\mu}=\frac{\partial\check{\mathcal{L}}}{\partial x^{\mu}}$ and
$\delta x=\rho(\text{Ad}_{\Phi}\eta)x$ as before. We find an alternative
equivalent expression for the generalised body force density

$\displaystyle T$ $\displaystyle=T_{i}B_{i}\in\mathfrak{g}^{*}$ (28)
$\displaystyle T_{i}$
$\displaystyle=\mathcal{T}_{\mu}\rho(\text{Ad}_{\Phi}b_{i})_{\nu}^{\mu}x^{\mu}$
$\displaystyle=\mathcal{T}_{\mu}\Pi(\Phi)_{\nu}^{\mu}\rho(b_{i})_{\sigma}^{\nu}x_{0}^{\sigma}.$

Let us now take a closer look at the transformation from $\mathcal{T}$ to $T$.
Recall that we may in general consider the configuration space as a subset of
some larger space $\mathbb{X}\subseteq\mathbb{R}^{m},$where $m>n$. Therefore,
we have that $\mathcal{T}=\frac{\partial\mathcal{U}}{\partial x}\in
T^{*}\mathbb{R}^{m}$. That is, at a given configurational point
$q\in\mathbb{X}$, the spatial generalised body force $\mathcal{T}\in
T_{q}^{*}\mathbb{R}^{m}$ may be “pointing outside” of the cotangent plane
$T_{q}^{*}\mathbb{X}$. However, as Eq. 28 is a pull-back from
$T^{*}\mathbb{R}^{m}$ to $\mathfrak{g}^{*}$, this leads to a generalised force
$T$ that is compatible with the geometry of the configuration space.

Considering the same example as before, we now find that
$\boldsymbol{\mathcal{T}}=-\frac{\partial\mathcal{U}}{\partial\boldsymbol{p}}=-a\mathbf{p}^{s}$.
From Eq. 28 we have that
$\boldsymbol{\mathcal{T}}^{T}R\hat{b}_{i}\mathbf{p}=\boldsymbol{\mathcal{T}}^{T}(\mathbf{e}_{i}\times\mathbf{p}^{s})=\mathbf{e}_{i}^{T}(\mathbf{p}\times\boldsymbol{\mathcal{T}})$,
such that
$\mathbf{T}=R^{T}(\mathbf{p}^{s}\times\boldsymbol{\mathcal{T}})=aR^{T}(\mathbf{p}\times\mathbf{p}^{s})$,
which is the same expression as earlier. Comparing the expressions for
$\boldsymbol{\mathcal{T}}$ and $\mathbf{T}$ we see that the latter is of a
form such that it is compatible with the geometry of the configuration space.
That is, where $\boldsymbol{\mathcal{T}}=-a\mathbf{p}$ is a _force_ ,
$\mathbf{T}=aR^{T}(\mathbf{p}\times\mathbf{p}^{s})$ is a _moment,_ in the
traditional sense of classical mechanics. The corresponding moment in the
spatial frame is then
$\mathbf{T}^{s}=R\mathbf{T}=a(\mathbf{p}\times\mathbf{p}^{s})$.

###  IV.3 Non-conservative dynamics and surface forces

Energy dissipation or gain signifies the presence of non-conservative dynamics
within a continuum system. This implies that, mathematically, the forces and
stresses that are acting on the system do not arise from a scalar principle.
In this section, we will generalise the results of the previous two
subsections to include such non-conservative dynamics.

In classical mechanics, Hamilton’s principle serves as a specialized instance
of the D’Alembert principle, which accommodates non-conservative dynamics. By
analogy, we will introduce a _generalized integral Lagrange-D’Alembert_
_principle_ tailored to Cartan media. We begin by considering classical
mechanics. In the framework presented in this paper, classical mechanical
systems are point-particles, corresponding to a $0$-dimensional material base
space $\text{dim}(M)=0$. As before, the configuration space is $\mathbb{X}$,
and the system configuration is denoted by $q$ (which in this context are
often called _generalised coordinates_). The conservative dynamics of such a
system is specified by a Lagrangian $\mathcal{L}(q,\dot{q})$, and from
Hamilton’s principle $\delta\int_{0}^{T}\mathcal{L}=0$ we find
$\int_{0}^{T}\left(\frac{\partial\mathcal{L}}{\partial\dot{q}^{\mu}}\delta\dot{q}^{\mu}+\frac{\partial\mathcal{L}}{\partial
q^{\mu}}\delta q^{\mu}\right)dt=0$. In most applications the Lagrangian is of
the form $\mathcal{L}(q,\dot{q})=\mathcal{K}(\dot{q})-\mathcal{U}(q)$, where
$\mathcal{K}$ and $\mathcal{U}$ are the kinetic and potential energy functions
of the system. In which case, we can identify the momentum
$p_{\mu}=\frac{\partial\mathcal{K}}{\partial\dot{q}^{\mu}}$ and
$f_{\mu}=-\frac{\partial\mathcal{U}}{\partial q^{\mu}}$ as the inertial and
external forces acting on the system respectively. For a point particle, we
can model non-conservative dynamics by simply relaxing the assumption that $f$
derives from a potential. In which case, we arrive at the variational
principle $\int_{0}^{T}(p_{\mu}\delta\dot{q}^{\mu}-f_{\mu}\delta
q^{\mu})dt=0$, which is the integral Lagrange-D’Alembert principle for a point
particle (Marsden and Ratiu, 2013; Holm  _et al._ , 1998). This same
conceptual leap can be extended to Cartan media. As we have already seen, the
dynamics of continua is substantially different from that of point-particles,
as the former can suffer constitutive stresses in addition to body forces. In
the non-conservative setting, there is a third possible component of the
dynamics. The system may suffer a surface force density, normal to the
material surface.

To construct a Lagrange-D’Alembert principle for Cartan media, we suspend the
requirement that the dynamical fields derive from the gradient of a potential.
That is, at the point when the generalised stress field $Q^{\alpha}$ and the
generalised body force density $T$ are introduced in Eq. 23 and Eq. 27
respectively, we allow for these to be general functions of the variables of
the kinematic configuration. Lastly, we also introduce a generalised surface
force density $P$, defined over the material surface of the system. The
resulting expression is

$\displaystyle\int_{W}\left\\{\langle\delta N,S\rangle-\langle\delta
X_{\alpha},Q^{\alpha}\rangle+\langle\eta,T\rangle\right\\}dt\wedge dV$ (29)
$\displaystyle\quad+\int_{[0,T]\times\partial M}PdS=0$

which is a generalised Lagrange-D’Alembert principle for Cartan media. Here,
$dS$ is a surface element induced by the pullback of the volume element $dV$
from $M$ to $\partial M$. The resulting equations of motion are

$\displaystyle\mathcal{D}_{t}^{*}S$
$\displaystyle=\mathcal{D}_{\alpha}^{*}Q^{\alpha}+T,$ (30a) $\displaystyle
n_{\alpha}Q^{\alpha}$ $\displaystyle=P,\quad\text{on }\partial M,$ (30b)

which, together with the kinematics equations of motion Eq. 7a, defines the
non-conservative mechanics of Cartan media.

###  IV.4 Dynamics with adapted structure fields

In Sec. III.5 we considered the case when $r>n$. That is, when the dimension
of the symmetry group $G$ is larger than that of the configuration space
$\mathbb{X}$. The additional $n-r$ degrees of freedom in the kinematic
description of the system can be seen as a gauge freedom, which is eliminated
by a constructing an adapted structure field. As a result, $X_{\alpha}$ and
$N$ will each suffer $n-r$ constraints on their components. That is, the
kinematic fields will now take value in vector subsets $X_{\alpha}\in
V^{\alpha}\subseteq\mathfrak{g}$ and $N\in V^{t}\subseteq\mathfrak{g}$
respectively, where $\text{dim}(V_{\alpha})=\text{dim}(N)=n$.

