# Intrinsic Langevin dynamics of rigid inclusions on curved surfaces

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

###### Abstract

The stochastic dynamics of a rigid inclusion constrained to move on a curved
surface has many applications in biological and soft matter physics, ranging
from the diffusion of passive or active membrane proteins to the motion of
phoretic particles on liquid-liquid interfaces. Here we construct intrinsic
Langevin equations for an oriented rigid inclusion on a curved surface using
Cartan’s method of moving frames. We first derive the Hamiltonian equations of
motion for the translational and rotational momenta in the body frame.
Surprisingly, surface curvature couples the linear and angular momenta of the
inclusion. We then add to the Hamiltonian equations linear friction, white
noise and arbitrary configuration-dependent forces and torques to obtain
intrinsic Langevin equations of motion in phase space. We provide the
integrability conditions, made non-trivial by surface curvature, for the
forces and torques to admit a potential, thus distinguishing between passive
and active stochastic motion. We derive the corresponding Fokker-Planck
equation in geometric form and obtain fluctuation-dissipation relations that
ensure Gibbsian equilibrium. We extract the overdamped equations of motion by
adiabatically eliminating the momenta from the Fokker-Planck equation, showing
how a peculiar cancellation leads to the naively expected Smoluchowski limit.
The overdamped equations can be used for accurate and efficient intrinsic
Brownian dynamics simulations of passive, driven and active diffusion
processes on curved surfaces. Our work generalises to the collective dynamics
of many inclusions on curved surfaces.

##  I Introduction

The stochastic dynamics of a rigid inclusion on a curved surface provides an
useful theoretical model for quantitatively describing several important
experimental phenomena in biological and soft matter physics, for instance the
diffusion of passive [1, 2] or active [3] proteins on the cell membrane or the
motion of phoretic particles confined to liquid-liquid interfaces [4, 5, 6].
In these situations, the inertia of the particle is typically negligible and a
theoretical description in terms of overdamped equations of motion is
appropriate. Accordingly, the existing literature on the stochastic motion of
particles on curved surfaces has focussed on the overdamped limit [7, 8, 9,
10]. There are, however, at least three separate mathematically subtle issues
in describing the generic stochastic motion of inclusions on curved surfaces.

The first issue in formulating dynamics is that local coordinates have no
intrinsic meaning on manifolds. Therefore, the physical content of the
equations of motion must be expressed in a manner that is invariant under
change of coordinates. This poses a problem that is peculiar to stochastic
processes, since there are at least two prescriptions for how the change of
variables should be implemented in the stochastic equations of motion. This is
the well-known Ito-Stratonovich dilemma [11, 12].

For the specific case of the motion of a rigid inclusion on a curved surface,
the natural description of the motion is in terms of a body frame aligned with
the principal axes of inertia and the position of the inclusion on the
surface. Therefore, there is a natural basis, provided by the body frame, in
which to express tensor fields on the manifolds. This is related to Cartan’s
method of moving frames [13, 14, 15] with the crucial difference being that
the frame represents a physical degree of freedom and does not have any
redundant ”gauge” degrees of freedom usually appearing in the classical
applications of the method. This leads to a formulation of the equations of
motion using moving frames, giving, as we shall show below, a clear conceptual
picture of the kinematics of an inclusion on a curved surface.

The second issue is that the overdamped limit is a singular limit, in the
sense that the highest time derivative in the problem is multiplied by a small
parameter which is eventually taken to zero [16]. For stochastic equations of
motion, this process of taking the singular limit usually introduces
additional terms (often called noise-induced spurious drift [17, 18]),
unanticipated by a naive realisation of the limit.

Therefore, it is essential to start from the inertial equations of motion,
which remain unambiguous if linear coordinate-dependent friction is present,
and systematically derive the overdamped limit by an adiabatic elimination
procedure [19, 20, 21]. This process can be expected to have additional
subtleties on curved spaces as geometric quantities associated with the
surface, appearing in the inertial equations of motion can remain in the
overdamped limit [18]. To formulate the intrinsic Langevin equations and to
derive the overdamped limit carefully is one of the main purposes of our work.

A third issue, of great contemporary interest, is identifying forces and
torques that do not derive from a potential. These are termed active forces
and torques in the contemporary literature, but have also been referred to
previously as circulatory forces [22]. Therefore, we need to identify when
force-torque pairs, given as a function of the body frame coordinates, derive
(or do not) from a potential.

Obtaining the integrability conditions in our setting is non-trivial because
the body frame velocities are non-holonomic, i.e. they are not time-
derivatives of coordinate functions. The integrability conditions, therefore,
have a non-trivial character which we elucidate using ideas from exterior
calculus [23].

The remainder of the paper is organised as follows. In Section II, we first
construct the kinematic equations of motion, in which the non-holonomic body
frame components of the velocity are used in preference to holonomic
velocities. It is natural to use non-holonomic velocities in the moving frame
method in which a preference is given to the simplicity of the kinematic
description over the choice of a coordinate basis. We then construct the
dynamical equations of motion in the Lagrangian picture for diffusive motion
in a potential landscape on the surface. These are transformed into the
Hamiltonian picture with non-canonical momenta and the equations of motion are
expressed in terms of Poisson brackets. To these Poisson bracket equations we
add linear friction, white noise and forces and torques of a general
character. We identify conditions for when the latter derive from a potential.
In Section III, having established the pathwise equations of motion, we focus
on the equation of motion for the probability densities and derive the Fokker-
Planck equation in phase space, expressing it in geometric form. From this we
obtain the fluctuation-dissipation relation when the forces and torques derive
from a potential. In Section IV we perform a systematic adiabatic elimination
procedure for the momenta in the case of large friction, and obtain the
Smoluchowski equation for the probability density in configuration space.
Identifying the corresponding stochastic equations of motion suggests a
reliable intrinsic method for performing Brownian dynamics simulations.
Finally, in Section V we present a detailed application of our general
formalism to the case of a rigid inclusion on a sphere, and we draw
conclusions in Section VI.

##  II Equations of motion

###  II.1 Method of moving frames

$S$
---
$\mathbf{e}_{1}$
---
$\mathbf{e}_{1}$
---
$\mathbf{e}_{1}$
---
$\mathbf{e}_{1}$
---
$\mathbf{e}_{1}$
---
$\mathbf{e}_{1}$
---
$\mathbf{e}_{1}$
---
$\mathbf{e}_{2}$
---
$\mathbf{e}_{2}$
---
$\mathbf{e}_{2}$
---
$\mathbf{e}_{2}$
---
$\mathbf{e}_{2}$
---
$\mathbf{e}_{2}$
---
$\mathbf{f}_{1}$
---
$\mathbf{f}_{1}$
---
$\mathbf{f}_{1}$
---
$\mathbf{f}_{1}$
---
$\mathbf{e}_{2}$
---
$\mathbf{f}_{2}$
---
$\mathbf{f}_{2}$
---
$\mathbf{f}_{2}$
---
$\mathbf{f}_{2}$
---
$x(t)$
---
$\psi$
---
$\psi$
---
$\psi$
---
$\psi$
---
Figure 1: Illustration of moving frames and coordinate systems on the surface
$S$.

In this section we establish some notation and recall useful concepts from the
classical differential geometry of surfaces via the method of moving frames
[24, 25].

Let $S$ be a smooth embedded oriented surface in $\mathbb{R}^{3}$. The
orthonormal frame bundle $FS$ of the surface $S$ is the set of tuples of the
form $\left(x,\mathbf{f}_{1},\mathbf{f}_{2}\right)$, where $x\in S$ and
$\left\\{\mathbf{f}_{1},\mathbf{f}_{2}\right\\}$ is a right-handed orthonormal
basis of the tangent space $T_{x}S$. It is a smooth manifold of dimension
three, on which we can choose local coordinates as follows. We first pick a
local coordinate system $x^{\alpha}$ on $S$, with the index $\alpha$ running
from $1$ to $2$ as well as a local orthonormal frame field $\mathbf{e}_{a}$ on
$S$, which can be done by e.g. performing the Gram-Schmidt process on the
coordinate basis vectors $\frac{\partial}{\partial x^{\alpha}}$. Now if
$q=\left(x,\mathbf{f}_{a}\right)\in FS$ is a point in the domain of the local
coordinate chart and frame field, we can parametrize $x$ by $x^{\alpha}$ and a
unique angle $\psi\in[0,2\pi)$ such that

$\begin{pmatrix}\mathbf{f}_{1}&\mathbf{f}_{2}\end{pmatrix}=\begin{pmatrix}\mathbf{e}_{1}&\mathbf{e}_{2}\end{pmatrix}\begin{pmatrix}\cos\psi&-\sin\psi\\\
\sin\psi&\cos\psi\end{pmatrix}.$ (1)

Thus the triplet $\left(x^{1},x^{2},\psi\right)$ forms a valid local
coordinate system on $FS$. It is also useful to introduce the dual basis
$\theta^{a}$ of one-forms to $\mathbf{e}_{a}$, defined by the relations

$\theta^{a}\left(\mathbf{e}_{b}\right)=\delta^{a}_{b}.$ (2)

The one-forms $\theta^{a}$ are also referred to as an orthonormal coframe
field since they are orthonormal with respect to the inverse metric. In a
local coordinate system we may expand both the frame and coframe fields as

$\displaystyle\mathbf{e}_{a}$ $\displaystyle=$ $\displaystyle
e_{a}^{\alpha}\frac{\partial}{\partial x^{\alpha}},$ (3)
$\displaystyle\theta^{a}$ $\displaystyle=$
$\displaystyle\theta^{a}_{\alpha}dx^{\alpha}.$ (4)

The coframe fields induce a Riemannian area element $\mathrm{vol}_{S}$ on the
surface as:

$\mathrm{vol}_{S}=\theta^{1}\wedge\theta^{2}=\sqrt{\det g}dx^{1}\wedge
dx^{2}.$ (5)

The Riemannian manifold $S$ can be equipped with a distinguished connection,
the Levi-Civita connection, defined as the unique metric-compatible torsion-
free connection. With respect to the local frame field $\mathbf{e}_{a}$, it
can be described by a connection one-form $\Gamma=\Gamma_{\alpha}dx^{\alpha}$
as follows:

$\displaystyle D\mathbf{e}_{1}$ $\displaystyle=$
$\displaystyle\mathbf{e}_{2}\Gamma,$ (6) $\displaystyle D\mathbf{e}_{2}$
$\displaystyle=$ $\displaystyle-\mathbf{e}_{1}\Gamma,$ (7)

where $D\mathbf{u}$ denotes the covariant derivative of the vector field
$\mathbf{u}$. It is important to note that both the coordinate $\psi$ and the
connection form $\Gamma$ depends on the arbitrary choice of moving frame
$\mathbf{e}_{a}$. In the physics literature, this choice is called a gauge,
and a change of the moving frame is a gauge transformation. Under this
transformation, the connection form $\Gamma$ changes to
$\Gamma^{\prime}=\Gamma+d\varphi$, where $\varphi(x)$ is an arbitrary smooth
real-valued function representing the angle between the new and the old gauge.
Notice, however, that the two-form

$d\Gamma^{\prime}=d\Gamma=-\kappa(x)\mathrm{vol}_{S}$ (8)

is unchanged, since the exterior derivative of the exact one-form $d\varphi$
vanishes. As the space of two-forms over $S$ is one-dimensional, $d\Gamma$ has
to be proportional to the Riemannian area form, and the constant of
proportionality —representing the gauge-invariant curvature of the connection
— is an intrinsic geometric quantity that is in fact equal to minus the
Gaussian curvature $\kappa(x)$ of the surface at the point $x$ [24].

