# Jointly Equivariant Dynamics for Interacting Particles

Alain Ajami  Université Saint Joseph de Beyrouth, Faculté d’ingégnierie, ESIB,
Lebanon<EMAIL_ADDRESS>Jean-Paul Gauthier  Université de Toulon, LIS,
UMR CNRS 7020, Campus de La Garde, 83041 Toulon, France<EMAIL_ADDRESS>Francesco Rossi  Dipartimento di Matematica “Tullio Levi-Civita”, Università
di Padova, Italy<EMAIL_ADDRESS>
###### Abstract

Let a finite set of interacting particles be given, together with a symmetry
Lie group $G$. Here we describe all possible dynamics that are jointly
equivariant with respect to the action of $G$. This is relevant e.g., when one
aims to describe collective dynamics that are independent of any coordinate
change or external influence.

We particularize the results to some key examples, i.e. for the most basic low
dimensional symmetries that appear in collective dynamics on manifolds.

###### keywords:

Large-scale systems, Lie groups, equivariant dynamics, systems with symmetry

##  1 Introduction

The aim of this article is to characterize all possible interaction models for
$N$ agents evolving on a manifold $M$, that are jointly equivariant with
respect to the action of a group $G$ on $M$. Equivariance means that an
identical action of the group on each agent provides a change of the
interaction vector field that is equivariant with respect to such action. As a
result, the corresponding dynamics is equivariant too, hence the relative
dynamics between agents is unchanged under the group action.

The simplest (and very interesting) example is the case in which a group of
agents evolve on the plane $M=\mathbb{R}^{2}$ and $G=SE(2)$ is the Lie group
of rototranslations: joint equivariance of interactions in this case means
that a rototranslation of the initial data induces the same rototranslation of
the trajectory. More deeply, this implies that one of the agents (or an
external observer subject to the same rototranslation) will perceive the same
trajectory. This example shows the relevance of the research here: to classify
all interactions based on some “natural” features that can be perceived by
agents and not on some “superimposed” structures (such as coordinates), that
are added to the model just for other purposes (e.g. for parametrization).

The case of dynamics that are invariant with respect to rototranslations, that
we will describe in full details in Section 3, has been studied in a
cornucopia of articles and examples. A nonexhaustive bibliography is the
following: [1, 3, 4, 5, 6, 8, 10, 12, 14, 16, 23, 24, 25, 28]. The main result
of our article in this setting is not to propose yet other models, but to
prove a much stronger result: we will provide all possible dynamics in full
generality. Moreover, such general dynamics will be written in some normal
forms, in which the role of each term is identified for the purpose of
modeling.

This example also shows the interest of the classification from a reverse
point of view. If we describe the dynamics of a group of agents with a given
interaction model and remark that it is not jointly equivariant with respect
to a natural group action (e.g. the rototranslations on the plane as above),
this means that an external factor breaks down the natural symmetry of the
group. As an example, several models describing flocks flying on large
distances on the Earth are not jointly equivariant with respect to the natural
group of rotations on the sphere, since they take into account the geomagnetic
field. If the description of the dynamics based on these models is efficient,
this can be seen as a (secondary) evidence of the fact that birds perceive the
magnetic field, i.e. their magnetoreceptivity. See e.g. [29, 31].

The main aim of this article is then twofold. On a more theoretical side, we
aim to find all possible interaction models, under some assumptions of
equivariance or joint equivariance. More precisely, given a manifold $M$ and a
Lie group $G$ acting on it, we want to describe all dynamics of $N$ particles
evolving on $M$ that are jointly equivariant with respect to the action of
$G$. In particular, we want to identify the minimal number of (functional)
parameters that characterize such dynamics. On the more applied side, we aim
to understand the role of such parameters in modeling, i.e. which “part” of
interactions they describe. With this goal, we will then find all interaction
models for key examples determined by triples $(N,M,G)$. Then, we will choose
simple functional parameters (e.g. constant or linear functions) to understand
their role via some simulations.

One of the interesting features of this study is that the results depend on
the number of agents $N$ in an unexpected way. On one side, the case $N=1$ is
often far from being trivial (see for instance Sections 5.3 and 6). On the
other side, some results are proved when $N$ is sufficiently large with
respect to the dimension of the manifold (see Theorem 2). As a consequence,
our methods do not cover “intermediate” numbers of agents, in general, (see
e.g. Remark 40).

All along the article, we will consider interacting agents that may have
distinct features, i.e. with distinct rules for dynamics. Also, in several
examples (starting from statistical mechanics), agents are supposed to satisfy
identical rules for the dynamics. In other terms, agents are only
distinguishable by their state (e.g. position, or position/velocity) and not
by other features. We will call such models permutation-equivariant
populations. From the mathematical viewpoint, this means that the dynamics is
equivariant with respect to permutation of agents: if $x_{i}(0)$ and
$x_{j}(0)$ are exchanged, their trajectories are exchanged too, while
trajectories of other agents do not vary. We will then study the family of
interaction models that satisfy equivariance with respect to both the action
of a connected Lie group and to permutations of agents. Clearly, these
families have much less parameters than the ones studied before, but their
structure is very similar.