This in turn has implications on the dynamic conjugate variables $S$ and
$Q^{\alpha}$. Recall that kinematic adaptation can be seen as a way to emulate
the system by kinematically constraining another Cartan system in a larger
configuration space, with the same shared symmetry group $G$ that accomodates
for all the degrees of freedom of $G$. This entails that the generalised
momentum balance equations must be consistent with these kinematic
constraints. That is, the generalised momentum $S$ must have the same degrees
of freedom as that of the generalised velocity $N$, and the generalised
stresses must be such that they enforce the kinematic constraint.

Continuing the discussion from Sec. III.5, let us again consider a spatially
adapted structure field. Equation 14b are $r-n$ constraints on $N$. Let
$N_{i},\ i=1,\dots,n$ designate independent degrees of freedom of the
velocity, and $N_{j}=N_{j}(X),\ j=n+1,\dots,r$ be components of the velocity
determined by Eq. 14b, such that
$N=N_{i}\tilde{b}_{i}+N_{j}\tilde{b}_{j}=N_{k}\tilde{b}_{k},\ k=1,\dots,r$,
where $\tilde{b}_{i}$ is a basis for $\mathfrak{g}$ that reflects this
separation. Let $\tilde{B}_{i}$ be the corresponding dual basis for
$\mathfrak{g}^{*}$, and we write $S=S_{k}\tilde{B}_{k}$. Then, as the
independent degrees of freedom in the velocity are $N_{i}$, the components of
the generalised momentum in this basis are
$S_{i}=\frac{\partial\check{\mathcal{L}}}{\partial N_{i}}$ and $S_{j}=0$. From
Eq. 30a we then find that $(\mathcal{D}_{\alpha}^{*}Q^{\alpha}+T)_{j}=0$,
which are $r-n$ constraints on the generalised stress, where the subscript
denotes the components in the dual basis $\tilde{B}_{j}$.

We now return to the example of the filament on the sphere, discussed in Sec.
III.5. As $\Omega_{2}$ is not a dynamical degree of freedom of the filament,
we have that $L_{2}=0$ of the angular momentum (or, in the general setting,
the generalised momentum). From Eq. 30 we then find that
$M_{1}=(L_{1}\Omega_{3}-L_{3}\Omega_{1}-M_{2}^{\prime}-m_{2})/\kappa$. We thus
see that kinematic adaptation introduces a fictitious moment
$\mathbf{M}^{f}=M_{1}\mathbf{e}_{1}$. We can interpret the fictitious moment
as acting on the “Cosserat rod” on the sphere so as to align $\mathbf{e}_{1}$
to be tangent with its center-line, and thus effectively reducing its degrees
of freedom to that of a filament on a sphere.

##  V Geometric integration

We have derived the geometric mechanics of Cartan media, encapsulated in Eq.
III and Eq. 30. The temporal evolution of a system under the influence of
given generalised stresses and body force densities, given initial boundary
conditions, can be found by integration of these equations of motion. Here we
derive integrators to this end.

Given an initial boundary condition $q_{0}:M\to\mathbb{X}$, and a
corresponding structure field $\Phi_{0}:M\to G$ that satisfies
$q_{0}=\Phi_{0}\cdot q_{r}$, we have that the structure generator at the
initial temporal boundary is $\xi_{M,0}=\Phi_{0}^{-1}d_{M}\Phi_{0}$. As the
equations of motions are of second order, we must also specify an initial
velocity $\dot{\Phi}_{0}:M\to TG$ (the dot does not denote a derivative here).
Given a chart $(U,\mathbf{u})\in\mathcal{A}_{M}$, we have that the initial
generalised strain and generalised velocity are
$X_{\alpha,0}=\Phi_{0}^{-1}\partial_{\alpha}\Phi_{0}$ and
$N_{0}=\Phi^{-1}\dot{\Phi}_{0}$. Then, over this chart, the structure
generator can be solved for by integrating

$\displaystyle\dot{X}_{\alpha}$ $\displaystyle=\mathcal{D}_{u}N$ (31a)
$\displaystyle\mathcal{D}_{t}^{*}S$
$\displaystyle=\mathcal{D}_{\alpha}^{*}Q^{\alpha}+T,$ (31b)
$\displaystyle\dot{\Phi}$ $\displaystyle=\Phi N,$ (31c)

with initial conditions $X_{\alpha}(0,\mathbf{u})=X_{\alpha,0}(\mathbf{u})$
and $N(0,\mathbf{u})=N_{0}(\mathbf{u})$, over a patchwork of charts that
collectively cover $M$. The generalised stress must satisfy
$Q^{\alpha}n_{\alpha}=P$ at the material boundary $\partial M$, where $n$ is a
normal covector field on $\partial M$. If the stress is a function of the
strain $Q^{\alpha}=Q^{\alpha}(X_{1},X_{2},\dots,X_{d})$, then these boundary
conditions will in turn induce conditions on $X_{\alpha}$ at the material
boundaries. Note that in many relevant applications, including $M=S^{2}$ (see
the methodological note at the end of Sec. VII.2), it is possible to simulate
systems with a single chart.

The solution of Eq. 31b-31b yields the structure generator
$\xi:W\to\mathfrak{g}$, and Eq. 31c yields the structure field $\Phi:W\to G$,
from which the spatio-temporal configuration $q:W\to\mathbb{X}$ can be found
using Eq. 1. It is only necessary to integrate Eq. 31c concurrently with Eq.
31b-31b if the dynamics couples directly with $\Phi$ (or $q$). That is, if the
system is purely constitutive then Eq. 31b-31b can be solved on its own.
Equation 31c can then be solved separately in order to reconstruct the
structure field.

Naively, the simplest method by which to numerically solve Eq. 31 is using the
_Forward-Euler integrator_ (FEI) (Ascher and Petzold, 1998), which assumes
that the right-hand sides are approximately constant over the interval $\Delta
t$ between time steps. This effectively linearises the equations of motion,
such that, for example, the propagator of Eq. 31c is $\Phi(t+\Delta
t,p)\approx\Phi(t,p)+\Phi(t,p)N(t,p)\Delta t$, evaluated point-wise at $p\in
M$, where the numerical error grows as $O(\Delta t^{2})$. A significant
drawback of the FEI, and similar but more sophisticated methods like Runge-
Kutta schemes (Ascher and Petzold, 1998), is that $\Phi(t+\Delta t,p)\not\in
G$ in general, for any finite $\Delta t$. That is, numerical errors pushes
$\Phi(t+\Delta t,p)$ out of the Lie group. Fundamentally, this is due to how
Lie group elements are parameterised as matrices $G\subset\mathbb{R}^{m\times
m}$, such that $T_{g}G\subset T_{g}\mathbb{R}^{m\times m}$ at any point $g\in
G$. This means that $\Phi(t,p)N(t,p)\Delta t\in
T_{\Phi(t,p)}\mathbb{R}^{m\times m}$, such that the sum
$\Phi(t,p)+\Phi(t,p)N(t,p)\Delta t$ is only an element of $G$ in general if
$\Delta t=0$. To circumvent this problem, we may instead solve the discretised
equation over the time-step. The result is known as a _Lie group integrator_
(LGI) (Iserles  _et al._ , 2005). In the case of Eq. 31c, the solution is
$\Phi(t+\Delta t,p)\approx\Phi(t,p)\exp_{G}(N(t,p)\Delta t)$, where
$\exp_{G}:\mathfrak{g}\to G$ is the expontential map. The propagator ensures
that $\Phi$ remains in $G$ regardless of $\Delta t$. That is, the error in the
numerical scheme, which remains $O(\Delta t^{2})$, take value in $G$, and not
in $\mathbb{R}^{m\times m}$.