The connection can be used to define the parallel transport of frames on $FS$.
Let $q=\left(x,\mathbf{f}_{1},\mathbf{f}_{2}\right)\in FS$ be an orthonormal
frame. The infinitesimal generator of parallel transport along the frame
vector $\mathbf{f}_{a}$ is a vector field $V_{a}$ on $FS$, whose coordinate
expression is given by

$V_{a}=f_{a}^{\alpha}\left(\frac{\partial}{\partial
x^{\alpha}}-\Gamma_{\alpha}\frac{\partial}{\partial\psi}\right).$ (9)

Equation (9) follows from the fact that infinitesimal parallel transport along
$\mathbf{f}_{a}$ shifts the basepoint $x$ of the frame by the vector
$\mathbf{f}_{a}=f_{a}^{\alpha}\frac{\partial}{\partial x^{\alpha}}$ while
rotating the frame by an amount
$-\Gamma\left(\mathbf{f}_{a}\right)=-f_{a}^{\alpha}\Gamma_{\alpha}$ so that
the frame itself remains covariantly constant, i.e. parallel. The vector field
generating infinitesimal rotations of the frame is simply

$V_{R}=\frac{\partial}{\partial\psi}.$ (10)

In the mathematical literature, the vector fields $V_{a}$ are called
horizontal vector fields, while $V_{R}$ is a vertical vector field [14, 23].
In the sequel, we are going to use the notation $V_{i}$ for the collection of
horizontal and vertical vector fields, with the convention that the index $i$
runs from $1$ to $3$ and $V_{3}\equiv V_{R}$.

Since the vectors $V_{i}$ form a global basis of vector fields for each
tangent space of $FS$, we can also construct the corresponding global basis of
one-forms $\eta^{i}$ defined by the relations

$\eta^{i}\left(V_{j}\right)=\delta^{i}_{j}.$ (11)

In particular, let

$\begin{pmatrix}\Theta^{1}\\\
\Theta^{2}\end{pmatrix}\triangleq\begin{pmatrix}\cos\psi&\sin\psi\\\
-\sin\psi&\cos\psi\end{pmatrix}\begin{pmatrix}\theta^{1}\\\
\theta^{2}\end{pmatrix}$ (12)

be the dual coframe basis to $\mathbf{f}_{a}$, we have:

$\displaystyle\eta^{a}$ $\displaystyle=$
$\displaystyle\Theta^{a}=\Theta^{a}_{\alpha}dx^{\alpha}$ (13)
$\displaystyle\eta^{3}$ $\displaystyle=$
$\displaystyle\Gamma+d\psi=\Gamma_{\alpha}dx^{\alpha}+d\psi$ (14)

###  II.2 Kinematics

$\mathbf{v}$
---
$\mathbf{f}_{3}=\mathbf{n}$
---
$\mathbf{f}_{1}$
---
$\mathbf{f}_{2}$
---
$\omega_{3}=\omega$
---
$S$
---
$\omega_{1}$
---
$\omega_{2}$
---
$x\in S$
---
$T_{x}S$
---
Figure 2: Representation of an inclusion on a curved surface $S$.

We model the inclusion as a rigid body with both its positional and
orientational degrees of freedom constrained. We assume that the centre of
mass of the inclusion is constrained to lie on a smooth embedded oriented
surface $S\subset\mathbb{R}^{3}$, while one of its principal axes of inertia
$\mathbf{f}_{3}$ is constrained to be normal to the surface. Therefore the
configuration the inclusion at any instance of time can be described by the
tuple $\left(x,\mathbf{f}_{1},\mathbf{f}_{2}\right)$, where $x\in S$ is the
position of its centre of mass and
$\left\\{\mathbf{f}_{1},\mathbf{f}_{2}\right\\}$ are the two other principal
axes of inertia, forming a right-handed orthonormal basis of the tangent plane
$T_{x}S$. As a result, the configuration space of the inclusion is precisely
the orthonormal frame bundle $FS$ of the surface.

The fundamental kinematical equations of motion are:

$\displaystyle dx$ $\displaystyle=$ $\displaystyle\mathbf{f}_{I}v^{I}dt,$ (15)
$\displaystyle d\mathbf{f}_{I}$ $\displaystyle=$
$\displaystyle\mathbf{f}_{J}\omega^{J}_{I}dt,\quad I=1,2,3,$ (16)

where $v^{I}(t)$ are the components of the translational velocity of the body
in the body frame, while $\omega^{I}_{J}(t)$ are the components of the angular
velocity of the body, represented by a $3\times 3$ antisymmetric matrix, also
with respect to the body frame. (Uppercase indices $I,J$ run from $1$ to $3$
and repeated indices are summed over.) The matrix $\omega_{I}^{J}(t)$ is
related to the usual angular velocity vector
$\left(\omega_{1}\quad\omega_{2}\quad\omega_{3}\right)^{T}$ via

$\omega^{I}_{J}=\begin{pmatrix}0&-\omega_{3}&\omega_{2}\\\
\omega_{3}&0&-\omega_{1}\\\ -\omega_{2}&\omega_{1}&0\end{pmatrix}.$ (17)

Since the curve $x(t)$ lies on $S$, the inclusion cannot have any normal
velocity component to the surface. Furthermore, the rate of change of the
normal must be tangential to the surface, giving, by the Weingarten equation,

$d\mathbf{f}_{3}=-s^{a}_{b}\mathbf{f}_{a}v^{b}dt,$ (18)

where $s^{a}_{b}(x)$ are the components of the second fundamental form of the
surface at $x$ with respect to the basis $\mathbf{f}_{a}$ of the local tangent
space. (Lowercase indices $a,b$ run from $1$ to $2$, they are raised and
lowered with the standard metric $\delta_{ab}$ and repeated indices are summed
over.) Combining (18) and (16), the kinematical constraints may be
encapsulated in the following algebraic form:

$\displaystyle v^{3}$ $\displaystyle=$ $\displaystyle 0,$ (19)
$\displaystyle\omega^{a}_{3}$ $\displaystyle=$ $\displaystyle-s^{a}_{b}v^{b}.$
(20)

It follows that the kinematics is fully specified by the pair of translational
velocity components $v^{1}(t),v^{2}(t)$ and the angular velocity component
$\omega(t)\triangleq\omega_{3}(t)$ normal to the surface.

The concepts introduced in the previous section allow us to connect the
angular velocity of the inclusion to the covariant derivative of the its
associated body frame as follows. By taking the dot product of (16) for $I=1$
with $\mathbf{f}_{2}$ and using (1) together with (6) we find

$\displaystyle\omega(t)$ $\displaystyle=$
$\displaystyle\mathbf{f}_{2}\cdot\frac{d\mathbf{f}_{1}}{dt}$ (21)
$\displaystyle=$
$\displaystyle\frac{d\psi}{dt}+\mathbf{e}_{2}\cdot\frac{d\mathbf{e}_{1}}{dt}$
$\displaystyle=$ $\displaystyle\frac{d\psi}{dt}+\mathbf{e}_{2}\cdot
D_{\mathbf{v}}\mathbf{e}_{1}$ $\displaystyle=$
$\displaystyle\frac{d\psi}{dt}+\Gamma(\mathbf{v})$ $\displaystyle=$
$\displaystyle\frac{d\psi}{dt}+v^{a}\Gamma_{\alpha}f_{a}^{\alpha}.$

Equation (21) provides another useful interpretation of $\omega(t)$: it is the
scalar-valued orthogonal projection to $S$ of the rate of change of the vector
field $\mathbf{f}_{1}(t)$ along the curve $x(t)$. One of the fundamental
results in the differential geometry of embedded surfaces in $\mathbb{R}^{3}$
states that this projection is independent of the embedding of the surface $S$
in ambient space and is given by the covariant derivative of the vector field
$\mathbf{f}_{1}(t)$ along $x(t)$ with respect to the Levi-Civita connection of
the Riemannian surface $S$ [26]. With the aid of (21), we can project out the
rate of change of surface normal to obtain the intrinsic form of the
kinematical equations (15)—(16):

$\displaystyle dx$ $\displaystyle=$ $\displaystyle\mathbf{f}_{a}v^{a}dt,$ (22)
$\displaystyle D\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\mathbf{f}_{b}\varepsilon^{b}_{a}\omega dt,$ (23)

where

$\varepsilon^{a}_{b}=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}.$ (24)

Therefore using (23) we can interpret $\omega$ as the covariant rate of
rotation of the frame $\left\\{\mathbf{f}_{1}(t),\mathbf{f}_{2}(t)\right\\}$.
Finally, observe that if $q(t):\left[t_{0},t_{1}\right]\to FS$ is the
trajectory of the inclusion, then we have:

$\dot{q}(t)=v^{i}(t)V_{i}\in T_{q(t)}FS,$ (25)

where we have introduced $v^{i}$ as the components of the generalized velocity
of the inclusion, with the convention $v^{R}\equiv\omega$. Thus we can
interpret $v^{i}(t)$ as the components of the tangent vector $\dot{q}(t)$
expanded in the basis $V_{i}$ of horizontal and vertical vector fields.

###  II.3 Lagrangian dynamics

The equations of motion for the inclusion are derived from a variational
principle. The standard Lagrangian is the sum of translational and rotational
kinetic energies minus a potential energy

$L=\frac{1}{2}\sum_{a=1}^{2}m\left(v^{a}\right)^{2}+\frac{1}{2}\sum_{l=1}^{3}\mathbb{I}_{l}\left(\omega_{l}\right)^{2}-U(q),$
(26)

where $m$ is the mass of the inclusion and $\mathbb{I}_{l}$ are its principal
moments of inertia. (Without loss of generality we may assume that the
$\mathbf{f}_{a}$-s are aligned with the principal axes of inertia). For
simplicity, we are going to assume that
$\mathbb{I}_{1}=\mathbb{I}_{2}\triangleq\mathbb{I}$ and use the notation
$J\triangleq\mathbb{I}_{3}$ for the moment of inertia along the surface
normal. Substituting (20) in (26) yields, using the summation convention:

$L=\frac{1}{2}\left(m\delta_{ab}+\mathbb{I}\delta_{cd}s^{c}_{a}s^{d}_{b}\right)v^{a}v^{b}+\frac{1}{2}J\omega^{2}-U(q).$
(27)

The intutitive interpretation of (27) is that the inclusion obtains extra
translational mass on top of $m$ because it has to rotate its orientation in
order to stay normal to the surface. However, this contribution is generally
small: if $l=\sqrt{J/m}$ is roughly the size of the inclusion and $k$ is the
typical curvature of the surface (having units of inverse length), then we
have

$\frac{\mathbb{I}\delta_{cd}s^{c}_{a}s^{d}_{b}}{m}\sim\frac{ml^{2}k^{2}}{m}=(lk)^{2}\ll
1,$ (28)

since for our model to be valid the inclusion size has to be much smaller than
the scale over which the manifold is curved. Therefore we may neglect the
second term in (27) to obtain

$L=\frac{1}{2}m\delta_{ab}v^{a}v^{b}+\frac{1}{2}J\omega^{2}-U(q).$ (29)

A remarkable property of the Lagrangian (29) is that it is purely intrinsic,
i.e. it does not involve any extrinsic quantity coming from the embedding of
the surface $S$ in ambient space. It is also useful to have a local coordinate
expression of (29):

$L=\frac{1}{2}mg_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}+\frac{1}{2}J\left(\dot{\psi}+\Gamma_{\alpha}\dot{x}^{\alpha}\right)^{2}-U\left(x^{\alpha},\psi\right).$
(30)

From equation (30) we could write down the Euler-Lagrange equations directly
in local coordinates. However, they would reveal little insight about the
intrinsic geometric structure of the equations motion, therefore we turn to
the Hamiltonian picture instead where the equations of motion can be derived
and analysed elegantly using Poisson brackets.