As already stated, in this article we deal with a finite number $N$ of
interacting agents. The limit $N\rightarrow+\infty$ of permutation-equivariant
populations, mathematically defined via the mean-field limit, has been hugely
studied in the literature, see e.g.[9, 13, 17, 20, 21, 27]. This setting plays
a key role as a good approximation of the dynamics when the number of agents
is very large, e.g. in bird flocks. We aim to describe the class of admissible
equivariant dynamics in the mean-field limit in a future work.

We will also consider the case of controlled dynamics, in which the dynamics
$F$ is constrained to belong to a subset of the vector fields on $M$. The most
interesting case here is given by nonholonomic systems, in which a constraint
on the velocity of agents is added to the interaction field. As an example,
the unicycle describes the dynamics of an oriented agent, that can move
forward or rotate on itself: this constrained dynamics is also used for the
modeling of pedestrians, see e.g. [2, 11].

All along the article, we will deal with first-order dynamics (eventually with
nonholonomic constraints as explained above). This choice, even though very
common in many interaction models (e.g. [5]), is in contrast with a large
number of other models, in which dynamics are of second-order [1, 3, 7, 30].
The second-order models being chosen for several different reasons:

  * 1.

when velocities play the role of “state variables” for the dynamics, e.g. when
one aims to describe alignment in bird flocks (e.g. in the celebrated Cucker-
Smale model [14]);

  * 2.

when the basic dynamics of agents is of second-order, e.g. when dealing with
classical equations of physics;

  * 3.

when one aims to describe orientation of agents via their velocity vectors.

In this last case, we will show that a different solution can be introduced,
via constrained dynamics. As we already recalled above, one can describe a
dynamics in which the trajectory is always tangent to the orientation of the
body, by adding an angle variable and a nonholonomic constraint. This
description has the advantage of directly describing the desired constraint,
without resorting to higher-order dynamics that are not justified by the
physical model. See more details in Section 7.

There is an enormous amount of mathematical contributions about equivariant
dynamical systems, which theory is well-developed. Among many of them, the
following references are particularly interesting for our setting: [19, 22,
26]. In the present work, we do not use the tools and results from this field.
The originality of our contribution is the choice of the specific diagonal
action for several agents. In a forthcoming paper, we will present (still in
the context of the diagonal action) several results about equilibria, relative
equilibria and stabilization. The present paper also aims to highlight the
mathematical interest of equivariant diagonal actions.

The structure of the article is the following. We give the main definition
related to equivariance in Section 2. We also prove here the main general
theorems about the structure of Families of Population Dynamics and their
properties. We then study several key examples, in which we also highlight the
usefulness of the theory we developed:

  * 1.

In Section 3 we study the most important case: the dynamics on the plane that
are equivariant under rototranslations;

  * 2.

In Section 4 we study the relativistic dynamics, both on the line and on the
plane;

  * 3.

In Section 5 we study the dynamics on both the circle $S^{1}$ and the sphere
$S^{2}$, where the group actions are the natural ones ($SO(2,\mathbb{R})$ and
$SO(3,\mathbb{R})$);

  * 4.

In Section 6, we briefly present the volume-preserving action of $SL(2)$ on
the plane;

  * 5.

In Section 7, we describe equivariant dynamics for a well-known non-holonomic
control system, namely the unicycle. We present both the classical and
relativistic cases.

  * 6.

In Section 8, we briefly introduce the generalization of our theory to quantum
systems and apply it to a system of two quantum agents.

##  2 Definitions and General Results

In this section we define the main objects under study, and provide a few
general results about Population Dynamics.

Notation: The space of smooth vector fields defined on a domain $D$ is denoted
by $\chi(D)$. The Lie derivative of a form $\omega$ along a vector field $v$
will be denoted by $L_{v}\omega$. The matrix multiplication $A.B$ is denoted
with a dot.

###  2.1 Lie theory

In this section, we recall some definitions related to Lie groups and Lie
algebras, also fixing the setting of our work. In particular, we need to work
in the analytic setting to ensure equivalence between integral definitions (in
terms of Lie groups) and their differential counterpart (in terms of Lie
algebras), as it will be clear in the next results.

######  Definition 1 ($G$-space).

Given $G$ a connected Lie group, an analytic manifold $X$ is a $G$-space if
$G$ acts analytically on $X$, with an action denoted by $\Phi(g,x)$.

Let $F$ be an analytic vector field over $X$ and $\mathrm{Exp}(tF(x_{0}))$ be
the associated flow, i.e. the unique solution of the Cauchy problem

$\begin{cases}\dot{x}=F(x),\\\ x(0)=x_{0},\end{cases}$

for times $t\in\mathbb{R}$ for which the solution exists.

The vector field $F$ is said to be $G$-equivariant if it holds

$\Phi(g,\mathrm{Exp}(tF(x)))=\mathrm{Exp}(tF(\Phi(g,x))),$ (2.1)

for all $t,g,x$ such that both sides are defined.

We now prove the equivalent differential version of (2.1), in the setting of
Lie algebras. Additionally, this will prove that the sets on which the left
and right hand sides of (2.1) are defined coincide. Given $l\in\mathfrak{L}$,
the Lie algebra of $G$, we identify it with the vector field $l(x)$ on $X$ as
the following: $l(x)=\frac{d}{dt}_{|t=0}\Phi(\mathrm{Exp}(tl),x)$.