Here we will derive LGIs for the mechanics of Cartan media. We begin with Eq.
31c, which outlines the usage of the above example in more explicit detail. We
then move on to Eq. 31b-31b, where we derive semi-analytical forms of the LGI
in terms of the Lie group operations $\exp_{G}$ and Ad. The results we present
here can be developed further to construct propagators with better error
scaling; for example, Runge–Kutta–Munthe–Kaas methods (Munthe-Kaas, 1999; Budd
and Iserles, 1999; Engø and Faltinsen, 2001) can be used, which incorporates
Runge-Kutta methods into LGIs.

###  V.1 Spatio-temporal reconstruction

For a given structure generator $\xi$, the frame field is given by solving the
reconstruction equation $d\Phi=\Phi\xi$, and by specifying some point
$\Phi(t_{0},u_{0})=\Phi_{0}$. Formally, the frame field at $(t,p)\in W$ can be
found by integrating the reconstruction equation over some curve
$\gamma:[0,1]\to W$ that satisfies $\gamma(0)=(t_{0},u_{0})$ and
$\gamma(1)=(t,p)$. That is,
$\Phi(\gamma(\alpha))=\Phi(t_{0},p_{0})\mathscr{A}\exp\left\\{\int_{0}^{1}\xi(\gamma(\alpha))\right\\}$,
which reconstructs the frame along $\gamma$, and where $\mathscr{A}$ signifies
a $\alpha$-ordered integral. The single-valuedness of
$\Phi(\gamma(1))=\Phi(t,p)$ under any choice of $\gamma$ is ensured by the
integrability conditions Eq. 6. Numerically, we may approximate this
integration by discretising the path along the parameter $\alpha$. We
therefore see that there are in principle an infinite amount of ways $\Phi$
can be numerically reconstructed, by the repeated integration of $\xi$ along a
set of curves. Below we will construct such an integration scheme.

Let $\Phi_{0}:M\to SE(3)$ be the structure field of the initial conditions of
the system. We discretise time uniformly as $t=k\Delta t$, where
$k=1,\dots,n_{t}$ and $n_{t}=T/\Delta t$. Given a generalised velocity
$N:W\to\mathfrak{se}(3)$, the numerical integration scheme is

$\Phi^{k+1}(p)=\Phi^{k}(p)\exp\left(N(i\Delta t,p)\Delta t\right),$ (32)

with initial conditions $\Phi^{0}(p)=\Phi_{0}(p)$, such that $\Phi(i\Delta
t,p)\approx\Phi^{k}(p)$, and where $k=1,\dots,n_{t}$, and where
$\exp:\mathfrak{g}\to G$ is the expontential map.

###  V.2 Geometric integrators for dynamical motion

The form of the mechanical equations of motion belie a structure that is
amenable to geometric Lie group integration. To see this, we can insert the
definition of $\mathcal{D}_{u}$ into Eq. 31a, and rewrite the kinematic
equation of motion as $\dot{X}=V-\text{ad}_{N}X$, where $V=N^{\prime}$ .
Similarly, by inserting the definitions of $\mathcal{D}_{t}^{*}$ and
$\mathcal{D}_{u}^{*}$ into Eq. 31b, we can rewrite the dynamical equation of
motion as $\dot{S}=W-\text{ad}_{N}^{*}S$, where $W=\mathcal{D}_{u}^{*}Q+T$.
Therefore, we can see that the equations of motion are in terms of
translations ($V$ and $W$) and adjoint actions ($-\text{ad}_{N}$ and
$-\text{ad}_{N}^{*}$) on $S$ and $X$ respectively. Let
$\tau_{V}:\mathfrak{se}(3)\to\mathfrak{se}(3)$ and
$\tau_{W}^{*}:\mathfrak{se}(3)^{*}\to\mathfrak{se}(3)^{*}$ be operators
defined such that $\tau_{V}X=V+X$ and $\tau_{W}^{*}S=W+S$. Then we have that
$\dot{X}=(\tau_{V}-\text{ad}_{N})X$ and
$\dot{S}=(\tau_{W}^{*}-\text{ad}_{N}^{*})S$. For short times $\Delta t$, we
can approximate as these operators as constant over the interval $[t,t+\Delta
t]$. Formally, we can then construct short-time propagators as

$\displaystyle X(t+\Delta t,p)$ $\displaystyle\approx e^{\Delta
t(\tau_{V}-\text{ad}_{N})}X(t,p),$ (33a) $\displaystyle S(t+\Delta t,p)$
$\displaystyle\approx e^{\Delta t(\tau_{W}^{*}-\text{ad}_{N}^{*})}S(t,p),$
(33b)

which would be evaluated point-wise over $p\in M$. The right-hand sides equate
to (Hale, 2009)

$\displaystyle e^{\Delta t(\tau_{V}-\text{ad}_{N})}X$
$\displaystyle=e^{-\Delta t\ \text{ad}_{N}}X+\left(\int_{0}^{\Delta
t}e^{-(\Delta t-\Delta t^{\prime})\text{ad}_{N}}d\Delta t^{\prime}\right)V,$
(34a) $\displaystyle e^{\Delta t(\tau_{W}^{*}-\text{ad}_{N}^{*})}S$
$\displaystyle=e^{-\Delta t\ \text{ad}_{N}^{*}}S+\left(\int_{0}^{\Delta
t}e^{-(\Delta t-\Delta t^{\prime})\text{ad}_{N}^{*}}d\Delta
t^{\prime}\right)W.$ (34b)

Using the identities (Rossmann, 2006) $\text{Ad}_{e^{Y}}=e^{\text{ad}_{Y}}$
and $\text{Ad}_{e^{Y}}^{*}=e^{-\text{ad}_{Y}^{*}}$ we can succintly write the
Lie group integrators of Eq. 31a and Eq. 31b as

$\displaystyle X(t+\Delta t,p)$ $\displaystyle\approx\text{Ad}_{e^{-\Delta
tN}}X(t,p)+\mathscr{E}(\Delta t,-N)V,$ (35a) $\displaystyle S(t+\Delta t,p)$
$\displaystyle\approx\text{Ad}_{e^{\Delta
tN}}^{*}S(t,p)+\mathscr{E}^{*}(\Delta t,N)W,$ (35b)

where we have defined the maps
$\mathscr{E}:\mathbb{R}_{+}\times\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$
and
$\mathscr{E}:\mathbb{R}_{+}\times\mathfrak{g}\times\mathfrak{g}^{*}\to\mathfrak{g}^{*}$
as $\mathscr{E}(\Delta t,Y)=\int_{0}^{\Delta t}\text{Ad}_{e^{(\Delta t-\Delta
t^{\prime})Y}}d\Delta t^{\prime}$ and $\mathscr{E}^{*}(\Delta
t,A)=\int_{0}^{\Delta t}\text{Ad}_{e^{Y(\Delta t-\Delta
t^{\prime})}}^{*}d\Delta t^{\prime}$, which acts as $\mathscr{E}(\Delta t,Y)C$
and $\mathscr{E}^{*}(\Delta t,Y)Z$ for any $\Delta t\in\mathbb{R}_{+}$,
$Y,C\in\mathfrak{g}$ and $Z\in\mathfrak{g}^{*}$. It should be noted that the
LGI for the dynamics can be mapped to that of the kinematics, meaning that Eq.
35a and Eq. 35b do not need to be implemented separately in numerics. The
mapping can be found by taking the transpose of the latter, leading to
$S(t+\Delta t,u)^{T}\approx\text{Ad}_{e^{N\Delta
t}}S(t,u)^{T}+\mathscr{E}(\Delta t,N)W^{T}$. We thus see that only
$\text{Ad}_{e^{Y}}$ and $\mathscr{E}(\Delta t,Y)$ need to be implemented
numerically.