###  II.4 Hamiltonian dynamics

In this subsection we derive the governing equations of motion in Hamiltonian
form in the body frame using Poisson brackets. Let
$q^{\mu}=\left(x^{\alpha},\psi\right)$ be our set of generalized coordinates
with corresponding generalized momenta $p_{\mu}=\left(p_{\alpha},s\right)$,
where the index $\mu$ runs from $1$ to $3$. We have:

$\displaystyle p_{\alpha}$ $\displaystyle=\frac{\partial
L}{\partial\dot{x}^{\alpha}}=$ $\displaystyle
mg_{\alpha\beta}\dot{x}^{\beta}+J\left(\dot{\psi}+\Gamma_{\beta}\dot{x}^{\beta}\right)\Gamma_{\alpha},$
(31) $\displaystyle s$ $\displaystyle=\frac{\partial L}{\partial\dot{\psi}}=$
$\displaystyle
J\left(\dot{\psi}+\Gamma_{\beta}\dot{x}^{\beta}\right)=J\omega.$ (32)

Performing a Legendre transformation, we find the Hamiltonian to be:

$\displaystyle
H=\frac{g^{\alpha\beta}\left(p_{\alpha}-s\Gamma_{\alpha}\right)\left(p_{\beta}-s\Gamma_{\beta}\right)}{2m}+\frac{s^{2}}{2J}+U(q).$
(33)

It is instructive to introduce the body frame linear momenta

$\pi_{a}=\frac{\partial L}{\partial v^{a}}=m\delta_{ab}v^{b}=mv_{a}.$ (34)

These are related to the canonical momenta as:

$\pi_{a}=f_{a}^{\alpha}\left(p_{\alpha}-s\Gamma_{\alpha}\right).$ (35)

From now on, we will refer to the triplet $\left(\pi_{1},\pi_{2},s\right)$
with $\left(\pi_{i}\right)$, where the index $i$ running from $1$ to $3$ with
the convention that $\pi_{3}=s$.

In terms of the body frame momenta, the Hamiltonian (33) assumes the standard
quadratic form

$H=\frac{\delta^{ab}\pi_{a}\pi_{b}}{2m}+\frac{s^{2}}{2J}+U(q)=\frac{1}{2}G^{ij}\pi_{i}\pi_{j}+U(q),$
(36)

where $G^{ij}$ denotes the inverse of the inertia metric

$G_{ij}=\begin{pmatrix}m&0&0\\\ 0&m&0\\\ 0&0&J\end{pmatrix}.$ (37)

The transformation (35) between the body frame momenta and the canonical
momenta can be succinctly captured using the components
$V_{i}=V_{i}^{\mu}(q)\frac{\partial}{\partial q^{\mu}}$ of the horizontal (9)
and vertical vector fields (10) as:

$\pi_{i}=V_{i}^{\mu}(q)p_{\mu}.$ (38)

The phase space carries a canonical Poisson structure that reads in canonical
coordinates $\left(q^{\mu},p_{\nu}\right)$ as:

$\left\\{q^{\mu},q^{\nu}\right\\}=\left\\{p_{\mu},p_{\nu}\right\\}=0,\quad\left\\{q^{\mu},p_{\nu}\right\\}=\delta^{\mu}_{\nu}.$
(39)

After some calculations detailed in the appendix, the Poisson structure in
terms of the body frame momenta are given by:

$\displaystyle\left\\{\pi_{1},\pi_{2}\right\\}$ $\displaystyle=$
$\displaystyle-\kappa(x)s,$ (40) $\displaystyle\left\\{\pi_{1},s\right\\}$
$\displaystyle=$ $\displaystyle\pi_{2},$ (41)
$\displaystyle\left\\{\pi_{2},s\right\\}$ $\displaystyle=$
$\displaystyle-\pi_{1}.$ (42)

Since these relations are linear in the momenta, we collect them as:

$\left\\{\pi_{i},\pi_{j}\right\\}=\Omega^{k}_{ij}(q)\pi_{k}.$ (43)

The Poisson brackets of the generalized coordinates and the body frame momenta
are given by the components of the horizontal and vertical vector fields
(9)—(10):

$\left\\{q^{\mu},\pi_{i}\right\\}=V^{\mu}_{i}$ (44)

The intuition behind the Poisson structure (40)—(42) and (44) can be
understood as follows. The body frame momenta $\pi_{a}$ are the infinitesimal
generators of translations along the body frame axes, while $s$ is the
generator of infinitesimal rotations of the body. In flat space, these
generators satisfy the same commutation relations (with a minus sign) as the
Lie algebra $\mathfrak{se}(2)$ of the Lie group $\mathrm{SE}(2)$ of
translations and rotations in the Euclidean plane [27]. However, on a curved
surface, infinitesimal translations no longer commute: parallel transport of a
frame along an infinitesimal loop results in the net rotation of the frame,
where the rotation angle is proportional to the local Gaussian curvature of
the surface. The Poisson brackets (44) follow from the fact that infinitesimal
parallel transport and rotation of the frame are generated by the vector
fields $V_{i}$ and the rate of change of generalized coordinates $q^{j}$ along
such infinitesimal motions are given by the components $V_{i}^{j}$ of the
corresponding vector fields.

For free motion $U(q)=0$, Hamilton’s equations of motion in terms of Poisson
brackets are:

$\displaystyle\dot{\pi}_{i}$ $\displaystyle=$
$\displaystyle\left\\{\pi_{i},H\right\\},$ (45) $\displaystyle\dot{q}^{\mu}$
$\displaystyle=$ $\displaystyle\left\\{q^{\mu},H\right\\}.$ (46)

These read explicitly as:

$\displaystyle\dot{\pi}_{a}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{J}-\frac{\kappa}{m}\right)\varepsilon^{b}_{a}s\pi_{b},$
(47) $\displaystyle\dot{s}$ $\displaystyle=$ $\displaystyle 0,$ (48)
$\displaystyle\dot{x}^{\alpha}$ $\displaystyle=$ $\displaystyle
v^{a}f_{a}^{\alpha},$ (49) $\displaystyle\dot{\psi}$ $\displaystyle=$
$\displaystyle\omega-v^{a}f_{a}^{\alpha}\Gamma_{\alpha}.$ (50)

Equations (49) and (50) are the same as the kinematic equations of motion
(22)—(23), with $v^{a}$ and $\omega$ expressed in terms of the momenta using
(34) and (32). Equation (48) expresses angular momentum conservation, which
follows from the fact that the Lagrangian (29) does not depend explicitly on
the angle $\psi$, hence the corresponding momentum is conserved.

Equation (47) describes the rate of change of the body frame momenta. Note
that the magnitude of the momentum $\pi_{a}\pi_{b}\delta^{ab}$ is conserved,
only its direction is rotated: this is because energy is conserved (as the
Hamiltonian does not depend explicitly on time), and since $s$ is conserved,
so is the magnitude of the momentum. The first term on the right-hand side of
(45) is a co-rotational derivative, originating from the fact that the
$\pi_{a}$-s are the components of the momentum in a rotating frame, while the
second term is a contribution from surface curvature, which acts as a
gyroscopic force that rotates the linear momentum of the inclusion depending
on its angular velocity and the local Gaussian curvature. We also mention that
similar equations have been studied in the context of rotating test bodies in
general relativity [28].

It is also instructive to write the free dynamical equations (47)—(48)
extrinsically in terms of the translational velocity vector
$\mathbf{v}=v^{1}\mathbf{f}_{1}+v^{2}\mathbf{f}_{2}$ and the angular velocity
vector $\bm{\omega}=\omega\mathbf{f}_{3}$. Putting together (47)—(48) and
(22)—(23) we find:

$\displaystyle m\frac{D\mathbf{v}}{dt}$ $\displaystyle=$ $\displaystyle\kappa
J\left(\bm{\omega}\times\mathbf{v}\right),$ (51) $\displaystyle
J\frac{d\omega}{dt}$ $\displaystyle=$ $\displaystyle 0.$ (52)

Equation (52) represents angular momentum conservation, while equation (51)
implies that on a curved surface, rotation and translation are coupled,
resulting in a gyroscopic force in the equations of motion that rotates the
direction of the velocity based on the local Gaussian curvature and the
angular velocity.

###  II.5 Non-conservative dynamics and integrability conditions

We will now generalize Hamilton’s equations (45)—(46) to include external
forces and damping, still in a determinstic setting. We are going to add
linear memoryless friction and forcing to the right-hand side of (45) while
keeping the kinematics (46) unchanged:

$\displaystyle\dot{\pi}_{i}$ $\displaystyle=$
$\displaystyle\left\\{\pi_{i},H\right\\}-\gamma_{ij}(q)v^{j}+F_{i}(q),$ (53)
$\displaystyle\dot{q}^{\mu}$ $\displaystyle=$
$\displaystyle\left\\{q^{\mu},H\right\\}.$ (54)

In equation (53), $\gamma_{ij}(q)$ denotes the positive definite friction
tensor (which may be anisotropic and could also depend on position) and
$F_{i}(q)$ are conjugate forces, assumed to be independent of momenta (using
the convention that $F_{3}(q)\equiv M(q)$ is an external torque). The
velocities $v^{i}$ in (53) should be expressed in terms of the momenta
$\pi_{i}$ via the Legendre transformation. The reason for writing the friction
in terms of the velocities is that the damping originates from a quadratic
Rayleigh dissipation in the velocities added to the right-hand side of the
Euler-Lagrange equations.

The conservative forces coming from the potential term $U(q)$ in the
Hamiltonian (36) are given by

$F_{i}^{(c)}(q)=\\{\pi_{i},U(q)\\}=\\{\pi_{i},q^{\mu}\\}\frac{\partial
U}{\partial q^{\mu}}=-V_{i}^{\mu}\frac{\partial{U}}{\partial q^{\mu}}.$ (55)

The work done by the forces $F_{i}^{(c)}$ is zero along any closed loop in the
configuration space.

Since we can include conservative forces in the first term on the right-hand
side of (53), it is natural to reserve the last term $F_{i}(q)$ for non-
conservative, e.g. active or ”self-propulsion” forces (these are called
”circulatory forces” in [22]). In order to make sure that a given force field
does not derive from a potential, we have to check that the work done by the
conjugate forces is zero along each closed loop in configuration space.
Restricting one’s attention to infinitesimal loops results in the celebrated
potential or curl conditions: a necessary condition for a force field in
Euclidean space to be conservative is that its curl has to vanish.

We can generalize the curl conditions to our setting as follows. Suppose that
a given force field $F_{i}(q)$ derives from a potential $U(q)$ as in (55).
Using the Jacobi identity for the Poisson bracket we get:

$\displaystyle\\{\pi_{i},F_{j}(q)\\}$ $\displaystyle=$
$\displaystyle\\{\pi_{i},\\{\pi_{j},U(q)\\}\\}$ (56) $\displaystyle=$
$\displaystyle\\{\pi_{j},\\{\pi_{i},U(q)\\}\\}+\\{\\{\pi_{i},\pi_{j}\\},U(q)\\}$
$\displaystyle=$
$\displaystyle\\{\pi_{j},F_{i}(q)\\}+\left\\{\Omega^{k}_{ij}(q)\pi_{k},U(q)\right\\}$

Expanding the Poisson brackets and rearranging, we obtain the generalized curl
conditions:

$V_{j}^{\mu}\frac{\partial F_{i}}{\partial q^{\mu}}-V_{i}^{\mu}\frac{\partial
F_{j}}{\partial q^{\mu}}=-\Omega^{k}_{ij}F_{k}.$ (57)

We also emphasize that the curl conditions are independent of local
coordinates, since they can be written as

$V_{i}\left[F_{j}\right]-V_{j}\left[F_{i}\right]=\Omega_{ij}^{k}F_{k},$ (58)

where $V_{i}\left[F_{j}\right]$ denotes the application of the vector field
$V_{i}$ to the function $F_{j}$. Substituting in the values of
$\Omega_{ij}^{k}$ from (40)—(42) we find:

$\displaystyle V_{1}\left[F_{2}\right]-V_{2}\left[F_{1}\right]$
$\displaystyle=$ $\displaystyle-\kappa(x)M$ (59) $\displaystyle
V_{2}\left[M\right]-V_{R}\left[F_{2}\right]$ $\displaystyle=$ $\displaystyle-
F_{1}$ (60) $\displaystyle V_{R}\left[F_{1}\right]-V_{1}\left[M\right]$
$\displaystyle=$ $\displaystyle-F_{2}$ (61)

On the left hand side of equation (58) we find a kind of antisymmetric
derivative of the force field $F_{i}(q)$, akin to the standard curl of a
vector field in Euclidean space. However, contrary to the usual curl
conditions, for a conservative force field this expression is in general non-
zero, which can be understood intuitively as follows. When traversing a closed
loop in configuration space consisting of infinitesimal translations along the
body frame vectors, the work done by the translational forces are given by the
expression on the left-hand side of (58). However, as we have seen before, on
a curved surface there is a net rotation of the frame at the end of the
process, and we need to do extra work against the external torque to rotate
the body back to its original position, captured by the term on the right-hand
side of (58).