######  Proposition 2.

The vector field $F$ is $G$-equivariant if and only if $[F,l(x)]=0$ for all
$l\in\mathfrak{L}$, $x\in X$.

###### Proof.

For simplicity of exposition, we assume that $G$ is an exponential connected
linear Lie group, $X$ is an open subset of $\mathbb{R}^{k}$, invariant under
the linear $G$-action, i.e. that $\Phi(g,x)=g.x$. In that case, the Lie
algebra $\mathfrak{L}$ of $G$ is a Lie subalgebra of the space
$\mathcal{M}_{k}$ of $k\times k$ real matrices. The equality (2.1) then
rewrites as

$g.\mathrm{Exp}(tF(x))=\mathrm{Exp}(tF(g.x)).$ (2.2)

Necessity: Differentiate (2.2) at $t=0$ to get $g.F(x)=F(g.x)$. Take
$g=\mathrm{Exp}(vl)$ and differentiate with respect to $v$ at $v=0$ to get the
result.

Sufficiency: Assume $[F,l.x]=0$ for all $l\in\mathfrak{L}$ and define

$A(t,x):=\mathrm{Exp}(-tl).F(\mathrm{Exp}(tl).x)=\sum_{r=0}^{+\infty}(ad(l)^{r}F)(x)\frac{t^{r}}{r!},$

where $ad(l)^{r}F:=[l,ad(l)^{r-1}F]$ for $r\in\mathbb{N}\setminus\\{0\\}$ and
$ad(l)^{0}F=F$. Here, the series is absolutely convergent in $t$, due to
analyticity. Then $A(t,x)=F(x)$, thus
$F(\mathrm{Exp}(tl).x)=\mathrm{Exp}(tl).F(x)$ for all $x,t$. It means that
$F(g.x)=g.F(x)$, or equivalently

$F(x)=g^{-1}.F(g.x)\mbox{~{}~{}for all~{}~{}}g\in G.$ (2.3)

For a fixed $g\in G$, set $B(t,x)=g^{-1}.\mathrm{Exp}(tF(g.x))$. We then have:

$\partial_{t}B=g^{-1}.F(\mathrm{Exp}(tF(g.x)))=g^{-1}.F(g.B)=F(B),$ (2.4)

where the last identity comes from (2.3). By observing that $B(0,x)=x$, we
have that (2.4) implies $B=\mathrm{Exp}(tF(x))$, that in turn proves (2.2),
for small $t$. Moreover, this also shows that both sides of the equality
extend to the same $t$. ∎

We now define radial functions and characterize them in differential terms.

######  Definition 3.

An analytic function $\phi:D_{\phi}\subset X\to\mathbb{R}$, with $D_{\phi}$
being an open dense $G$-space, is radial if it is constant on the $G$-orbits
in $X$.

######  Proposition 4.

The function $\varphi$ is a radial function if and only if

$L_{lx}\varphi=0\mbox{~{}~{}for all~{}}l\in\mathfrak{L}.$

The space of radial functions is an Abelian ring, that we denote by
$\mathcal{R}$.

###### Proof.

For $l\in\mathfrak{L}$ and for all $t,x$, it holds
$\varphi(\mathrm{Exp}(t.lx))=\sum_{r=0}^{+\infty}L_{lx}^{r}(\varphi)(x)\frac{t^{r}}{r!}$.
Independence of both sides with respect to $t$ is equivalent to have
$L_{lx}^{r}(\varphi)(x)=0$ for all $x\in D_{\phi}$ and $r\geq 1$, that is in
turn equivalent to have $L_{lx}(\varphi)(x)=0$ for all $x\in D_{\phi}$.

It is easy to prove that $\phi,\psi$ radial implies both $\phi+\psi$ radial,
by linearity of the derivative, and $\phi\psi$ radial, by the Leibnitz rule.
Commutativity is standard. ∎

###  2.2 Population Dynamics

We now focus on Population Dynamics, i.e. on dynamics of a set of interacting
agents all moving on the same manifold $M$. All along the article, we will
denote by $n$ the dimension of such manifold, while
$N\in\mathbb{N}\backslash\\{0\\}$ will be the number of agents in the
population. The space $X$ on which the dynamics take place will then be an
open dense analytic submanifold of $M^{N}$.

We first define the diagonal action, both of a Lie group and of its Lie
algebra, as well as diagonal $G$-spaces.

######  Definition 5.

Let $G$ be a connected Lie group acting on a manifold $M$ with action $\Phi$,
and $\mathfrak{L}$ its Lie algebra. Given a number of agents
$N\in\mathbb{N}^{\ast}$, we define:

  * 1.

the diagonal action of $G$ as the action of $G$ on $M^{N}$ given by

$\Phi(g)(X_{1},\ldots,X_{N}):=(\Phi(g)X_{1},\ldots,\Phi(g)X_{N});$

  * 2.

the diagonal extension of the infinitesimal action of $\mathfrak{L}$ as
follows: any $l\in\mathfrak{L}$ defines a vector field
$l(X)\frac{\partial}{\partial X}$ on $M$, and we consider the corresponding
vector field on $M^{N}$ given by $\hat{l}=l_{1}+\ldots+l_{n}$ with
$l_{i}=l(X_{i})\partial_{X_{i}}$. The Lie algebra of such vector fields is
denoted by $\hat{\mathfrak{L}}$;