##  VI Cosserat media

Cosserat media, itself a generalisation of classical continuum mechanics, can
be seen as a specific instance of Cartan media. In full generality, the
configuration at a material point $p\in M$ is specified by a displacement
$\mathbf{r}(p)\in\mathbb{E}^{3}$, the external configuration, and a set of
directors $\mathbf{v}_{i}(p)\in T\mathbb{E}^{3},\ i=1,\dots,s$, the internal
configuration, where $s$ is the number of directors. The directors are fully
independent degrees of freedom, and as such the symmetry group of the internal
configuration is the general linear group $GL(3)$ (or, if $s>3$, products of
$GL(3)$). In many applications (Renda  _et al._ , 2014; Boyer  _et al._ ,
2012; Sack  _et al._ , 2016; Grazioso  _et al._ , 2019; Obrezkov  _et al._ ,
2022; Aydin  _et al._ , 2019; Naughton  _et al._ , 2021; Neff, 2004; Simo and
Fox, 1989; Ericksen, 1974; Krishnaswamy, 1996; Rangamani  _et al._ , 2014;
Stefanou  _et al._ , 2008; Jog, 2004; Dong  _et al._ , 2014; Rubin and
Benveniste, 2004), it suffices to consider inextensible and and orthonormal
directors, such that the collective symmetry group of the configuration space
is $SE(3)$. Here we apply the theoretical framework we developed in previous
sections to Cosserat systems of this kind, in $d=\text{dim}(M)=3$ (Cosserat
solid) $d=2$ (Cosserat surface), and $d=1$ (Cosserat rod) material dimensions.
In all three examples, the discussion will follow the same general sequence:
After contextualising the model in the broader physics literature, we define
the material base space, configuration space and symmetry group, and then
construct a structure field that reconstructs the spatio-temporal
configuration. This allows us to define the generalised strain and velocity
fields of the system, and we use Eq. 4 to show their relation to the
derivatives of the spatio-temporal configuration. We also briefly
contextualise the generalised strain and velocity fields for the system in
related concepts within differential geometry. The kinematics and dynamics of
the system follow as a straightforward application of the machinery developed
in Sec. III and Sec. IV, and we find the physical dimensions of all quantities
involved using dimensional analysis. As the mechanics of the three systems
share many similarities, we will abbreviate the exposition where possible to
avoid repetition.

###  VI.1 Cosserat solids

A Cosserat solid can be considered a continuum solid, where the point-continua
possess micropolar degrees of freedom. Here, we consider the case where the
micropolarity is described by a triad of orthonormal directors. Such systems
have been used to model, for example, liquid crystals (Epstein and de León,
2001; Lee and Eringen, 2003), electromagnetic and ferromagnetic media
(Ivanova, 2022; Paria, 1978; Ivanova, ; Kotera  _et al._ , 2000), biological
materials (Onck, 2002), ceramics (Stefanou  _et al._ , 2008), rocks and
granular media (Besdo, 1985, 1991; Ebrahimian  _et al._ , 2021; Mohan  _et
al._ , 1999), and other applications in material science (Stefanou  _et al._ ,
2017, 2008; Ieşan, 2011; Forest  _et al._ , 2000).

The Cosserat solid is an instance of a Cartan system in three material
dimensions $d=3$, with configuration space
$\mathbb{X}=\mathcal{F}(\mathbb{E}^{3})$ and symmetry group $G=SE(3)$. The
configurations of the point continua of the system comprises a displacement
$\mathbf{r}\in\mathbb{E}^{3}$, and a micropolar degree of freedom represented
as an orthonormal triad of directors $E=(\mathbf{e}_{1}\ \mathbf{e}_{2}\
\mathbf{e}_{3}),\ \mathbf{e}_{i}\in T\mathbb{E}^{3}$. The material base space
is a connected and bounded subset $M\subset\mathbb{E}^{3}$, equipped with
coordinates $(u,v,w):M\to\mathbb{R}^{3}$. Configurations and group elements
will be represented as matrices

$\left(\begin{array}[]{cc}1&\mathbf{0}^{T}\\\
\mathbf{f}&F\end{array}\right)\in\mathcal{F}(\mathbb{E}^{3}),\
\left(\begin{array}[]{cc}1&\mathbf{0}^{T}\\\ \mathbf{a}&A\end{array}\right)\in
SE(3),$ (36)

where $A\in SO(3)$, and in short-hand we write
$(\mathbf{f},F)\in\mathcal{F}(\mathbb{E}^{3})$ and $(\mathbf{a};A)\in SE(3)$.
The group action can then be written as a matrix multiplication
$(\mathbf{a};A)\cdot(\mathbf{f},F)=(\mathbf{a};A)(\mathbf{f},F)=(A\mathbf{f}+\mathbf{a},AF)$.
The spatio-temporal configuration is
$q=(\mathbf{r},E):W\to\mathcal{F}(\mathbb{E}^{3})$. Let
$q_{r}=(\mathbf{0},D)$, where $D$ is an orthonormal triad of fixed basis
vectors, and let $R:W\to SO(3)$ satisfy $RD=E$. Then $\Phi=(\mathbf{r};R):W\to
SE(3)$ is a structure field satisfying $q=\Phi q_{r}$. For the sake of
simplicity, we will let $D=\mathbbm{1}_{3\times 3}$, such that
$q_{r}=\mathbbm{1}_{4\times 4}$, implying that $R=E$, and so effectively
identifying $\Phi$ and $q$. However, we will still distinguish between $E$ and
$R$, the former referring to the micropolar degree of freedom of the Cosserat
solid, and the latter to the rotation between the stationary basis $D$ and the
moving basis of $E$. Such that
$\mathbf{v}^{s}=R\mathbf{v}=v_{i}\mathbf{e}_{i}$, relating tangent vectors in
the spatial and body frames of reference. As we have noted in previous
examples, $q_{r}=(\mathbf{0},D)$ can be seen as the observed position and
orientation of a material point $p\in M$ at time $t\in[0,T]$, of an observer
that is located at $\mathbf{r}(t,p)$ and in a reference frame $E(t,p)$.

We write the generalised velocity and strain fields as
$N=\Phi^{-1}\dot{\Phi}=\\{\mathbf{V};\boldsymbol{\Omega}\\}$ and
$X_{\alpha}=\Phi^{-1}\partial_{\alpha}\Phi=\\{\boldsymbol{\theta}_{\alpha},\boldsymbol{\pi}_{\alpha}\\},\
\alpha=u,v,w$, where
$\mathbf{V},\boldsymbol{\Omega},\boldsymbol{\theta}_{\alpha},\boldsymbol{\pi}_{\alpha}:W\to
T\mathbb{E}^{3}$, and where we have introduced the short-hand for the matrix
representation of $\mathfrak{se}(3)$

$\\{\mathbf{a};\mathbf{b}\\}:=\left(\begin{array}[]{cc}0&\mathbf{0}^{T}\\\
\mathbf{a}&\hat{b}\end{array}\right)\in\mathfrak{se}(3),$ (37)

for any $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$, where the hat map is defined
in Eq. 47. The translational and rotational components of $N$ and $X_{\alpha}$
can be contextualised in terms of more familiar expressions using Eq. 4. We
have that $\dot{q}=(\text{Ad}_{\Phi}N)q$, from which we find that
$\dot{\mathbf{r}}=\mathbf{V}^{s}=V_{i}\mathbf{e}_{i}$ and
$\dot{\mathbf{e}}_{i}=\boldsymbol{\Omega}^{s}\times\mathbf{e}_{i}=(\Omega_{i}\mathbf{e}_{i})\times\mathbf{e}_{i}=\mathbf{e}_{j}\hat{\Omega}_{ji}$.
We see that $\mathbf{V}$ and $\boldsymbol{\Omega}$ is a velocity and angular
velocity on the displacement field $\mathbf{r}$ and the director frame $E$
respectively. Along the material coordinates, we have
$\partial_{\alpha}q=(\text{Ad}_{\Phi}X_{\alpha})q$, from which we find
$\partial_{\alpha}\mathbf{r}=\boldsymbol{\theta}^{s}=\theta_{\alpha,i}\mathbf{e}_{i}$
and
$\partial_{\alpha}\mathbf{e}_{i}=\boldsymbol{\pi}_{\alpha}^{s}\times\mathbf{e}_{i}=\mathbf{e}_{j}\hat{\pi}_{\alpha,ji}$.
We see that $\boldsymbol{\theta}_{\alpha}$ and $\boldsymbol{\pi}_{\alpha}$
encode the strains of $\mathbf{r}$ and $E$ respectively. We should emphasise
that $\mathbf{V}$, $\boldsymbol{\Omega}$, $\boldsymbol{\theta}_{\alpha}$ and
$\boldsymbol{\pi}_{\alpha}$ are all vector fields in the body frame of
reference, and comprise therefore the intrinsic spatio-temporal geometry of
the Cosserat solid.