###  II.6 Intrinsic Langevin dynamics

In this subsection we are going to transform the Hamiltonian equations of
motion (53)—(54) by adding thermal noise to the right-hand side, turning the
deterministic ODEs into stochastic differential equations (SDEs). The
resulting system of SDEs

$\displaystyle d\pi_{i}$ $\displaystyle=$
$\displaystyle\left(\left\\{\pi_{i},H\right\\}-\gamma_{ij}(q)v^{j}+F_{i}(q)\right)dt$
(62) $\displaystyle+\sum_{n=1}^{3}\sigma^{n}_{i}(q)\circ dW_{n},$
$\displaystyle dq^{i}$ $\displaystyle=$
$\displaystyle\left\\{q^{i},H\right\\}dt,$ (63)

is a generalization of the celebrated Ornstein-Uhlenbeck process [29] for a
non-Euclidean phase space. In equation (62) $\sigma_{i}^{n}(q)$ governs the
amplitude of the noise, with the noise index $n$ running from $1$ to $3$. The
$dW_{n}(t)$ are standard Wiener noise increments that satisfy

$\left\langle dW_{n}(t)\right\rangle=0,\quad\left\langle
dW_{n}(t)dW_{n^{\prime}}(t^{\prime})\right\rangle=\delta_{nn^{\prime}}\delta\left(t-t^{\prime}\right).$
(64)

Observe that there are no ambiguities in the injection of noise into (45)
because the body frame momenta are defined irrespective of any coordinate
system. For convenience, we summarize the indices appearing in the paper,
their range and role in the table below.

Table 1: Summary of indices appearing in the paper, their range and purpose Index | Range | Purpose
---|---|---
$\alpha$ | $1,2$ | Local coordinates on the surface
$\mu$ | $1,2,3$ | Local coordinates on the frame bundle
$a$ | $1,2$ | Moving frames, body frame velocities, etc.
$i$ |  $1,2,3$111$i=3$ always refers to a rotational quantity or object (angular velocity, angular momentum, torque etc.) To emphasize this, we use the letter $R$ interchangeably with the index $i=3$. | Generalized velocities, momenta, forces
$I$ | $1,2,3$ | Extrinsic body frame vectors
$n$ | $1,2,3$ | Wiener noise index

Even though we allow $\sigma^{n}_{i}(q)$ and $\gamma_{ij}(q)$ to depend on
positions $q^{\mu}$, the noise in (62)—(63) is still additive — just as for a
classical Ornstein-Uhlenbeck process — therefore we are free to interpret the
random terms in either the Ito or Stratonovich sense. In this paper we opt for
the Stratonovich interpretation, because it allows us to treat the equations
in a covariant manner, without reference to any particular coordinate system.
The geometric form of the system (62)—(63) reads:

$dy=\left(X_{H}+F+K\right)dt+\sum_{n=1}^{3}N^{n}\circ dW_{n},$ (65)

where $y=(q,\pi)$ is the point in phase space describing the state of the
inclusion, $X_{H}$ is the Hamiltonian vector field (also called the
Liouvillian), defined by its action on phase space functions as

$\displaystyle X_{H}[f]=\\{f,H\\}$ $\displaystyle=$ $\displaystyle
V_{i}^{\mu}\left(\frac{\partial f}{\partial q^{\mu}}\frac{\partial
H}{\partial\pi_{i}}-\frac{\partial f}{\partial\pi_{i}}\frac{\partial
H}{\partial q^{\mu}}\right)$ (66)
$\displaystyle+\Omega^{k}_{ij}\pi_{k}\frac{\partial
f}{\partial\pi_{i}}\frac{\partial H}{\partial\pi_{j}},$

We call

$K=-\gamma_{ij}v^{j}(\pi)\frac{\partial}{\partial\pi_{i}}=-\lambda^{j}_{i}\pi_{j}\frac{\partial}{\partial\pi_{i}}$
(67)

the dissipative vector field (with the matrix of inverse momentum relaxation
times defined via $\lambda^{i}_{j}=\gamma_{kj}G^{ki}$),

$N^{n}=\sigma(q)^{n}_{i}\frac{\partial}{\partial\pi_{i}}$ (68)

are the noise vector fields for $n=1,\ldots,3$ and

$F=F_{i}(q)\frac{\partial}{\partial\pi_{i}}$ (69)

the generalized force vector field. The geometric formulation of the
stochastic equations of motion will allow us to deduce important properties
about the stochastic process in an elegant, coordinate-free fashion.

##  III Fokker-Planck equation

In this section we are going to derive the Fokker-Planck equation (FPE)
corresponding to the SDE (65) in coordinate-free form on phase space, and
obtain the fluctuation-dissipation relation for our system from it.

The phase space of the inclusion carries a natural measure, the symplectic
measure [23], which, in canonical coordinates, reads as:

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle dp_{1}\wedge dp_{2}\wedge
dp_{3}\wedge dq^{1}\wedge dq^{2}\wedge dq^{3}$ $\displaystyle=$ $\displaystyle
dp_{1}\wedge dp_{2}\wedge ds\wedge dx^{1}\wedge dx^{2}\wedge d\psi$

Our first aim is to find the expression of $\rho$ in terms of the body frame
momenta $\pi_{i}$. To that end, recall that the transformation between
$\pi_{i}$ and $p_{\mu}$ is linear and invertible, therefore we can write

$p_{\mu}=\eta_{\mu}^{i}(q)\pi_{i}$ (71)

for some matrix of coefficients $\eta_{\mu}^{i}(q)$, where, in fact,

$\eta^{j}=\eta^{j}_{\mu}(q)dq^{\mu}$ (72)

are the dual one-forms on $FS$ to the basis afforded by the horizontal and
vertical vector fields $V_{i}$, introduced in (13) and (14). We thus have:

$\eta^{j}_{\mu}=\begin{pmatrix}\Theta^{1}_{1}&\Theta^{1}_{2}&0\\\
\Theta^{2}_{1}&\Theta^{2}_{2}&0\\\ \Gamma_{1}&\Gamma_{2}&1\end{pmatrix}$ (73)

Substituting (71) and (73) into (III), while using the fact that the
$\Theta^{a}$ are orthonormal, we find:

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\left(\det\eta^{j}_{\mu}\right)d\pi_{1}\wedge d\pi_{2}\wedge
d\pi_{3}\wedge dx^{1}\wedge dx^{2}\wedge d\psi$ (74) $\displaystyle=$
$\displaystyle\left(\det g\right)^{\frac{1}{2}}d\pi_{1}\wedge d\pi_{2}\wedge
d\pi_{3}\wedge dx^{1}\wedge dx^{2}\wedge d\psi$ $\displaystyle=$
$\displaystyle d\pi_{1}\wedge d\pi_{2}\wedge
d\pi_{3}\wedge\mathrm{vol}_{S}\wedge d\psi$

The infinitesimal generator (also called the backward Kolmogorov operator) of
the stochastic process (65) is given by the following second-order linear
differential operator:

$L_{B}=C+\frac{1}{2}\sum_{n=1}^{3}N^{n}N^{n},$ (75)

where $C=X_{H}+F+K$ is the drift vector field of the process [30].

The operator which governs the forward time evolution of the probability
distribution of the system is similarly a second-order linear operator, but
acting on probability measures rather than functions. Nevertheless, any
sufficiently well-behaved time-dependent probability measure $\zeta$ can be
represented by a function $P=P(t,y)$ as $\zeta=P(t,y)\chi$ on phase space and
of time, called the probability density function, once a reference measure
$\chi$ is chosen on phase space. The Fokker-Planck equation or forward
Kolmogorov equation

$\frac{\partial P}{\partial t}=L_{F}P$ (76)

governs the time evolution of $P$, where the Fokker-Planck operator $L_{F}$ is
the adjoint of $L_{B}$ with respect to the following inner product

$\langle f,g\rangle_{\chi}=\int_{T^{*}FS}fg\chi,$ (77)

defined for square-integrable functions on phase space, denoted by
$L^{2}(\chi)$ in what follows. It is important to stress that the Fokker-
Planck operator $L_{F}$ depends on the choice of $\chi$ through the inner
product (77). In most applications, the state space of the stochastic process
is a trivial manifold (usually $\mathbb{R}^{n}$), on which there are global
coordinates $x^{1},\ldots,x^{n}$ and $\chi=dx^{1}\wedge\dots\wedge dx^{n}$ is
implicitly chosen to obtain the standard form of the FPE. For a more in-depth
discussion of the issue, see e.g. [31, 32].

For our case, a natural candidate for the reference $\chi$ is the symplectic
measure $\rho$. The physical implication of this choice is that $P$ will
describe the phase space density of our stochastic system, while the forward
operator will be

$L_{F}=\lambda_{i}^{i}-\left(X_{H}+F+K\right)+\frac{1}{2}\sum_{n=1}^{3}N^{n}N^{n},$
(78)

with the details of the calculation given in the appendix. For convenience, we
let $\Lambda=\mathrm{Tr}\left[\lambda^{i}_{j}\right]=\lambda^{i}_{i}$. Using
the fact that for any function $f$ on phase space we have
$X_{H}[f]=\\{f,H\\}$, the explicit form of the Fokker-Planck equation (76)
reads

$\frac{\partial P}{\partial
t}+\\{P,H\\}=\frac{\partial}{\partial\pi_{i}}\left[\left(\lambda^{j}_{i}(q)\pi_{j}-F_{i}(q)\right)P\right]+\frac{1}{2}\sum_{n=1}^{3}\sigma_{i}^{n}(q)\sigma_{j}^{n}(q)\frac{\partial^{2}P}{\partial\pi_{i}\partial\pi_{j}}.$
(79)

Equation (79) is the generalization of the Klein-Kramers equation to our
Hamiltonian system on a curved space with orientational degrees of freedom. On
the left-hand side, we have the total derivative of the distribution function
$P$ along the flow on phase space generated by the Hamiltonian $H$. In the
deterministic setting, the right-hand side would be zero by Liouville’s
theorem, therefore the Fokker-Planck equation (79) can also be interpreted as
the generalization of Liouville’s equation to dissipative and noisy
Hamiltonian dynamics, an observation which has already made by Chandrasekhar
[33].