  * 3.

a diagonal $G$-space as a $G$-space $D\subset M^{N}$ being invariant under the
diagonal action of $G$;

  * 4.

the ring of $\hat{\mathfrak{L}}$-jointly radial functions (eventually
$\hat{\mathfrak{L}}_{N}$-jointly radial functions if $N$ needs to be
specified) is the Abelian ring of analytic functions
$\varphi:D_{\varphi}\subset M^{N}\rightarrow\mathbb{R}$, with open dense
diagonal $G$-space $D_{\varphi}$ that are radial with respect to the diagonal
action of $G$. Equivalently, by Proposition 4, jointly radial functions are
solutions of

$L_{\widehat{l}}\varphi=0\mbox{~{}~{}for
all~{}~{}}\hat{l}\in\hat{\mathfrak{L}}.$ (2.5)

We now define Population Dynamics (PD) and the corresponding Family of
Population Dynamics (FPD).

######  Definition 6.

A Population Dynamics (PD for short) is defined by the elements $(N,M,G,D,F)$,
where:

  * 1.

$N\in\mathbb{N}\backslash\\{0\\}$ is the number of agents in the population;

  * 2.

$M$ is a differentiable manifold, and each agent $X_{i}$ with $i=1,\ldots,N$
evolves on it;

  * 3.

$G$ is a connected Lie group, acting on $M$ via an action $\Phi$;

  * 4.

$D\subset M^{N}$ is a diagonal $G$-space;

  * 5.

$F$ is an analytic vector field over $D$, that is $G$-equivariant with respect
to the diagonal action of $G$, i.e. each component $F_{i}$ of the vector field
$F$ (acting on the $i$-th agent) satisfies

$T\Phi(g)F_{i}(x_{1},\ldots,x_{N})=F_{i}(\Phi(g)x_{1},\ldots,\Phi(g)x_{N})\text{
for all}\in G.$ (2.6)

  * 6.

the domain of $F$ is maximal, i.e. there exists no vector field $\hat{F}$ with
a domain $\hat{D}\supset D$ being a diagonal $G$-space such that
$F(x)=\hat{F}(x)$ for all $x\in D$.

Given $(N,M,G)$ as above, the corresponding Family of Population Dynamics (FPD
for short) is the largest family $\mathbb{F}$ of smooth vector fields such
that for all $F\in\mathbb{F}$ it holds:

  * 1.

the domain of $F$ is a diagonal $G$-space $D_{F}\subset M^{N}$;

  * 2.

$(N,M,G,D_{F},F)$ is a population dynamics.

The main goal of the article is to characterize FPD’s: given $(N,M,G)$ fixed,
we aim to compute and describe all the possible associated Population
Dynamics. We start by characterizing them in terms of Lie algebras. This
provides a Lie bracket condition for a vector field $F$ to be a PD.

######  Corollary 7.

Let $(N,M,G,D)$ be given as in Definition 6, with $D$ connected. Let $F$ be a
smooth vector field in ${\LARGE\chi}(D)$. Let $\mathfrak{L}$ be the Lie
algebra of $G$ and $\hat{\mathfrak{L}}$ its diagonal extension. Then, $F$ is
jointly equivariant with respect to the action of $G$ if and only if

$[F,\hat{l}]=0,\text{ for all }\hat{l}\in\hat{\mathfrak{L}}.$ (2.7)

######  Remark 8.

Even though we will mainly deal with a whole domain $D=M^{N}$, the case of $D$
smaller is important too. The most relevant example is given by interaction
based on potentials with singularities to prevent particle collisions. In this
case, it holds $D=M^{N}\setminus\cup_{i\neq j}\\{X_{i}=X_{j}\\}$.

As a consequence, the FPD associated with $(N,M,G)$ is the centralizer of
$\hat{\mathfrak{L}}$ in the space of vector fields defined on open dense
diagonal $G$-spaces of $M^{N}$. In particular it has the structure of a Lie
Algebra.

###### Proof.

Apply Proposition 2 to the integral formulation (2.6). ∎

We now aim to describe the components of FPD.

######  Definition 9.

Denote by $\SS$ the space of solutions of FPD by components, i.e. the space of
functions $f:M^{N-1}\to\chi(M)$ such that

$F(X_{1},\ldots,X_{N})=(F_{1}(\cdot,X_{2},\ldots,X_{N}),F_{2}(X_{1},\cdot,X_{3},\ldots,X_{N}),\ldots,F_{N}(X_{1},\ldots,X_{N-1},\cdot))$

is a PD and $f=F_{i}$ for one of the components, considered as a vector field
on $M$ for the $X_{i}$ variable, with
$X_{1},\ldots,X_{i-1},X_{i+1},\ldots,X_{N}$ seen as parameters.

We have the following main result.

######  Theorem 1.

The FPD is a module over the ring of $\hat{\mathfrak{L}}$-jointly radial
functions. The space of solutions $\SS$ is also a module over the same ring.

Given $M$ a manifold of dimension $n$ and $f^{1},\ldots,f^{n}$ solutions,
independent at each point of an open dense $G$-space, then the module is free
and all solutions are of the form $\sum_{i=1}^{n}\phi^{i}f^{i}$ with
$\phi^{i}$ jointly radial functions.