The strains of the displacement and director fields,
$\boldsymbol{\theta}_{\alpha}$ and $\boldsymbol{\pi}_{\alpha}$, can be related
to corresponding objects in Riemannian geometry. At a given time $t$, the
spatial configuration of the Cosserat solid can be seen as a manifold
$\mathcal{R}_{t}=\mathbf{r}(t,M)\subset\mathbb{E}^{3}$ and vector fields
$\mathbf{e}_{i}:W\to T\mathcal{R}_{t},\ i=1,2,3$. As
$d_{M}\mathbf{r}=\boldsymbol{\theta}_{\alpha}^{s}du^{\alpha}$, we have that
$d_{M}\mathbf{r}\cdot d_{M}\mathbf{r}=g_{\alpha\beta}du^{\alpha}du^{\beta}$,
where
$g_{\alpha\beta}=\boldsymbol{\theta}_{\alpha}^{T}\boldsymbol{\theta}_{\beta}$
is a time-dependent Riemannian metric on $\mathcal{R}_{t}$. Furthermore, the
Christoffel symbols (Misner  _et al._ , 2017) are defined as
$\partial_{\alpha}\mathbf{e}_{j}=\Gamma_{j\alpha}^{i}\mathbf{e}_{i}$, we can
thus identify $\hat{\pi}_{\alpha,ij}=\Gamma_{j\alpha}^{i}$, and $D_{\alpha}$
as the covariant derivative on $\mathbf{r}(t,M)$. If we extend these same
arguments to include time, now using the exterior derivative $d$ rather than
$d_{M}$, we can conclude that $D_{t}$ is a covariant derivative on the spatio-
temporal manifold $\mathbf{r}(W)$ in the curvi-linear coordinates defined by
the orthonormal frame field $E$.

As for all Cartan media, the geometrised kinematics of the Cosserat solid is
given by Eq. 7a. Substituting $N=\\{\mathbf{V};\boldsymbol{\Omega}\\}$ and
$X_{\alpha}=\\{\boldsymbol{\theta}_{\alpha},\boldsymbol{\pi}_{\alpha}\\}$ into
the equations of motion, we find

$\displaystyle D_{t}\boldsymbol{\theta}_{\alpha}$
$\displaystyle=D_{\alpha}\mathbf{V}$ (38a)
$\displaystyle\dot{\boldsymbol{\pi}}_{\alpha}$
$\displaystyle=D_{\alpha}\boldsymbol{\Omega}$ (38b)

for $\alpha=u,v,w$, where $D_{t}=\partial_{t}+\hat{\Omega}$ and
$D_{\alpha}=\partial_{\alpha}+\hat{\pi}_{\alpha}$. The differential operators
$D_{t}$ and $D_{\alpha}$ should be interpreted as covariant derivatives on the
manifold $\mathbf{r}(W)$, which we will show below. The spatial integrability
conditions Eq. 7b yields
$D_{\alpha}\boldsymbol{\theta}_{\beta}=D_{\beta}\boldsymbol{\theta}_{\alpha}$
and
$\partial_{\alpha}\boldsymbol{\pi}_{\beta}=D_{\beta}\boldsymbol{\pi}_{\alpha}$,
for $\alpha,\beta=u,v,w$, and must be satisfied at all times $t\in[0,T]$.
However, as was shown explicitly in Sec. III.3, it suffices that they hold at
$t=0$ as the kinematic equations of motion preserves spatial integrability.

Given a kinetic energy density $\mathcal{K}(N)$ density, we write the
generalised momentum as $S=\frac{\partial\mathcal{K}}{\partial
N}=\\{\mathbf{P};\mathbf{L}\\}^{*}$, where the matrix derivative is taken
using the numerator-layout convention (see Eq. 52), and where we introduced a
short-hand for the matrix representation of $\mathfrak{se}(3)^{*}$ as
$\\{\mathbf{y};\mathbf{z}\\}^{*}=\\{\mathbf{y};\mathbf{z}\\}^{T}\in\mathfrak{se}(3)^{*}$,
for any $\mathbf{y},\mathbf{z}\in\mathbb{R}^{3}$. We write the generalised
stress and body force density as
$Q^{\alpha}=\\{\mathbf{F}^{\alpha};\mathbf{M}^{\alpha}\\}^{*}$ and
$T=\\{\mathbf{f};\mathbf{m}\\}^{*}$, where
$\mathbf{F}^{\alpha},\mathbf{M}^{\alpha},\mathbf{f},\mathbf{m}:W\to
T\mathbb{E}^{3},\ \alpha=u,v,w$, Substituting these expressions into the
generalised momentum balance equations Eq. III.3, we find

$\displaystyle D_{t}\mathbf{P}$
$\displaystyle=D_{\alpha}\mathbf{F}^{\alpha}+\mathbf{f},$ (39a) $\displaystyle
D_{t}\mathbf{L}$
$\displaystyle=D_{\alpha}\mathbf{M}^{\alpha}+\boldsymbol{\theta}_{\alpha}\times\mathbf{F}^{\alpha}+\mathbf{m}.$
(39b)

and $n_{\alpha}\mathbf{F^{\alpha}}=n_{\alpha}\mathbf{M}^{\alpha}=0$ on
$\partial M$. Given conservative dynamics specified by a Lagrangian density
$\mathcal{L}(q,\dot{q},\partial_{\alpha}q)$, with corresponding left-
trivialisation $\check{\mathcal{L}}=\mathcal{K}-\mathcal{U}$, the components
of the generalised stress
$Q^{\alpha}=\\{\mathbf{F}^{\alpha};\mathbf{M}^{\alpha}\\}^{*}$ can be found as
$\mathbf{F}^{\alpha}=-\frac{\partial\check{\mathcal{L}}}{\partial\boldsymbol{\theta}_{\alpha}}=\frac{\partial\mathcal{U}}{\partial\boldsymbol{\theta}_{\alpha}}$
and
$\mathbf{M}^{\alpha}=-\frac{\partial\check{\mathcal{L}}}{\partial\boldsymbol{\pi}_{\alpha}}=\frac{\partial\mathcal{U}}{\partial\boldsymbol{\pi}_{\alpha}}$.
An expression for the components of generalised body force density
$T=\\{\mathbf{f};\mathbf{m}\\}^{*}$ can be found by evaluating Eq. 27, to find

$\displaystyle\mathbf{f}$
$\displaystyle=R^{T}\frac{\partial\mathcal{L}}{\partial\mathbf{r}},$ (40a)
$\displaystyle\mathbf{m}$
$\displaystyle=R^{T}\sum_{i=1}^{3}\frac{\partial\mathcal{L}}{\partial\mathbf{e}_{i}}\times\mathbf{e}_{i}.$
(40b)

Equation 39 and Eq. 39 together fully define the mechanics of Cosserat solids,
given that the map $S=S(N)$ is invertible.