In thermal equilibrium and the absence of external forces, we require the
Gibbs measure $e^{-\beta H}\rho$ to be a steady state solution of the Fokker-
Planck equation, implying a relationship between the dissipative coefficients
and the noise amplitudes known as the fluctuation-dissipation relation (FDR).
In our setting, we demand that the function $P_{G}=e^{-\beta H}$ on phase
space is annihilated by the right-hand side of (79). This gives the equation:

$\lambda_{i}^{i}-\beta\lambda^{i}_{k}G_{ij}\pi^{j}\pi^{k}=\frac{\beta}{2}\sum_{n=1}^{3}\sigma_{j}^{n}\sigma_{k}^{n}\left(G^{jk}-\beta\pi^{j}\pi^{k}\right),$
(80)

where the indices of $\pi_{i}$ are raised by the inertia metric $G_{ij}$.
Since in equation (80) the momenta can take arbitrary values, using
$\gamma_{kj}=\lambda^{i}_{k}G_{ij}$ we obtain:

$\displaystyle 2k_{B}T\gamma_{jk}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{3}\sigma_{j}^{n}\sigma_{k}^{n},$ (81) $\displaystyle
2k_{B}T\lambda^{i}_{i}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{3}\sigma_{j}^{n}\sigma_{k}^{n}G^{jk}.$ (82)

The second condition follows from the first by taking the trace, hence (81) is
the FDR for our system. To make contact with standard notation, let us assume
for simplicity that both $\gamma_{ij}$ and $\sigma_{i}^{n}$ are diagonal:

$\gamma_{ij}=\begin{pmatrix}\gamma_{1}&0&0\\\ 0&\gamma_{2}&0\\\
0&0&\gamma_{R}\end{pmatrix},\quad\sigma_{i}^{n}=\begin{pmatrix}\sigma_{1}&0&0\\\
0&\sigma_{2}&0\\\ 0&0&\sigma_{R}\end{pmatrix}.$ (83)

In this case, (81) gives:

$2k_{B}T\gamma_{a}=\sigma_{a}^{2},\quad 2k_{B}T\gamma_{R}=\sigma_{R}^{2}$ (84)

Introducing the translational $D_{a}$ and rotational diffusion coefficients
$D_{R}$ as

$D_{a}=\frac{\sigma_{a}^{2}}{2\gamma_{a}^{2}},\quad
D_{R}=\frac{\sigma_{R}^{2}}{2\gamma_{R}^{2}},$ (85)

the fluctation-dissipation relation (81) assumes the standard form:

$D_{a}=\frac{k_{B}T}{\gamma_{a}},\quad D_{R}=\frac{k_{B}T}{\gamma_{R}}.$ (86)

##  IV Smoluchowski limit

###  IV.1 Adiabatic elimination of momenta

In this section we are going to derive the overdamped or Smoluchowski limit of
the process (62)—(63) by adiabatically eliminating the momenta from the
equations by a standard procedure, albeit we concentrate on the backward
Kolmogorov operator rather than the forward one [30, 34, 12, 20, 19]. We
assume that the FDR (81) holds and introduce the rescaled momenta and inertia
metric as follows:

$\tilde{\pi}_{i}=\frac{\pi_{i}}{\sqrt{m}},\quad\tilde{G}_{ij}=\frac{G_{ij}}{m}.$
(87)

Subsequently, $\tilde{G}^{ij}$ denotes the inverse metric in (87) and indices
$i,j$ are raised and lowered with respect to $\tilde{G}_{ij}$. The backward
Kolmogorov operator (75) can be split as a sum of two terms:

$L_{B}=\frac{1}{m}L_{1}+\frac{1}{\sqrt{m}}L_{2},$ (88)

where we have (using the definition of the friction tensor and the
fluctuation-dissipation relations):

$\displaystyle L_{1}=$
$\displaystyle-\gamma_{ij}\tilde{G}^{kj}\tilde{\pi}_{k}\frac{\partial}{\partial\tilde{\pi}_{i}}+\frac{1}{2}\sum_{n=1}^{3}\sigma_{i}^{n}\sigma_{j}^{n}\frac{\partial^{2}}{\partial\tilde{\pi}_{i}\partial\tilde{\pi}_{j}}$
(89)
$\displaystyle=\gamma_{ij}\left(-\tilde{\pi}^{j}\frac{\partial}{\partial\tilde{\pi}_{i}}+k_{B}T\frac{\partial^{2}}{\partial\tilde{\pi}_{i}\partial\tilde{\pi}_{j}}\right),$

while

$L_{2}=F_{i}\frac{\partial}{\partial{\tilde{\pi}_{i}}}+\sqrt{m}X_{H}.$ (90)

For the purposes of this derivation, we include all force contributions (both
conservative and non-conservative) in the first term of equation (90) and work
with a purely kinetic Hamiltonian. The operator $L_{1}$ admits the invariant
momentum density

$\xi\left(\tilde{\pi}_{i}\right)=\mathcal{N}\exp\left(-\frac{\beta\tilde{G}^{ij}\tilde{\pi}_{i}\tilde{\pi}_{j}}{2}\right),$
(91)

where $\mathcal{N}$ is a normalization factor. In what follows, we are going
to work in the function space $L^{2}(\xi)$ of square-integrable functions with
respect to the weight $\xi\left(\tilde{\pi}_{i}\right)$. Assume that the
solution $f\left(q^{\mu},\tilde{\pi}_{i},t\right)\in L^{2}(\xi)$ of the
backward Kolmogorov equation

$\frac{\partial f}{\partial t}=L_{B}f$ (92)

has an asymptotic expansion of the form:

$f=f_{0}+\sqrt{m}f_{1}+mf_{2}+\dots$ (93)

Substituting (93) into (92) using (88) and equating powers of $\sqrt{m}$ we
find the following hierarchy:

$\displaystyle L_{1}f_{0}$ $\displaystyle=$ $\displaystyle 0,$ (94)
$\displaystyle L_{1}f_{1}$ $\displaystyle=$ $\displaystyle-L_{2}f_{0},$ (95)
$\displaystyle\frac{\partial f_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle L_{1}f_{2}+L_{2}f_{1}.$ (96)

From equation (94) and examining the null space of the operator $L_{1}$ in
$L^{2}(\xi)$ it follows that $f_{0}$ has to be independent of momenta, so that
$f_{0}=f_{0}(q,t)$. Equation (95) then reads:

$\displaystyle L_{1}f_{1}$ $\displaystyle=$
$\displaystyle-\sqrt{m}\\{f_{0},H\\}=-\sqrt{m}V_{i}\left[f_{0}\right]\pi_{j}G^{ij}$
(97) $\displaystyle=$ $\displaystyle-\tilde{\pi}^{i}V_{i}[f_{0}].$ (98)

The general solution to (95) in $L^{2}(\xi)$ is therefore given by:

$f_{1}=\left(\gamma^{-1}\right)^{ij}\tilde{\pi}_{i}V_{j}\left[f_{0}\right]+g_{0}(q,t),$
(99)

where we have assumed that the friction tensor $\gamma_{ij}$ is invertible
with inverse $\left(\gamma^{-1}\right)^{ij}$ and $g_{0}(q,t)$ is an
integration constant which, in this case, is an arbitrary function of
positions and time. Substituting (99) into equation (96) we obtain:

$\displaystyle\frac{\partial f_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle L_{1}f_{2}+\left(\gamma^{-1}\right)^{ij}F_{i}V_{j}[f_{0}]$
(100)
$\displaystyle+\left\\{\left(\gamma^{-1}\right)^{ij}\pi_{i}V_{j}[f_{0}],H\right\\}+\tilde{\pi}^{i}V_{i}\left[g_{0}\right]$

Expanding the Poisson bracket yields:

$\displaystyle\frac{\partial f_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle
L_{1}f_{2}+\tilde{\pi}^{i}V_{i}\left[g_{0}\right]+\left(\gamma^{-1}\right)^{ij}F_{i}V_{j}[f_{0}]$
(101)
$\displaystyle+\tilde{G}^{kl}\tilde{\pi}_{i}\tilde{\pi}_{l}V_{k}\left[\left(\gamma^{-1}\right)^{ij}V_{j}\left[f_{0}\right]\right]$
$\displaystyle+\Omega^{k}_{il}\tilde{G}^{lr}\left(\gamma^{-1}\right)^{ij}\tilde{\pi}_{k}\tilde{\pi}_{r}V_{j}\left[f_{0}\right]$

To eliminate $f_{2}$ and $g_{0}$ from equation (101) we first multiply both
sides by the the invariant density $\xi$ then integrate over momenta. The term
containing $f_{2}$ will vanish as

$\int_{T^{*}FS}\xi L_{1}f_{2}\rho=\int_{T^{*}FS}L_{1}^{\dagger}\xi
f_{2}\rho=0,$ (102)

since $L_{1}^{\dagger}$ (adjoint taken with respect to $\rho$) annihilates the
stationary momentum density $\xi$, while the term containing $g_{0}$ is odd in
momenta and will also vanish upon integration. Hence we find the limiting form
of the backward equation for $f_{0}$:

$\frac{\partial f_{0}}{\partial
t}=\left(F_{i}+\frac{\Omega^{k}_{ik}}{\beta}\right)\left(\gamma^{-1}\right)^{ij}V_{j}[f_{0}]+V_{i}\left[\frac{\left(\gamma^{-1}\right)^{ij}}{\beta}V_{j}\left[f_{0}\right]\right].$
(103)

The second term in the parentheses in (103) — not anticipated by a naive
overdamped limit where all inertia terms are neglected — vanishes owing to the
structure of the coefficients $\Omega^{k}_{ij}$. Let us introduce the inverse
noise amplitude coefficients $\Sigma(q)^{i}_{n}$ via the relations

$\Sigma(q)^{i}_{n^{\prime}}\sigma(q)_{i}^{n}=\delta^{n}_{n^{\prime}},\quad\sum_{n=1}^{3}\Sigma(q)^{i}_{n}\sigma(q)^{n}_{j}=\delta^{i}_{j}.$
(104)

Using the fluctuation-dissipation relation (81) we have:

$\left(\gamma^{-1}\right)^{ij}=2k_{B}T\sum_{n=1}^{3}\Sigma^{i}_{n}\Sigma^{j}_{n}.$
(105)

Let us define the noise vector fields $Y_{n}$ as

$Y_{n}=2k_{B}T\Sigma(q)^{i}_{n}V_{i},$ (106)

which allows us to write the infinitesimal generator $\Xi$ of the limiting
process as:

$\displaystyle\Xi$ $\displaystyle=$
$\displaystyle\left(\left(\gamma^{-1}\right)^{ij}F_{i}+2\left(k_{B}T\right)^{2}\sum_{n=1}^{3}\Sigma^{j}_{n}V_{i}\left[\Sigma^{i}_{n}\right]\right)V_{j}$
(107) $\displaystyle+\frac{1}{2}\sum_{n=1}^{3}Y_{n}Y_{n}.$

###  IV.2 Intrinsic Brownian dynamics

Using the correspondence between the infinitesimal generator of stochastic
processes and stochastic differential equations [30] we are now in a position
to write down the overdamped limit of the stochastic equations of motion:

$\displaystyle dq^{\mu}$ $\displaystyle=$
$\displaystyle\left(\left(\gamma^{-1}\right)^{ij}F_{i}+2\left(k_{B}T\right)^{2}\sum_{n=1}^{3}\Sigma^{j}_{n}V_{i}\left[\Sigma^{i}_{n}\right]\right)V_{j}^{\mu}dt$
(108) $\displaystyle+2k_{B}T\sum_{n=1}^{3}\Sigma^{i}_{n}V_{i}^{\mu}\circ
dW^{n}.$

Note the presence of the extra drift term coming from the position dependence
of the friction and diffusion tensors. To simplify the exposition, in what
follows we will assume that the $\Sigma^{i}_{n}$ are independent of position,
but our results can be readily extended for the case of state-dependent
friction and diffusivity.