###### Proof.

We prove the first statement. Let $F$ be a PD and $\phi$ a jointly radial
function. It holds

$[\phi F,\hat{l}]=\phi[F,\hat{l}]-L_{\hat{l}}\phi F=\phi\cdot 0-0\cdot F=0,$

i.e. $\phi F$ is a PD. The proof of the second statement is a direct
consequence: given $F=(F_{1},\ldots,F_{N})$ a PD and $f=F_{i}$ for some $i$ be
a solution, then $\phi F$ with $\phi$ jointly radial is a PD and $\phi f=\phi
F_{i}$ is a solution too.

We prove the last statement. Let $F=(F_{1},\ldots,F_{n})$ be a PD and
$f=F_{i}$ for some $i$ be a solution. Since it is of the form
$f:M^{N-1}\to\chi(M)$ and we have a family of $n$ independent
$f^{i}:M^{N-1}\to\chi(M)$, we can always write in a unique way
$f=\sum_{i=1}^{n}\phi^{i}f^{i}$ with $\phi^{i}:M^{N}\to\mathbb{R}$ being some
functions. We need to prove that such $\phi^{i}$ are jointly radial. We write
$[F,\hat{l}]=0$ and consider its $i$-th component, that reads as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{N}\partial_{X_{k}}\left(\sum_{j=1}^{n}\phi^{j}f^{j}\right).l(X_{k})-\partial_{X_{i}}l.\left(\sum_{j=1}^{n}\phi^{j}f^{j}\right)=$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{N}\sum_{j=1}^{n}(\partial_{X_{k}}(\phi^{j}).l(X_{k}))f^{j}+\sum_{j=1}^{n}\phi^{j}\left(\sum_{k=1}^{N}\partial_{X_{k}}f^{j}.l(X_{k})-\partial_{X_{i}}l.f^{j}\right)$

The last term is zero, since $f^{j}$ are solutions. By independence of the
$f^{j}$, the first term now becomes
$\sum_{k=1}^{N}(\partial_{X_{k}}(\phi^{j}).l(X_{k}))=0$, i.e. $\phi^{j}$ is
jointly radial. ∎

From now on, we will consider solutions of (2.7) as smooth vector fields
$F\in{\LARGE\chi}(D_{F})$ defined on a diagonal $G$-space $D_{F}\subset M^{N}$
that is open and dense. We also identify solutions of (2.7) if they coincide
over an open dense connected diagonal $G$-space.

As recalled in the introduction, the case of $N$ agents that are
indistinguishable with respect to the dynamics is very important for modeling
of population dynamics. We will say Permutation-Equivariant Population
Dynamics from now on. In mathematical terms, this means that permutation of
agents $i,j$ induces the permutation of their trajectories and no variation in
the dynamics of other agents. The formal definition is given here.

######  Definition 10.

We say that a Population Dynamics $(N,M,G,D,F)$ is Permutation-Equivariant
(PEPD for short) if for each $i\neq j\in\\{1,\ldots,N\\}$ and $k\neq i,j$ it
holds

$\displaystyle F_{i}(X_{1},\ldots,X_{i},\ldots,X_{j},\ldots,X_{n})$
$\displaystyle=F_{j}(X_{1},\ldots,X_{j},\ldots,X_{i},\ldots,X_{n}),$ (2.8)
$\displaystyle F_{k}(X_{1},\ldots,X_{i},\ldots,X_{j},\ldots,X_{n})$
$\displaystyle=F_{k}(X_{1},\ldots,X_{j},\ldots,X_{i},\ldots,X_{n}),$

Given $(N,M,G)$ as above, the corresponding Family of Permutation-Equivariant
Population Dynamics (FPEPD for short) is the largest family $\mathbb{F}$ of
smooth vector fields such that for all $F\in\mathbb{F}$ it holds:

  * 1.

the domain of $F$ is a diagonal $G$-space, also being invariant under
permutations;

  * 2.

$(N,M,G,D,F)$ is a permutation-equivariant population dynamics.

Examples of Permutation-Equivariant Population Dynamics are ubiquitous in
literature, see e.g. [1, 3, 5, 7, 8, 9, 14, 24, 30]. We will study examples of
PEPDs in the remaining of the article.

We now define Controlled Population Dynamics, i.e. dynamics in which an
additional constraint on the dynamics is imposed. This is often the case of
control systems, in which the admissible vector fields are just a subset of
all vector fields on a manifold. As already stated, this is the case of
nonholonomic constraints in the context of mechanical engineering, robotics,
or human locomotion models [2, 11, 18].

######  Definition 11.

We say that $(N,M,G,D,F,\Omega)$ is a controlled population dynamics if:

  * 1.

$(N,M,G,D,F)$ is a population dynamics;

  * 2.

$\Omega$ is a subset of the Lie Algebra of (open-densely defined) vector
fields over $M$ and $F\in\Omega$.

Given $(N,M,G,\Omega)$ as above, the corresponding Family of Controlled
Population Dynamics (controlled FPD for short) is the largest family
$\mathbb{F}$ of smooth vector fields such that for all $F\in\mathbb{F}$ it
holds:

  * 1.

the domain of $F$ is a diagonal $G$-space;

  * 2.

$(N,M,G,D,F)$ is a controlled population dynamics.