Physical interprations of the components of the generalised strain, velocity,
momentum, stress and body force fields can be found using dimensional
analysis. Let $L$, $T$ and $M$ refer to the dimensions of material length,
time and mass respectively. For all Cartan media, the dimensions of all
quantities can be inferred from Eq. 4, given the dimensions of the spatio-
temporal configuration $q=(\mathbf{r},E)$ and its derivatives, as well as the
dimensions of the kinetic energy density $\mathcal{K}(N)$. We thus let
$[\mathbf{r}]=L$, $[E]=1$, $[\partial_{\alpha}]=L^{-1}$ and
$[\partial_{t}]=T^{-1}$. Now, as $\dot{\mathbf{r}}=V_{i}\mathbf{e}_{i}$, for
example, we find that $[\mathbf{V}]=LT^{-1}$, as is expected of a velocity.
Continuing in the same vein, we find that $[\boldsymbol{\theta}_{\alpha}]=1$,
$[\boldsymbol{\Omega}]=\mathrm{T}^{-1}$ and
$[\boldsymbol{\pi}_{\alpha}]=\mathrm{L}^{-1}$. Now, let the kinetic energy
density have units of energy per unit material volume
$[\mathcal{K}(N)]=ML^{-1}T^{-2}$. From
$\mathbf{P}=\frac{\partial\mathcal{K}}{\partial\mathbf{V}}$ and
$\mathbf{L}=\frac{\partial\mathcal{K}}{\partial\boldsymbol{\Omega}}$ we find
that $[\mathbf{P}]=MLT^{-1}$ and $[\vec{L}]=ML^{-1}T^{-1}$. From Eq. 39 we
find that $[\mathbf{F}^{\alpha}]=ML^{-1}T^{-2}$,
$[\mathbf{M}^{\alpha}]=MT^{-2}$, $[\mathbf{f}]=ML^{-2}T^{-2}$ and
$[\mathbf{m}]=ML^{-1}T^{-2}$. We thus see that $\mathbf{P}$ and $\mathbf{L}$
have units of momentum and angular momentum per unit material volume
respectively, and $\mathbf{F}^{\alpha}$ and $\mathbf{M}^{\alpha}$ force and
moment per unit material area respectively. Finally, if we assume a form
$\mathcal{K}(N)=\frac{1}{2}\rho_{0}^{V}|\mathbf{V}|^{2}+\frac{1}{2}\boldsymbol{\Omega}^{T}\mathbb{I}\boldsymbol{\Omega}$,
where $\rho_{0}^{V}$ is the mass volume density of the body and
$\mathbb{I}\in\mathbb{R}^{3\times 3}$ the moment of inertia of the director
frame, then
$\mathbf{P}=\frac{\partial\mathcal{K}}{\partial\mathbf{V}}=\rho_{0}^{V}\mathbf{V}$
and $\mathbf{L}=\mathbb{I}\boldsymbol{\Omega}$. We thus interpret
$\mathbf{P}$, $\mathbf{F}^{\alpha}$ and $\mathbf{f}$ respectively as the
linear momentum, stress and body force density on the displacement field
$\mathbf{r}$. Similarly, $\mathbf{L}$, $\mathbf{M}^{\alpha}$ and $\mathbf{m}$
are respectively the angular momentum, moment-stress and moment density on the
director frame $E$.

We now relate the mechanics of Cosserat solids to that of non-micropolar
continuum mechanics. As $\dot{\mathbf{v}}^{s}=RD_{t}\mathbf{v}$ and
$\partial_{\alpha}\mathbf{v}^{s}=RD_{\alpha}\mathbf{v}$ for any vector field
$\mathbf{v}:W\to T\mathbb{E}^{3}$, we find the momentum balance equations in
the spatial frame of reference
$\dot{\mathbf{P}}^{s}=\partial_{\alpha}\mathbf{F}^{s,\alpha}+\mathbf{f}^{s}$
and
$\dot{\mathbf{L}}^{s}=\partial_{\alpha}\mathbf{M}^{s,\alpha}+\partial_{\alpha}\mathbf{r}\times\mathbf{F}^{s,\alpha}+\mathbf{m}^{s}$,
which are consistent with the literature (Rubin, 2000; Altenbach and Eremeyev,
2013a). The former of these can be identified as the Cauchy momentum equation
_(Basar and Weichert, 2013) of the displacement field $\mathbf{r}$, where
$\Sigma_{\alpha i}=F_{i}^{s,\alpha}$ is the _first Piola-Kirchhoff stress
tensor _._ The latter equation is the angular momentum balance of the director
field $E$, which should be seen as a micropolar analogue of the Cauchy
momentum equation. We can thus see Eq. 39 as a generalised and curvilinear
Cauchy momentum equation, expressed in terms of the moving frame $E$.

__

###  VI.2 Cosserat surfaces

Figure 3: The director field $\mathbf{p}:W\to S^{2}$ (blue arrows) and the
midsurface $\mathbf{r}:W\to\mathbb{E}^{2}$ (brown surface) of the Cosserat
surface approximates a thin shell by constraining the material fibres along
the director to be fixed. The upper and lower boundaries of the shell
(transparent red surfaces) are given by
$\mathbf{r}\pm(L_{0}^{1}/2)\mathbf{p}_{1}$ respectively.

The theory of Cosserat surface was primarily developed for applications to
thin elastic shells (Sansour and Bednarczyk, 1995; Altenbach  _et al._ , 2010;
Altenbach and Eremeyev, 2013b; Naghdi, 1972) and plates (Steinberg and Kvasov,
2022; Altenbach and Eremeyev, 2010; Naghdi, 1972). The material base space has
dimension $d=2$, and the external degree of freedom is a mid-surface
$\mathbf{r}:W\to\mathbb{E}^{3}$, such that $\mathbf{r}(t,M)$ is a two-
dimensional surface in $\mathbb{E}^{3}$, at each time $t$. As opposed to
Cosserat solids, the micropolar degree of freedom of Cosserat surfaces is
typically a single director $\mathbf{p}^{s}:W\to T\mathbb{E}^{3}$, which is
often (Ericksen, 1974; Jog, 2004; Dong  _et al._ , 2014; Rubin and Benveniste,
2004; Green  _et al._ , 1971) inextensible, satisfying
$\mathbf{p}^{s}\cdot\mathbf{p}^{s}=1$. In the context of shell theory, the
director field represents rigid fibres from
$\mathbf{r}(t,p)-(L_{w}/2)\mathbf{p}^{s}(t,p)$ to
$\mathbf{r}(t,p)+(L_{w}/2)\mathbf{p}^{s}(t,p)$ at each point $u\in M$, where
$L_{w}$ is the width of the shell. See Fig. 3 for an illustration. The
Cosserat surface can therefore be be considered the result of a dimensional
reduction of a three-dimensional continuum body that is thin in one material
dimension. What remains of the coarse-grained dimension, the director field,
accomodates for transverse shear deformations (Sansour and Bednarczyk, 1995)
of the shell boundary relative to the mid-surface. The omitted $SO(2)$
rotational degree of freedom around the axis of $\mathbf{p}^{s}$, so-called
drill rotations (Sansour and Bednarczyk, 1995), are in many applications
(Krishnaswamy, 1996; Ericksen, 1974; Jog, 2004; Rangamani  _et al._ , 2014)
excluded as they are unmotivated physically (ant, 2005; Mohammadi Saem  _et
al._ , 2021; Neff, 2004). In addition to shells, Cosserat surfaces have also
been used to model cell membranes (Rangamani  _et al._ , 2014; Krishnaswamy,
1996; Ericksen, 1979), liquid crystal and material interphases (Ericksen,
1974; Dong  _et al._ , 2014; Rubin and Benveniste, 2004).