For diagonal noise and friction tensors (83) obeying the FDT, in local
coordinates the system (108) takes the following familiar form:

$\displaystyle dx^{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{2}\frac{F_{a}f_{a}^{\alpha}}{\gamma_{a}}dt+\sum_{a=1}^{2}\sqrt{2D_{a}}f_{a}^{\alpha}\circ
dW_{a}$ (109) $\displaystyle d\psi$ $\displaystyle=$
$\displaystyle-\sum_{a=1}^{2}f_{a}^{\alpha}\Gamma_{\alpha}\left(\frac{F_{a}}{\gamma_{a}}dt+\sqrt{2D_{a}}\circ
dW_{a}\right)$ (110) $\displaystyle+\frac{M}{\gamma_{R}}dt+\sqrt{2D_{R}}\circ
dW_{R}$

Equation (109) governs the translational motion in the overdamped limit. The
first term on the right-hand side describes the effect of forces resulting in
a deterministic drift. In the active Brownian particle model, $F_{a}$ is taken
to be constant, and if $\gamma_{a}$ is also constant then
$u_{a}=\frac{F_{a}}{\gamma_{a}}$ becomes a constant swimming velocity with
which the particle translates in its body frame. Observe that the
translational noise, which is injected in the body frame, is always
multiplicative as $f_{a}^{\alpha}$ generally depends on $x^{\alpha}$ as well
as $\psi$, therefore it is essential to fix a noise interpretation
(Stratonovich in our case). Equation (110) governs the rotation of the frame,
comprised of three contributions: one from the covariant derivative (the first
term on the right-hand side), one from the torque and one from rotational
Wiener noise. The intuitive interpretation is that the covariant derivative of
the frame is given by sum of rotation from external torques and thermal noise,
consistent with previous proposals [8, 9, 10].

###  IV.3 Smoluchowski equation

In this section, we derive the Fokker-Planck or Smoluchowski equation
corresponding to the stochastic process (108). As discussed previously, the
form of the FPE depends on the choice of reference measure. In our setting, it
is natural to work with the Riemannian volume element

$\mu_{R}=\mathrm{vol}_{S}\wedge d\psi=\eta^{1}\wedge\eta^{2}\wedge\eta^{3}$
(111)

on $FS$, also appearing in the symplectic volume element in (74). The adjoint
$\Xi_{F}$ of the infinitesimal generator $\Xi$ with respect to the inner
product $L^{2}\left(\mu_{R}\right)$ on $FS$ is given by:

$\Xi_{F}[P]=-\left(\gamma^{-1}\right)^{ij}\left(V_{i}\left[F_{j}P\right]-k_{B}TV_{i}\left[V_{j}[P]\right]\right)$
(112)

The Smoluchowski equation reads:

$\frac{\partial P}{\partial
t}=-\left(\gamma^{-1}\right)^{ij}\left(V_{i}\left[F_{j}P\right]-k_{B}TV_{i}\left[V_{j}[P]\right]\right)$
(113)

For this special case when the friction and diffusion tensors are independent
of position we recover Brownian motion on the frame bundle $FS$ with respect
to a metric whose orthonormal vector fields are given by a scaled versions
$\sqrt{2D_{a}}V_{a}$ and $\sqrt{2D_{R}}V_{R}$ of the horizontal and vertical
vector fields, together with a drift vector field
$\left(\gamma^{-1}\right)^{ij}F_{j}V_{i}$.

This is consistent with the more general result shown in [18], where the
authors proved that the small mass limit of stochastic geodesic equations on
the tangent bundle of general manifolds perturbed by linear state-independent
friction and white noise is Brownian motion generated by the Laplace-Beltrami
operator of the metric [35]. In our case, the extremal trajectories of the
Lagrangian (29) are in fact geodesics of a metric on $FS$ whose orthonormal
vector fields are $V_{i}$ (up to scale), and the stochastic equations of
motion (62)—(63) are stochastic geodesic equations.

##  V Rigid inclusion on a sphere

###  V.1 Kinematics in local coordinates

As a concrete example, we compute explicitly the equations of motion for the
particular case of the sphere $S^{2}\subset\mathbb{R}^{3}$ of radius $R$
embedded in three-dimensional space as
$S^{2}=\left\\{(x,y,z)|x^{2}+y^{2}+z^{2}=R^{2}\right\\}$. We use spherical
polar coordinates $(\vartheta,\varphi)\in(0,\pi)\times[0,2\pi)$ on a
coordinate patch excluding the north and south poles as follows:

$\displaystyle x$ $\displaystyle=$ $\displaystyle R\sin\vartheta\cos\varphi,$
(114) $\displaystyle y$ $\displaystyle=$ $\displaystyle
R\sin\vartheta\sin\varphi,$ (115) $\displaystyle z$ $\displaystyle=$
$\displaystyle R\cos\vartheta.$ (116)

The restriction of the Euclidean metric to the sphere is given by
$g=R^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta d\varphi^{2}\right)$. It
follows that the vector fields

$\mathbf{e}_{1}=\frac{1}{R}\frac{\partial}{\partial\vartheta},\quad\mathbf{e}_{2}=\frac{1}{R\sin\vartheta}\frac{\partial}{\partial\varphi}$
(117)

form an orthonormal moving frame away from the poles, with the corresponding
coframes

$\theta^{1}=Rd\vartheta,\quad\theta^{2}=R\sin\vartheta d\varphi.$ (118)

The connection form $\Gamma$ can be computed by calculating the covariant
derivative of the frame vectors $\mathbf{e}_{a}$ or using the structure
equations (142)—(143). The result is

$\Gamma=\cos\vartheta d\varphi=\frac{\cot\vartheta}{R}\theta^{2}.$ (119)

The Gaussian curvature of the sphere is constant and equal to
$\kappa=\frac{1}{R^{2}}$. This can also be verified using equation (8):

$d\Gamma=-\sin\vartheta d\vartheta\wedge
d\varphi=-\frac{1}{R^{2}}\theta^{1}\wedge\theta^{2}=-\frac{1}{R^{2}}\mathrm{vol}_{S}.$

With respect to our choice of moving frame, any frame field
$\mathbf{f}_{a}(x,\psi)$ can be written as:

$\displaystyle\mathbf{f}_{1}$ $\displaystyle=$
$\displaystyle\mathbf{e}_{1}\cos\psi+\mathbf{e}_{2}\sin\psi=\frac{\cos\psi}{R}\frac{\partial}{\partial\vartheta}+\frac{\sin\psi}{R\sin\vartheta}\frac{\partial}{\partial\varphi},$
$\displaystyle\mathbf{f}_{2}$ $\displaystyle=$
$\displaystyle\mathbf{e}_{2}\cos\psi-\mathbf{e}_{1}\sin\psi=\frac{\cos\psi}{R\sin\vartheta}\frac{\partial}{\partial\varphi}-\frac{\sin\psi}{R}\frac{\partial}{\partial\vartheta}.$

The horizontal (9) and vertical (10) vector fields are given by:

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\frac{\cos\psi}{R}\frac{\partial}{\partial\vartheta}+\frac{\sin\psi}{R\sin\vartheta}\frac{\partial}{\partial\varphi}-\frac{\sin\psi}{R\tan\vartheta}\frac{\partial}{\partial\psi},$
(120) $\displaystyle V_{2}$ $\displaystyle=$
$\displaystyle\frac{\cos\psi}{R\sin\vartheta}\frac{\partial}{\partial\varphi}-\frac{\sin\psi}{R}\frac{\partial}{\partial\vartheta}-\frac{\cos\psi}{R\tan\vartheta}\frac{\partial}{\partial\psi},$
(121) $\displaystyle V_{R}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\psi}.$ (122)

Given the translational $v^{1}(t),v^{2}(t)$ and angular velocity $\omega(t)$
of the inclusion, its position can be integrated via:

$\displaystyle\frac{d\vartheta}{dt}$ $\displaystyle=$
$\displaystyle\frac{v^{1}(t)\cos\psi-v^{2}(t)\sin\psi}{R}$ (123)
$\displaystyle\frac{d\varphi}{dt}$ $\displaystyle=$
$\displaystyle\frac{v^{1}(t)\sin\psi+v^{2}(t)\cos\psi}{R\sin\vartheta}$ (124)
$\displaystyle\frac{d\psi}{dt}$ $\displaystyle=$
$\displaystyle\omega(t)-\frac{v^{1}(t)\sin\psi+v^{2}(t)\cos\psi}{R\tan\vartheta}$
(125)

###  V.2 Dynamics in local coordinates

In local coordinates, the Lagrangian (30) and Hamiltonian (33) read:

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{1}{2}mR^{2}\left(\dot{\vartheta}^{2}+\sin^{2}\vartheta\dot{\varphi}^{2}\right)+\frac{1}{2}J\left(\dot{\psi}+\cos\vartheta\dot{\varphi}\right)^{2},$
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{p_{\vartheta}^{2}}{2mR^{2}}+\frac{\left(p_{\varphi}-s\cos\vartheta\right)^{2}}{2mR^{2}\sin^{2}\vartheta}+\frac{s^{2}}{2J}.$
(126)

the underdamped stochastic equations of motion (62)—(63) are given by
(assuming that the noise and friction tensors are diagonal as in (83)):

$\displaystyle d\pi_{1}$ $\displaystyle=$
$\displaystyle\left(\left(\frac{1}{J}-\frac{1}{mR^{2}}\right)s\pi_{2}-\frac{\gamma_{1}}{m}\pi_{1}+F_{1}\right)dt$
(127) $\displaystyle+\gamma_{1}\sqrt{2D_{1}}\circ dW_{1}$ $\displaystyle
d\pi_{2}$ $\displaystyle=$
$\displaystyle\left(-\left(\frac{1}{J}-\frac{1}{mR^{2}}\right)s\pi_{1}-\frac{\gamma_{2}}{m}\pi_{2}+F_{2}\right)dt$
(128) $\displaystyle+\gamma_{2}\sqrt{2D_{2}}\circ dW_{2}$ $\displaystyle ds$
$\displaystyle=$
$\displaystyle-\left(\frac{\gamma_{R}}{J}s+M\right)dt+\gamma_{R}\sqrt{2D_{R}}\circ
dW_{R}$ (129) $\displaystyle d\vartheta$ $\displaystyle=$
$\displaystyle\frac{\pi_{1}\cos\psi-\pi_{2}\sin\psi}{mR}dt$ (130)
$\displaystyle d\varphi$ $\displaystyle=$
$\displaystyle\frac{\pi_{1}\sin\psi+\pi_{2}\cos\psi}{mR\sin\vartheta}dt$ (131)
$\displaystyle d\psi$ $\displaystyle=$
$\displaystyle\left(\frac{s}{J}-\frac{\pi_{1}\sin\psi+\pi_{2}\cos\psi}{mR\tan\vartheta}\right)dt$
(132)

Further assuming that the friction and noise tensors are constant, the
overdamped stochastic equations of motion (109)—(110) on the sphere read:

$\displaystyle d\vartheta$ $\displaystyle=$
$\displaystyle\left(\frac{F_{1}\cos\psi}{R\gamma_{1}}-\frac{F_{2}\sin\psi}{R\gamma_{2}}\right)dt$
$\displaystyle+\frac{\sqrt{2D_{1}}\cos\psi}{R}\circ
dW_{1}-\frac{\sqrt{2D_{2}}\sin\psi}{R}\circ dW_{2},$ $\displaystyle d\varphi$
$\displaystyle=$
$\displaystyle\left(\frac{F_{1}\sin\psi}{R\gamma_{1}\sin\vartheta}+\frac{F_{2}\cos\psi}{R\gamma_{2}\sin\vartheta}\right)dt$
$\displaystyle+\frac{\sqrt{2D_{1}}\sin\psi}{R\sin\vartheta}\circ
dW_{1}+\frac{\sqrt{2D_{2}}\cos\psi}{R\sin\vartheta}\circ dW_{2},$
$\displaystyle d\psi$ $\displaystyle=$
$\displaystyle\left(\frac{M}{\gamma_{R}}-\frac{\cot\vartheta}{R}\left(\frac{F_{1}\sin\psi}{\gamma_{1}}+\frac{F_{2}\cos\psi}{\gamma_{2}}\right)\right)dt$
$\displaystyle-\frac{\cot\vartheta}{R}\left(\sqrt{2D_{1}}\sin\psi\circ
dW_{1}+\sqrt{2D_{2}}\cos\psi\circ dW_{2}\right)$
$\displaystyle+\sqrt{2D_{R}}\circ dW_{R}.$