######  Remark 12.

It is clear that a controlled FPD is not a Lie algebra in general. However, it
is the case for a controlled FPD associated with $(N,M,G,\Omega)$ if the
constraint $\Omega$ is a Lie algebra itself.

###  2.3 Families of Population Dynamics with linear group actions

In this section, we study the algebraic condition (2.7) when $G$ is a linear
group acting on $M=\mathbb{R}^{n}$ (or an open dense $G$-invariant subset of
it). In this setting, the Lie algebra $\mathfrak{L}$ of $G$ is a Lie
subalgebra of $\mathcal{M}_{n}$, the Lie algebra of $n\times n$ matrices.
Given $l\in\mathfrak{L}$, we denote by $l.X$ the corresponding linear vector
field over $\mathbb{R}^{n}$. We then write $X=(X_{1},\ldots,X_{N})$ and
$F=(F_{1},\ldots,F_{n})$ with $F_{i}=f_{i}(X)\partial_{X_{i}}$ defined by $N$
functions $f_{i}:D_{F}\rightarrow\mathbb{R}^{n}$. Condition (2.7) is
nowrestated as the following set of $n$ first order linear partial
differential equations on $\mathbb{R}^{nN}$, parametrized by
$l\in\mathfrak{L}$: each $f_{1},\ldots,f_{N}$ is a solution $f$ of the
following equation:

$\partial_{X_{1}}f(X).lX_{1}+\partial_{X_{2}}f(X).lX_{2}+\ldots+\partial_{X_{N}}f(X).lX_{N}=l.f(X)\qquad\mbox{~{}for
all }l\in\mathfrak{L}.$ (2.9)

Following the same requirements for $F$, we only consider smooth solutions
$(f_{1},\ldots,f_{N})$, and we identify two solutions if they coincide over an
open dense diagonal $G$-space.

We now aim to describe the set of solutions of (2.9), that is the main objects
under study in this article. We then define three important sets.

######  Definition 13.

Given $\mathcal{M}_{n}$ the Lie algebra of $n\times n$ matrices, let
$\mathfrak{L}$ be a Lie subalgebra and $\mathfrak{C}$ its centralizer.

  * 1.

We recall that $\mathcal{R}$ is the ring of $\hat{\mathfrak{L}}$-jointly
radial functions and $\SS$ is the $\mathcal{R}$-module of solutions, that we
identify with the set of functions $\varphi:D_{\varphi}\to\mathbb{R}^{n}$ that
are solutions of (2.9).

  * 2.

We denote by $\mathcal{C}$ the set of solutions given by the centralizer
$\mathfrak{C}$, i.e. the $\mathcal{R}$-module generated by functions

$\left\\{c.X_{i}\mbox{~{}~{}such that~{}~{}}i\in\\{1,\ldots,N\\}\mbox{ and
}c\in\mathfrak{C}\right\\},$

where we denote by $c.X_{i}$ the function $(X_{1},\ldots,X_{N})\mapsto
c.X_{i}$.

  * 3.

We denote by $\mathcal{E}$ the set of elementary solutions, i.e. the
$\mathcal{R}$-module generated by $\left\\{X_{1},\ldots,X_{N}\right\\},$ where
we denote by $X_{i}$ with the function $(X_{1},\ldots,X_{N})\mapsto X_{i}$.

We now state some useful properties for these sets.

######  Proposition 14.

Let $\mathfrak{L},\mathfrak{C},\mathcal{R},\SS,\mathcal{C},\mathcal{E}$ be
defined as in Definition 13. Then, it holds
$\mathcal{E}\subset\mathcal{C}\subset\SS$, i.e. elementary solutions and
solutions given by the centralizer are actually solutions.

###### Proof.

The inclusion $\mathcal{E}\subset\mathcal{C}$ is a direct consequence of the
fact that $\mathrm{Id}\in\mathfrak{C}$. For $\mathcal{C}\subset\SS$, fix
$c\in\mathfrak{C}$ and define $f=c.X_{i}$. Then, we rewrite (2.9) for $f$ as
$c.l.X_{i}-l.(c.X_{i})=0$ for all $l\in\mathfrak{L}$. This is exactly the
condition for $c$ being in the centralizer of $\mathfrak{L}$ in
$\mathcal{M}_{n}$. As a consequence, the generators of $\mathcal{C}$ belong to
$\SS$, thus the inclusion is a consequence of the fact that $\SS$ is a
$\mathcal{R}$-module. ∎

We recall that our goal is to describe all FPD associated to a given triple
$(N,M,G)$, that is equivalent to describe the set $\SS$ of all solutions of
(2.9). It is then very useful to study the case in which all solutions are
reduced either to solutions given by the centralizer (i.e. $\mathcal{C}=\SS$)
or, even better, to elementary solutions (i.e. $\mathcal{E}=\SS$). In both
cases, the advantage is that the description of $\mathcal{C}$ or $\mathcal{E}$
is somehow explicit. The main result of this section, that we now state,
describes some important cases in which one of the two identities holds.