The Cosserat surface is a Cartan system with a material base space in $d=2$
dimensions, and configuration space $\mathbb{X}=\mathbb{E}^{3}\times S^{2}$,
on which $G=SE(3)$ is the symmetry group. If $M$ is homeomorphic to a bounded
and closed subset in $\mathbb{R}^{2}$, such as $M=[0,1]\times[0,1]$, then the
Cosserat surface called _open_. If $M$ is homeomorphic to $S^{2}$, then the
surface is _closed_ , as in Fig. 2. There are some subtleties involved in the
study of closed surfaces; see the end of Sec. VII.2 for a discussion. If the
surface is open then $M$ admits global coordinates, but in general we have
local charts $(u,v):U\to\mathbb{R}^{2}$, where $U\subseteq M$. We write the
spatio-temporal configuration as

$q=(\mathbf{r},\mathbf{p}^{s}):=\left(\begin{array}[]{cc}1&0\\\
\mathbf{r}&\mathbf{p}^{s}\end{array}\right)\in\mathbb{R}^{4\times 2}.$ (41)

Let $\Phi:W\to SE(3)$ be $\Phi=(\mathbf{r};R)$, and let
$q_{r}=(\mathbf{0},\mathbf{p})$, where $R:W\to SO(3)$ satisfies
$R\mathbf{p}=\mathbf{p}^{s}$ and $\mathbf{p}\in S^{2}$ is a constant unit
vector, then $\Phi$ is a structure field satisfying $\Phi q_{r}=q$. As before,
we will identify $R$ with an orthonormal triad in $T\mathbb{E}^{3}$, writing
$R=(\mathbf{e}_{1}\ \mathbf{e}_{2}\ \mathbf{e}_{3})=E$. As the notation
indicates, $\mathbf{p}$ should be considered the orientation of any point-
continua on the Cosserat surface, relative to the moving basis $E$.
Correspondingly, $\mathbf{p}^{s}(t,p)$ is the actual orientation of the point-
continua at $p\in M$ at time $t$ in the spatial frame of reference. For the
sake of simplicity, we will therefore set $\mathbf{p}=(0\ 0\ 1)^{T}$, such
that $\mathbf{p}^{s}=\mathbf{e}_{3}$.

The generalised velocity and strain are of identical form as those of the
Cosserat solid, but where the strain fields
$X_{\alpha}=\\{\boldsymbol{\theta}_{\alpha},\boldsymbol{\pi}_{\alpha}\\}$ are
now defined with respect to the two material directions $\alpha=u,v$. From Eq.
4 we find that $\dot{\mathbf{r}}=V_{i}\mathbf{e}_{i}$,
$\dot{\mathbf{p}}^{s}=\boldsymbol{\Omega}^{s}\times\mathbf{p}^{s}=\mathbf{e}_{i}\hat{\Omega}_{ij}p_{j}=\mathbf{e}_{i}\hat{\Omega}_{i3}$,
$\partial_{\alpha}\mathbf{r}=\theta_{\alpha,i}\mathbf{e}_{i}$ and
$\partial_{\alpha}\mathbf{p}^{s}=\boldsymbol{\pi}_{\alpha}^{s}\times\mathbf{p}^{s}=\mathbf{e}_{i}\hat{\pi}_{\alpha,i3}$.

Briefly, we will now relate the strains of the displacement field,
$\boldsymbol{\theta}_{u}$ and $\boldsymbol{\theta}_{v}$, to some standard
concepts from the differential geoemetry of surfaces (Chern  _et al._ , 1999;
Clelland, 2017). Let $\mathcal{R}_{t}=\mathbf{r}(t,M)$, which is a two-
dimensional surface in $\mathbb{E}^{3}$, for each time $t$. The _first
fundamental form_ of $\mathcal{R}_{t}$ is $\mathrm{I}=d_{M}\mathbf{r}\cdot
d_{M}\mathbf{r}:=g_{\alpha\beta}du^{\alpha}du^{\beta}$, where
$g_{\alpha\beta}=\boldsymbol{\theta}_{\alpha}^{T}\boldsymbol{\theta}_{\beta}$
is a time-dependent Riemannian metric on $\mathcal{R}_{t}$ induced from the
Euclidean metric on $\mathbb{E}^{3}$. The _second fundamental form_ is
$\mathrm{I\\!I}=-d_{M}\mathbf{n}\cdot
d_{M}\mathbf{r}:=b_{\alpha\beta}du^{\alpha}du^{\beta}$, where
$b_{\alpha\beta}=\partial_{\alpha}\mathbf{n}\cdot\partial_{\beta}\mathbf{r}$
and $\mathbf{n}:W\to S^{2}$ is the _Gauss map_ , given by
$\mathbf{n}=\boldsymbol{\theta}_{u}^{s}\times\boldsymbol{\theta}_{v}^{s}/\left|\boldsymbol{\theta}_{u}^{s}\times\boldsymbol{\theta}_{v}^{s}\right|$,
which is a vector field that is normal to $\mathcal{R}_{t}$. The _Gaussian
curvature_ of $\mathcal{R}_{t}$ is given by $K=\text{det}(b)/\text{det}(g)$,
and the _mean curvature_ as $H=\text{tr}(g^{-1}b)$.

The kinematic equations of motion of the Cosserat surface in terms of the
translational and rotational components of $X_{\alpha}$ and $N$ are indentical
to those of the Cosserat solid, Eq. 38, but where $\alpha=u,v$. Furthermore
the spatial integrability conditions reduce to
$D_{u}\boldsymbol{\theta}_{v}=D_{v}\boldsymbol{\theta}_{u}$ and
$\partial_{u}\boldsymbol{\pi}_{v}=D_{v}\boldsymbol{\pi}_{u}$. Similarly, the
generalised momentum balance equation of the Cosserat surface is given by Eq.
39, with $\alpha=u,v$, and the explicit expressions for the body force and
moment densities are given by Eq. 40. Given $[\mathbf{r}]=L$,
$[\mathbf{p}^{s}]=1$ and assuming that the kinetic energy density has units of
energy per unit material area $[\mathcal{K}(N)]=MT^{-2}$, we find that
$\mathbf{P}$ and $\mathbf{L}$ have units of momentum and angular momentum per
unit material area respectively, and $\mathbf{F}^{\alpha}$ and
$\mathbf{M}^{\alpha}$ units of force and moment per unit material length
respectively. The body force and moment densities $\mathbf{f}$ and
$\mathbf{m}$ have units of force and moment per unit material area
respectively.