###  V.3 The Fokker-Planck and Smoluchowski equations

The Fokker-Planck equation (79) for the phase space density
$P(\vartheta,\varphi,\psi,\pi_{1},\pi_{2},s)\sin\vartheta d\pi_{1}\wedge
d\pi_{2}\wedge ds\wedge d\vartheta\wedge d\varphi\wedge d\psi$ of the
inclusion can be obtained by using the local coordinate expressions of the
vector fields (120)—(122). Using the purely kinetic Hamiltonian (36) and (85)
we find:

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle+$
$\displaystyle\frac{\pi_{1}\cos\psi-\pi_{2}\sin\psi}{mR}\frac{\partial
P}{\partial\vartheta}+\frac{\pi_{1}\sin\psi+\pi_{2}\cos\psi}{mR\sin\vartheta}\frac{\partial
P}{\partial\varphi}+\left(\frac{s}{J}-\frac{\pi_{1}\sin\psi+\pi_{2}\cos\psi}{mR\tan\vartheta}\right)\frac{\partial
P}{\partial\psi}=$ (133) $\displaystyle=$
$\displaystyle\sum_{a=1}^{2}\frac{\partial}{\partial\pi_{a}}\left(\left(\frac{\gamma_{a}}{m}\pi_{a}-F_{a}\right)P\right)+\frac{\partial}{\partial
s}\left(\left(\frac{\gamma_{R}}{J}s-M\right)P\right)+\sum_{a=1}^{2}\gamma_{a}^{2}D_{a}\frac{\partial^{2}P}{\partial\pi_{a}^{2}}+\gamma_{R}^{2}D_{R}\frac{\partial^{2}P}{\partial
s^{2}}.$

The Smoluchowski equation (113) for the configuration space density
$P(\vartheta,\varphi,\psi)\sin\vartheta d\vartheta\wedge d\varphi\wedge d\psi$
can also be obtained by putting together (120)—(122) and (113). In the case of
constant diagonal and friction tensors, the result is:

$\displaystyle\frac{\partial P}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\cos\psi}{\gamma_{1}R}\frac{\partial\left(F_{1}P\right)}{\partial\vartheta}-\frac{\sin\psi}{\gamma_{1}R\sin\vartheta}\frac{\partial\left(F_{1}P\right)}{\partial\varphi}+\frac{\sin\psi}{\gamma_{1}R\tan\vartheta}\frac{\partial\left(F_{1}P\right)}{\partial\psi}$
(134)
$\displaystyle-\frac{\cos\psi}{\gamma_{2}R\sin\vartheta}\frac{\partial\left(F_{2}P\right)}{\partial\varphi}+\frac{\sin\psi}{\gamma_{2}R}\frac{\partial\left(F_{2}P\right)}{\partial\vartheta}+\frac{\cos\psi}{\gamma_{2}R\tan\vartheta}\frac{\partial\left(F_{2}P\right)}{\partial\psi}-\frac{1}{\gamma_{R}}\frac{\partial\left(MP\right)}{\partial\psi}$
$\displaystyle+D_{1}\left(\frac{\cos\psi}{R}\frac{\partial}{\partial\vartheta}+\frac{\sin\psi}{R\sin\vartheta}\frac{\partial}{\partial\varphi}-\frac{\sin\psi}{R\tan\vartheta}\frac{\partial}{\partial\psi}\right)\left(\frac{\cos\psi}{R}\frac{\partial
P}{\partial\vartheta}+\frac{\sin\psi}{R\sin\vartheta}\frac{\partial
P}{\partial\varphi}-\frac{\sin\psi}{R\tan\vartheta}\frac{\partial
P}{\partial\psi}\right)$
$\displaystyle+D_{2}\left(\frac{\cos\psi}{R\sin\vartheta}\frac{\partial}{\partial\varphi}-\frac{\sin\psi}{R}\frac{\partial}{\partial\vartheta}-\frac{\cos\psi}{R\tan\vartheta}\frac{\partial}{\partial\psi}\right)\left(\frac{\cos\psi}{R\sin\vartheta}\frac{\partial
P}{\partial\varphi}-\frac{\sin\psi}{R}\frac{\partial
P}{\partial\vartheta}-\frac{\cos\psi}{R\tan\vartheta}\frac{\partial
P}{\partial\psi}\right)+D_{R}\frac{\partial^{2}P}{\partial\psi^{2}}$

In the case of isotropic diffusion $D_{1}=D_{2}\equiv D$ and no drift, for the
spherical part we recover the standard form of the Smoluchowski equation on
the sphere [36]:

$\frac{\partial P}{\partial
t}=\frac{D}{R^{2}}\left(\cot\vartheta\frac{\partial
P}{\partial\vartheta}+\frac{\partial^{2}P}{\partial\vartheta^{2}}+\frac{\partial^{2}P}{\partial\varphi^{2}}\right)+\dots,$
(135)

where the omitted terms contain derivatives with respect to the orientational
coordinate $\psi$.

###  V.4 Active Brownian particle on the sphere

So far the forces $F_{a}$ and torque $M$ have been arbitrary functions of
position and orientation. To obtain a generalization of the standard active
Brownian particle model [37] to a rigid inclusion constrained to the surface
of the sphere, we make the choice

$F_{1}=\gamma_{1}v_{S}\equiv\text{const.},\quad F_{2}=M=0,$ (136)

where we assume that $\gamma_{1}$ is constant and we have introduced the so-
called swimming velocity $v_{S}$ of the inclusion. The physical interpretation
of this constitutive force and torque law is that the inclusion self-propels
with a constant force along a fixed direction in the body frame. Using results
from Section II, we can easily see that this constitutive force and torque law
does not derive from a potential, as in (60) the left-hand side is zero while
the right-hand side is not. We stress that this conclusion remains valid for
an arbitrary surface because the integrability conditions (59)—(61) hold for
general surfaces. With the choice (136), the overdamped stochastic equations
of motion simplify to:

$\displaystyle d\vartheta$ $\displaystyle=$
$\displaystyle\frac{v_{S}\cos\psi}{R}dt$
$\displaystyle+\frac{\sqrt{2D_{1}}\cos\psi}{R}\circ
dW_{1}-\frac{\sqrt{2D_{2}}\sin\psi}{R}\circ dW_{2},$ $\displaystyle d\varphi$
$\displaystyle=$ $\displaystyle\frac{v_{S}\sin\psi}{R\sin\vartheta}dt$
$\displaystyle+\frac{\sqrt{2D_{1}}\sin\psi}{R\sin\vartheta}\circ
dW_{1}+\frac{\sqrt{2D_{2}}\cos\psi}{R\sin\vartheta}\circ dW_{2},$
$\displaystyle d\psi$ $\displaystyle=$
$\displaystyle-\frac{v_{S}\cot\vartheta\sin\psi}{R}dt+\sqrt{2D_{R}}\circ
dW_{R}$ $\displaystyle-\frac{\cot\vartheta}{R}\left(\sqrt{2D_{1}}\sin\psi\circ
dW_{1}+\sqrt{2D_{2}}\cos\psi\circ dW_{2}\right).$

Alternatively, one can also study the corresponding Smoluchowski equation
(134). When translational noise is neglected (i.e. $D_{1}=D_{2}=0$), we
recover the previous proposal of an active swimmer on the sphere in [10].

##  VI Conclusion

In this paper the theory of intrinsic Langevin dynamics of rigid inclusions
has been formulated. Starting from the kinematics of a rigid oriented
inclusion constrained on a curved surface, the deterministic inertial
equations of motion have been derived in Hamiltonian form from a standard
Lagrangian via the method of moving frames, uncovering an interesting Poisson
structure along the way. Linear coordinate-dependent friction together with
general forces and torques have been added to Hamilton’s equation of motion
and integrability conditions for them have been obtained. By injecting
Gaussian white noise to Hamilton’s equations, stochastic differential
equations have been proposed that model the motion of the inclusion under the
influence of thermal noise. The Fokker-Planck equation of the inclusion has
been given in phase space in geometric form and the momenta have been
adiabatically eliminated to obtain the Smoluchowski limit of the process. An
example calculation in local coordinates has also been presented on the
sphere.

Our work can be applied to a wide range of different problems and extended in
various dimensions. The geometric structures uncovered in the equations
suggest structure-preserving numerical schemes for computer simulations [38].
The intrinsic stochastic differential equations presented in our paper can
readily be integrated in local coordinates using standard methods [39] for
numerical solutions of SDEs. An immediate advantage of an intrinsic method is
that the solution automatically satisfies the constraints, contrary to an
extrinsic solver that usually has to employ some kind of projection to achieve
that. However, since any coordinate system is only locally valid on a non-
trivial manifold, it is challenging to extend these methods globally, as one
generally has to change charts during integration. It would therefore be
desirable to develop a numerical scheme based on the method of moving frames,
which could combine the benefits of both the intrinsic and extrinsic
viewpoints, as it would respect the geometry of the problem but also
facilitate global integration by utilizing frames in the ambient space.

It is straightforward to extend our theory to many interacting rigid
inclusions on a surface by considering a multi-body Hamiltonian and an
interaction potential. It would be interesting to gain further insight into
how the fascinating collective effects produced by active particles such as
motility-induced phase separation [40, 41] or intriguing patterns in polymeric
systems [42, 43] manifest on a curved surface [44]. By a suitable coarse-
graining procedure, field theories could be written down and studied over
curved surfaces, where the solutions of the resulting PDEs would also be
dramatically altered by the geometry and topology of the surface [8, 45, 10].
Another exciting area of future research could be the extension of the well-
established linear response theory [46] to the manifold setting — it is quite
likely that curvature terms would influence e.g. the Green-Kubo relations,
connecting transport coefficients and correlation functions. The integrability
conditions for the forces and torques and the geometric form of the Fokker-
Planck equation could shed further light on questions related to
irreversibility [47, 32].

Large deviations and rare events is an important branch of statistical physics
[48, 49, 50, 51] that has found many applications in soft and active matter.
Our formalism is very well suited to study such problems in a manifold
setting, by looking at for example escape and stability problems at stationary
points of potentials on surfaces [52, 53]. Control of active matter [54] is
also an emerging field of research that has received considerable attention
lately and the investigation of optimal protocols on surfaces has already been
initiated [55]. We hope to extend these results with the aid of the theory
developed above. Finally, we aim to generalize our setup to the intriguing and
challenging scenario when there is feedback between the motion of the
inclusion and the surface, as is the case for most problems in biology.

###### Acknowledgements.

We thank Prof. Dr. Tanja Schilling for helpful and interesting discussions.
The first author thanks the organisers of the DPG Spring Meeting (17—22 March
2024, Berlin, Germany) for an opportunity to present this work. This research
was supported by the Engineering and Physical Sciences Research Council (B.N.,
award no. EP/W524141/1).