######  Theorem 2.

Let $G$ be a Lie group acting linearly on $M=\mathbb{R}^{n}$. Let
$\mathfrak{L},\mathfrak{C},\mathcal{R},\SS,\mathcal{C},\mathcal{E}$ defined as
in Definition 13. Then:

  1. Item 1.

If the space $\\{c.X,c\in\mathfrak{C}\\}$ has rank $n$ at some point
$X\in\mathbb{R}^{n}$, then $\SS=\mathcal{C}$ and it is a free
$\mathcal{R}$-module; moreover, given $\\{c_{1},\ldots,c_{n}\\}$ independent
generators of $\mathfrak{C}$ and $i\in\\{1,\ldots,N\\}$ a fixed index, a basis
of $\SS$ is given by $\\{c_{1}X_{i},\ldots,c_{n}X_{i}\\}$.

  2. Item 2.

If $N\geq n$, then $\SS=\mathcal{E}$ and it is a free $\mathcal{R}$-module,
with basis $\\{X_{i_{1}},\ldots,X_{i_{n}}\\}$ with $i_{1},\ldots,i_{n}$ being
any set of $n$ distinct indexes in $\\{1,\ldots,N\\}$.

######  Remark 15.

The choice of the basis in Item 2 has a direct impact on the ring
$\mathcal{R}$: its elements are functions whose domain is a $G$-space, that
depends on the selection of the basis. In any case, the FPD is completely
characterized on such domain.

###### Proof.

Item 1. Let $c_{1},\ldots,c_{n}$ be independent generators of $\mathfrak{C}$
such that $\\{c_{i}.X\\}$ has rank $n$ at some $X$, then on an open set
$\mathcal{O}\subset\mathbb{R}^{n}$, which is also dense due to the algebraic
character of linear dependence. Since $c_{1}\cdot X_{i},\ldots,c_{n}\cdot
X_{i}$ are all solutions due to Proposition 14 and are independent by
hypothesis, we apply Theorem 1 and get the result.

Item 2. For simplicity of notation, we choose $\\{i_{1},\ldots,i_{n}\\}$ to be
the first $n$ indexes. Consider the set $\mathcal{O}\subset\mathbb{R}^{nN}$ of
$X=(X_{1},\ldots,X_{n},X_{n+1},\ldots,X_{N})$ such that
$X_{1},\ldots,X_{n}\in\mathbb{R}^{n}$ are $n$ independent vectors. It is clear
that the set $\mathcal{O}$ is open dense. Restrict $F$ to $\mathcal{O}$ and,
similarly to Item 1, observe that $X_{1},\ldots,X_{n}$ are independent and are
solutions. Apply again Theorem 1 and get the result. ∎

######  Remark 16.

We see that, for small $N$, it may happen that $\SS$ properly contains
$\mathcal{E}$. It is the case for instance if $N=1$ and $\mathfrak{C}$ is not
reduced to $\mathbb{R}.\mathrm{Id}$, which holds for instance with $G$ the
group $SO(2,\mathbb{R})$ acting on the space $\mathbb{R}^{2}\setminus\\{0\\}$.
See Remark 44 below.

####  2.3.1 Time reparametrization

We now discuss the problem of time reparametrization of PD. Indeed, it is
clear that, if $F$ is a PD, then $\psi F$ is not a PD, in general. However,
Proposition 14 implies that $\psi F$ is a PD as soon as $\psi$ is
$\hat{\mathfrak{L}}$-jointly radial. This is particularly interesting for
Euclidean and pseudo-Euclidean manifolds, that admit a natural arclength
parametrization of trajectories.

######  Corollary 17.

Let $G$ be a linear group acting on $M=\mathbb{R}^{n}$. Let $F$ be a
Population Dynamics, and $\psi$ be a $\hat{\mathfrak{L}}$-jointly radial
function. Then $\psi F$ is a Population Dynamics.

Moreover, denote by $\|F\|$ the standard product Euclidean norm on
$\mathbb{R}^{nN}$ and let $G$ be a group of linear isometries. Then, the
functions $\phi=\|F\|^{\alpha}$ are jointly radial for any
$\alpha\in\mathbb{Z}$. In particular, the vector field
$\tilde{F}:=\frac{F}{\|F\|}$ is again a PD, outside equilibria of $F$. The
trajectories $\tilde{X}(s)$ of $\tilde{F}$ (that geometrically coincide with
trajectories $X(t)$ of $F$ outside equlibria) are parametrized by arclength
$ds=\sqrt{\sum_{i=1}^{N}||F_{i}(X(t))||^{2}}dt$.

The same results hold when $M$ is a pseudo-Euclidean space and $G$ is a group
of linear pseudo-isometries. Reparametrization of trajectories only holds if
$\|F\|^{2}>0$.

###### Proof.

The first statement is a consequence of the fact that $\SS$ is a
$\mathcal{R}$-module, proved in Theorem 1.

We prove the second statement for $\alpha=2$ only, the proof for other values
being a direct consequence. Denote by $<.,.>$ the standard Euclidean scalar
product on $\mathbb{R}^{nN}$. Since $G$ is a linear isometry, then its Lie
algebra is given by elements of the form $AX$ with $A$ skew-symmetric. It then
holds

$L_{\widehat{AX}}(\|F\|^{2})=L_{\widehat{AX}}(<F,F>)=2<L_{\widehat{AX}}F,F>=2\sum_{i=1}^{N}<A\cdot
F_{i},F_{i}>=0.$

∎

####  2.3.2 Permutation-Equivariant Population Dynamics with linear group
action

We now discuss the structure of Permutation-Equivariant Population Dynamics,
when $G$ is a linear group acting on $M=\mathbb{R}^{n}$. By using the notation
above, we write $X=(X_{1},\ldots,X_{N})$ and $F=(F_{1},\ldots,F_{n})$ with
$F_{i}=f_{i}(X)\partial_{X_{i}}$ defined by $N$ functions
$f_{i}:D_{F}\rightarrow\mathbb{R}^{n}$. It is clear that the vector field $F$
must be a PD, hence that (2.7) has to be satisfied. Moreover, it is also clear
that, as soon as one among the components $f_{i}$ is defined, all other
components $f_{j}$ with $j\neq i$ are directly defined by (2.8). For a similar
reason, the structure of $f_{i}$ has to be permutation-equivariant with
respect to indexes different from $i$, i.e.