The above fully determines the mechanics of Cosserat surfaces. However, note
that in the process of replacing the configurational variables $q$ and
$\dot{q}$ with the Lie algebraic strain and velocity fields $X_{\alpha}$ and
$N$, we have implicitly introduced a superfluous degree of freedom. These are
the aformentioned drill rotations, which is an $SO(2)$ gauge symmetry of
$(\mathbf{e}_{1}\ \mathbf{e}_{2})$ around the axis of the director
$\mathbf{p}^{s}=\mathbf{e}_{3}$. Fundamentally, this is reflected in the fact
that $\text{dim}(G)-\text{dim}(\mathbb{X})=r-n=1$. For certain systems, as
those of the subsequent section, the superfluous degrees of freedom lead to
inconveniences for the formulation of the mechanics. For example, it may be
difficult, or impractical, to construct Lagrangians that do not couple with
the superfluous degrees of freedom. However, for the Cosserat surface, this is
generally not an issue. To illustrate this, we consider a simple example. Let
the Lagrangian be separable as
$\mathcal{L}(q,\dot{q},\partial_{\alpha}q)=\mathcal{L}_{\mathbf{r}}(\mathbf{r},\dot{\mathbf{r}},\partial_{\alpha}\mathbf{r})+\mathcal{L}_{\mathbf{p}}(\mathbf{p}^{s},\dot{\mathbf{p}}^{s},\partial_{\alpha}\mathbf{p}^{s})$,
and let
$\mathcal{L}_{\mathbf{p}}=\frac{1}{2}r|\dot{\mathbf{p}}^{s}|^{2}+\frac{1}{2}k_{u}|\partial_{u}\mathbf{p}^{s}|+\frac{1}{2}k_{v}|\partial_{v}\mathbf{p}^{s}|$,
where $r,k_{u},k_{v}\in\mathbb{R}$ are constants. We have shown previously
that
$|\dot{\mathbf{p}}^{s}|^{2}=\boldsymbol{\Omega}^{T}P_{0}\boldsymbol{\Omega}$
and
$|\partial_{\alpha}\mathbf{p}^{s}|^{2}=\boldsymbol{\pi}_{\alpha}^{T}P_{0}\boldsymbol{\pi}_{\alpha}$,
where $P_{0}=\text{diag}\left\\{1,1,0\right\\}$, such that the reduced
Lagrangian is
$\ell_{\mathbf{p}}(\boldsymbol{\Omega},\boldsymbol{\pi}_{\alpha})=\frac{1}{2}r\boldsymbol{\Omega}^{T}P_{0}\boldsymbol{\Omega}+\frac{1}{2}k_{u}\boldsymbol{\pi}_{u}^{T}P_{0}\boldsymbol{\pi}_{u}+\frac{1}{2}k_{v}\boldsymbol{\pi}_{v}^{T}P_{0}\boldsymbol{\pi}_{v}$.
We then find that $\mathbf{L}=(r\Omega_{1}\ r\Omega_{2}\ 0)^{T}$,
$\mathbf{M}_{\alpha}=(k_{\alpha}\pi_{\alpha,1}\ k_{\alpha}\pi_{\alpha,2}\
0)^{T}$ and $\mathbf{m}=\mathbf{0}$. Substituting these into Eq. 39b we find
that the third components of both the left- and right-hand sides vanish; that
is, $(D_{t}\mathbf{L})_{3}=(D_{\alpha}\mathbf{M}^{\alpha})_{3}=0$. The angular
rotation around $\mathbf{p}$ may therefore be set to any constant
$\Omega_{3}(t,p)=\bar{\Omega}_{3}\in\mathbb{R}$ with impunity, and
$\bar{\Omega}_{3}=0$ is therefore a natural choice. This is consistent with
the fact that drill rotations do not enter into the dynamics. In general, it
is not possible for Lagrangians $\mathcal{L}(q,\dot{q},\partial_{\alpha}q)$
to, upon left-trivialisation, yield dynamics that couple with drill rotations.
See (Munthe-Kaas and Verdier, 2016) for further discussions on the ambiguities
of Lie group actions on homogeneous spaces when $r>n$, and how to resolve
them.

###  VI.3 Cosserat rods

The Cosserat rod can be conceived as a continuum limit of connected rigid
bodies; that is, the system consists of a center-line curve and an orthonormal
frame of directors, where the latter represents the rigid body cross-sections
of rod. As opposed to the pure center-line mechanics of filaments (see Sec.
VII.1), Cosserat rods can _shear_ ; that is, the normal of the cross-section
need not be tangent to the center-line. See Fig. 2 for an illustration. Such
models are prominently used in soft robotics (Renda  _et al._ , 2018; Grazioso
_et al._ , 2019; Caasenbrood  _et al._ , 2022; Boyer  _et al._ , 2006, 2012;
Renda  _et al._ , 2014; Naughton  _et al._ , 2021; Verl  _et al._ , 2015; Pai,
2002), the modelling of muscles and ligaments (Sack  _et al._ , 2016; Zhang
_et al._ , 2019; Kier and Smith, 1985), biological growth models (Moulton  _et
al._ , 2020; Goriely, 2017) and active filaments (Moulton  _et al._ , 2016;
Oliveri  _et al._ , 2021; Kaczmarski  _et al._ , 2022). In a forthcoming
publication we will be treating the geometrisation of the Cosserat rod in
further detail 333Lukas Kikuchi, Ronojoy Adhikari. In preparation..

The exposition of Cosserat solids in Sec. VI.1 largely subsumes that of
Cosserat rods, where the primary difference is that the latter is defined in
one material dimenion $d=1$. A __ rod is _open_ if the material base space is
an interval $M=[0,L_{0}]$, and _closed_ if the interval is periodic; that is,
if $M\cong\mathbb{T}^{1}$, where $\mathbb{T}^{1}$ is the $1$-torus. The frame
field, and the kinematic and dynamical equations of motion of the Cosserat rod
is identical to that of the Cosserat solid, but now in a single $\alpha=u$
dimension. Assuming that the kinetic energy density has units of energy per
unit material length $[\mathcal{K}(N)]=ML^{-1}T^{-2}$, we can find by
dimensional analysis that $\mathbf{P}$ and $\mathbf{L}$ have units of momentum
and angular momentum per unit material length respectively, and $\mathbf{F}$
and $\mathbf{M}$ units of force and moment respectively. The body force and
moment densities $\mathbf{f}$ and $\mathbf{m}$ have units of force and moment
per unit material length respectively.

##  VII Cartan media with adapted structure fields

Kinematic adaption, which we introduced in Sec. III.5, is a method by which we
reduce superfluous degrees of freedom that arise due to the geometrisation
process, when the dimension of the symmetry group $G$ is larger than that of
the configuration space $\mathbb{X}$. Furthermore, where the unadapted
kinematics does not capture the intrisic geometry of the system, an
appropiately chosen adapted structure field leads to a structure generator
that is intrinsic. As previously mentioned, the kinematic adaptation discussed
draws heavily from the theory of moving frames (Clelland, 2017; Cartan, 1951;
Darboux, 1887; Frenet, 1852; Fels and Olver, 1998, 1999; Olver, ; Olver,
2005), which can be consulted for more detailed treatments.

###  VII.1 Filaments

Filaments, also known as _Kirchoff rods_ (Kirchhoff, 1859; Dill, 1992), can be
viewed as thin, slender tubes in the limit of a vanishing cross-sectional
radius. Such systems appear in many applications (Eloy and Lauga, 2012;
Tornberg and Shelley, 2004; Goldstein and Langer, 1995; Soda, 1973; Nordgren,
1974; Hasimoto, 1972; Laskar and Adhikari, 2017, 2015; Goldstein  _et al._ ,
2000; Kaczmarski  _et al._ , 2022; Betchov, 1965; Liu  _et al._ , 2020;
Goldstein and Goriely, 2006; Lenz  _et al._ , 2009). As opposed to the
Cosserat rod, the filament does not posses microstructure; and can therefore
bend and extend, but not shear. However, in some applications (Goldstein and
Langer, 1995; Hasimoto, 1972; Goldstein  _et al._ , 2000; Parker, 1984) a
‘fictitious’ frame is introduced that is adapted to the curvature of the
filament. This is akin to a Cosserat rod whose microstructure is aligned so as
to be tangent to its center-line. The result of this is an intrinsic
description of the kinematics, where the filament is parameterised in terms of
the rotations of the frame along its material length.

We consider a system with $M=[0,L_{0}]$, $\mathbb{X}=\mathbb{E}^{3}$ and
$G=SE(3)$. We write its the spatio-temporal configuration in ‘homogeneous
coordinates’ as $q=(1\ \mathbf{r}^{T})^{T}$, where
$\mathbf{r}:W\to\mathbb{E}^{3}$ is a space curve
$\mathbf{r}(t,M)\subset\mathbb{E}^{3}$ for every time $t\in[0,T].$ We define
the structure field $\Phi:W\to SE(3)$ as $\Phi=\left(\mathbf{r};R\right)$,