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##  Appendix A Derivation of Poisson brackets

In this section we are going to derive the Poisson bracket structure (40)—(44)
in terms of the body frame momenta and positional coordinates. Using (71) and
standard properties of Poisson brackets we find:

$\left\\{q^{\mu},p_{\nu}\right\\}=\delta^{\mu}_{\nu}=\left\\{q^{\mu},\pi_{j}\eta^{j}_{\nu}(q)\right\\}=\eta^{j}_{\nu}(q)\\{q^{\mu},\pi_{j}\\}.$
(137)

Hence:

$\\{q^{\mu},\pi_{j}\\}=V^{\mu}_{j}(q)$ (138)

Similarly, we have:

$\displaystyle\\{\pi_{i},\pi_{j}\\}$ $\displaystyle=$
$\displaystyle\left\\{V_{i}^{\mu}(q)p_{\mu},V_{j}^{\nu}(q)p_{\nu}\right\\}$
(139) $\displaystyle=$ $\displaystyle V_{j}^{\nu}\frac{\partial
V_{i}^{\mu}}{\partial q^{\nu}}p_{\mu}-V_{i}^{\mu}\frac{\partial
V_{j}^{\nu}}{\partial q^{\mu}}p_{\nu}$ $\displaystyle=$
$\displaystyle-\left[V_{i},V_{j}\right]^{\mu}p_{\mu}$ $\displaystyle=$
$\displaystyle-\left[V_{i},V_{j}\right]^{\mu}\eta^{k}_{\mu}\pi_{k}$
$\displaystyle=$ $\displaystyle\Omega^{k}_{ij}\pi_{k}$

where square brackets denote the commutator or Lie bracket of vector fields.
There are various ways to finish the calculation from this point: for example,
one can compute the commutators using the coordinate expressions (9)—(10).
Here we simplify the calculations using the following exterior calculus
identity [23]:

$d\eta^{i}\left(V_{j},V_{k}\right)=V_{j}\left[\eta^{i}\left(V_{k}\right)\right]-V_{k}\left[\eta^{i}\left(V_{j}\right)\right]-\eta^{i}\left(\left[V_{j},V_{k}\right]\right).$
(140)

As $\eta^{i}\left(V_{j}\right)=\delta^{i}_{j}$, the first two terms on the
right-hand side vanish, leading to:

$d\eta^{i}\left(V_{j},V_{k}\right)=-\eta^{i}\left(\left[V_{j},V_{k}\right]\right)=\Omega^{i}_{jk}.$
(141)

Therefore it suffices to compute the exterior derivatives of the $\eta^{i}$-s
to find the Poisson brackets. For that purpose, we recall the structure
equations of the connection [23]:

$\displaystyle d\theta^{1}-\Gamma\wedge\theta^{2}$ $\displaystyle=$
$\displaystyle 0,$ (142) $\displaystyle d\theta^{2}+\Gamma\wedge\theta^{1}$
$\displaystyle=$ $\displaystyle 0,$ (143)

which represent the fact that the Levi-Civita connection has zero torsion.
After some calculations using these equations and (13)—(14), we get:

$\displaystyle d\eta^{1}$ $\displaystyle=$ $\displaystyle
d\Theta^{1}=\eta^{3}\wedge\eta^{2},$ (144) $\displaystyle d\eta^{2}$
$\displaystyle=$ $\displaystyle d\Theta^{2}=\eta^{1}\wedge\eta^{3},$ (145)
$\displaystyle d\eta^{3}$ $\displaystyle=$ $\displaystyle
d\Gamma=-\kappa\theta^{1}\wedge\theta^{2}=-\kappa\eta^{1}\wedge\eta^{2}.$
(146)

It follows that the Poisson brackets between the body frame linear and angular
momenta are:

$\displaystyle\\{\pi_{1},\pi_{2}\\}$ $\displaystyle=$
$\displaystyle\Omega^{i}_{12}\pi_{i}=\Omega^{3}_{12}\pi_{3}=-\kappa s,$ (147)
$\displaystyle\\{\pi_{2},\pi_{3}\\}$ $\displaystyle=$
$\displaystyle\Omega^{i}_{23}\pi_{i}=\Omega^{1}_{23}\pi_{1}=-\pi_{1},$ (148)
$\displaystyle\\{\pi_{3},\pi_{1}\\}$ $\displaystyle=$
$\displaystyle\Omega^{i}_{31}\pi_{i}=\Omega^{2}_{31}\pi_{2}=-\pi_{2}.$ (149)

##  Appendix B Derivation of the forward Fokker-Planck operator

In this section we summarize the relationship between the infinitesimal
generator of a continuous and time-homogeneous Markov process $X_{t}$ on a
manifold and the forward Kolmogorov operator in the Fokker-Planck equation,
for a more detailed account, consult e.g. [30].

Let $X_{t}$ evolve on a smooth manifold $M$ of dimension $m$, starting from
the deterministic initial condition $X_{0}=x_{0}$ for some $x_{0}\in M$. The
infinitesimal generator $L_{B}$ of $X_{t}$, also called the backward
Kolmogorov operator, is a differential operator defined by the following
property:

$\frac{d}{dt}\bigg{|}_{t=0}\mathbb{E}\left[f\left(X_{t}\right)\right]=L_{B}\left(x_{0}\right)[f],$
(150)

where $f:M\to\mathbb{R}$ is an arbitrary smooth function on $M$. In this
article, we restrict our attention to diffusion processes, where $L_{B}$ is a
second-order differential operator. In local coordinates $x^{i}$ it can always
be expressed in the form:

$L_{B}(x)=a^{i}(x)\frac{\partial}{\partial
x^{i}}+\frac{1}{2}b^{ij}(x)\frac{\partial^{2}}{\partial x^{i}\partial x^{j}},$
(151)

where the $a^{i}(x)$ and $b^{ij}(x)=b^{ji}(x)$ are smooth functions on $M$,
called the drift and diffusion coefficients, respectively. It is important to
stress that under a change of coordinates, the components $b^{ij}(x)$
transform as the components of a symmetric tensor field contrary to $a^{i}(x)$
that do not transform as the components of a vector field. This is related to
the fact that the drift of an Ito SDE is not a vector field.

To work out the forward Kolmogorov or Fokker-Planck equation of the process,
we introduce the time-dependent probability measure $p_{t}\chi$ on $M$
describing the distribution of $X_{t}$ at time $t$, where $\chi$ is an
arbitrary reference measure on $M$ and $p_{t}:M\to\mathbb{[}0,1]$ is the
probability density function with respect to $\chi$. For processes taking
place on $\mathbb{R}^{d}$, a natural choice is the Lebesgue measure
$dx^{1}\wedge\dots\wedge dx^{d}$, but, as we have seen previously, on general
manifolds different choices might be more appropriate. We may write (150) as:

$\int_{M}\frac{\partial p_{t}}{\partial
t}f\chi=\int_{M}L_{B}[f]p_{t}\chi=\int_{M}L_{B}^{\dagger}[p_{t}]f\chi,$ (152)

where $L_{B}^{\dagger}$ denotes the adjoint of $L_{B}$ in $L^{2}(\chi)$. Since
in (152) the function $f$ is arbitrary, upon localizing we obtain the forward
Kolmogorov or Fokker-Planck equation

$\frac{\partial p_{t}}{\partial t}=L_{F}\left[p_{t}\right],$ (153)

where $L_{F}=L_{B}^{\dagger}$.

The computation of $L_{F}$ simplifies when the infinitesimal generator is in
so-called Hörmander form, i.e. it is of the form

$L_{B}=V+\sum_{i=1}^{n}Y_{i}Y_{i},$ (154)

for some vector fields $V$ and $Y_{1},\dots,Y_{n}$ on $M$. This is because for
any vector field $A$ and real-valued functions $f,g$ on $M$ we have:

$\displaystyle\int_{M}A[f]g\chi$ $\displaystyle=$
$\displaystyle\int_{M}A^{i}(x)\frac{\partial f}{\partial
x^{i}}g(x)\chi(x)dx^{1}\wedge\dots\wedge dx^{m}$ $\displaystyle=$
$\displaystyle-\int_{M}f(x)\frac{\partial\left(A^{i}(x)g(x)\chi(x)\right)}{\partial
x^{i}}dx^{1}\wedge\dots\wedge dx^{m},$

after performing an integration by parts, neglecting boundary terms. Expanding
the derivative, we get:

$\displaystyle\int_{M}A[f]g\chi=\left\langle A[f],g\right\rangle_{\chi}$
$\displaystyle=$
$\displaystyle-\int_{M}f(x)\chi(x)\left[\frac{1}{\chi(x)}\frac{\partial\left(A^{i}(x)\chi(x)\right)}{\partial
x^{i}}g(x)+A^{i}\frac{\partial g}{\partial
x^{i}}\right]dx^{1}\wedge\dots\wedge dx^{m}$ (155) $\displaystyle=$
$\displaystyle-\int_{M}f\left(\left(\mathrm{div}_{\chi}A\right)g+A[g]\right)\chi=-\left\langle
f,\left(\mathrm{div}_{\chi}A\right)g+A[g]\right\rangle_{\chi},$ (156)

where we have introduced the divergence

$\mathrm{div}_{\chi}A=\frac{1}{\chi(x)}\frac{\partial\left(A^{i}(x)\chi(x)\right)}{\partial
x^{i}}$ (157)

of the vector field $A$ with respect to the measure $\chi$. The intrinsic
definition of the divergence uses the Lie derivative: since
$\mathcal{L}_{A}\chi$ is also a volume form, it has to be proportional to the
volume form $\chi$, and this factor of proportionality is the divergence of
the vector field $A$ with respect to $\chi$ [23]. It follows that the
$L^{2}(\chi)$-adjoint of the linear operator $f\mapsto A[f]$ is given by

$g\mapsto-\left(\mathrm{div}_{\chi}A\right)g-A[g].$ (158)

This relation allows us to compute the adjoint of any operator in Hörmander
form iteratively. For the underdamped system (65), we have:

$\mathrm{div}_{\mu}F=\mathrm{div}_{\mu}X_{H}=\mathrm{div}_{\mu}N^{n}=0,\quad\mathrm{div}_{\mu}K=-\lambda^{i}_{i}.$
(159)

These relations follow from a short explicit computation using (74). The fact
that $\mathrm{div}_{\mu}X_{H}=0$ is essentially the statement of Liouville’s
theorem, while a generic feature of dissipative vector fields is that they
shrink volume in phase space, hence the negative divergence of $K$ is
expected.

To compute the forward operator (112) of the Smoluchowski equation, we
essentially have to calculate the divergence of the vector fields $V_{i}$ with
respect to the Riemannian volume form (111). This can again be done in
coordinates, since the coordinate expressions of both the vector fields
$V_{i}$ and the dual frames $\eta^{j}$ are known from Subsection II.1. A
faster but more abstract method is to use the definition of divergence in
terms of Lie derivatives and Cartan’s magic formula [23]:

$\displaystyle\mathcal{L}_{V_{1}}\mu_{R}$ $\displaystyle=$
$\displaystyle\left(\mathrm{div}_{\mu_{R}}V_{1}\right)\mu_{R}$ (160)
$\displaystyle=$ $\displaystyle
d\iota_{V_{1}}\left(\eta^{1}\wedge\eta^{2}\wedge\eta^{3}\right)$
$\displaystyle=$ $\displaystyle d\left(\eta^{2}\wedge\eta^{3}\right)$
$\displaystyle=$ $\displaystyle d\left(-d\eta^{1}\right)$ $\displaystyle=$
$\displaystyle 0$

Similarly,
$\mathcal{L}_{V_{2}}\mu_{R}=-d\left(\eta^{1}\wedge\eta^{3}\right)=-dd\eta^{2}=0$
and

$\displaystyle\mathcal{L}_{V_{3}}\mu_{R}$ $\displaystyle=$ $\displaystyle
d\left(\eta^{1}\wedge\eta^{2}\right)=-d\left(\frac{1}{\kappa}d\eta^{3}\right)$
(161) $\displaystyle=$
$\displaystyle-\frac{1}{\kappa}(\mathbf{f}_{1}[\kappa]\eta^{1}+\mathbf{f}_{2}[\kappa]\eta^{2})\wedge\left(\eta^{1}\wedge\eta^{2}\right)$
$\displaystyle=$ $\displaystyle 0.$

Thus all $V_{i}$-s are divergence-free with respect to $\mu_{R}$, which
simplifies greatly the calculation of the forward Smoluchowski operator.