$f_{i}(X_{1},\ldots,X_{j},\ldots,X_{k},\ldots,X_{N})=f_{i}(X_{1},\ldots,X_{k},\ldots,X_{j},\ldots,X_{N})\hskip
8.53581pt\mbox{for all}\hskip 2.84526ptj\neq k\hskip
2.84526pt\mbox{with}\hskip 2.84526ptj,k\neq i.$ (2.10)

A direct consequence of this discussion is the following result.

######  Corollary 18.

Let $G$ be a Lie group acting linearly on $M=\mathbb{R}^{n}$. Let
$\mathfrak{L},\mathfrak{C}$ be defined as in Definition 13. Let the space
$\\{c.X,c\in\mathfrak{C}\\}$ have rank $n$ at some point, with generators
$c_{1}X,\ldots,c_{n}X$.

Denote by $\check{X}_{i}:=(X_{1},\ldots,X_{i-1},X_{i+1},\ldots,X_{N})$, i.e.
the configuration of all agents except $X_{i}$. Then, the Family of PEPD is
given by $F=(F_{1},\ldots,F_{n})$ with $F_{i}=f_{i}(X)\partial_{X_{i}}$, where

$f_{i}(X_{i},\check{X}_{i})=c_{1}X_{i}\psi_{1}(X_{i},\check{X}_{i})+\ldots+c_{n}X_{i}\psi_{n}(X_{i},\check{X}_{i})$

with $\psi_{k}$ being analytic functions of $N$ variables, jointly radial on
all variables and permutation invariant in the last $N-1$ ones.

###  2.4 Translation equivariance in Euclidean and pseudo-Euclidean spaces

In this section, we study two key aspects of the dynamics on Euclidean and
pseudo-Euclidean spaces, in which the group $G$ of equivariance contains
translations. The case is very relevant for models of population dynamics, as
already described in the introduction. In this setting, we extend the previous
results, moreover reducing the number of variables from $N$ to $N-1$.

We then assume to have a group $G=K\ltimes\mathbb{R}^{n}$ being the semi-
direct product of a connected Lie group $K$ by the Euclidean space
$M=\mathbb{R}^{n}$. We assume the following action: given $g=(M,b)$ and
$x\in\mathbb{R}^{n}$, it holds $g.x:=Mx+b$. As a consequence, equivariance
with respect to $G$ includes equivariance with respect to translations. This
in turn forces all $F_{i}$ to be of the following form:

$F_{i}=F_{i}(X_{1}-X_{j},\ldots,X_{N}-X_{j})\text{ for some }j.$ (2.11)

This already has a simple consequence.

######  Proposition 19.

For $N=1$ (a single particle), the FPD associated to
$(1,\mathbb{R}^{n},K\ltimes\mathbb{R}^{n})$ contains constant vector fields
only (eventually being smaller).

###### Proof.

A Population Dynamics $F$ necessarily has to be independent on $X_{1}$, due to
(2.11), i.e. constant.∎

From now on in this section, for simplicity in exposition, we chose $j=1$ in
(2.11). We then define the difference variables with respect to $X_{1}$, that
are coordinates in $\mathbb{R}^{n(N-1)}$, as follows:

$Z=(Z_{2},\ldots,Z_{N}):=(X_{2}-X_{1},\ldots,X_{N}-X_{1}).$ (2.12)

We are now ready to restate condition (2.7) for vector fields $F$ being
jointly equivariant with respect to the action of $K$ only, since equivariance
with respect to translations is taken into account by (2.11). We again write
$F=(F_{1},\ldots,F_{n})$ with $F_{i}=f_{i}(X)\partial_{X_{i}}$ defined by $N$
functions $f_{i}:D_{F}\rightarrow\mathbb{R}^{n}$. Since $K$ acts linearly on
$\mathbb{R}^{N}$, then each $f_{i}$ solves (2.9) for all $l\in\mathcal{K}$,
where $\mathcal{K}$ is the Lie algebra of $K$. We now recall that $f=f(Z)$,
hence it is easy to observe that
$\partial_{X_{1}}f=-\sum_{i=2}^{n}\partial_{Z_{i}}f$, while
$\partial_{X_{j}}f=\partial_{Z_{j}}f$ for $j\neq 1$. Then (2.9) in this
context reads as

$\left(-\sum_{i=2}^{n}\partial_{Z_{i}}f(Z).lX_{1}\right)+\partial_{Z_{2}}f(Z).lX_{2}+\ldots+\partial_{Z_{N}}f(Z).lX_{N}=l.f(Z)\qquad\mbox{~{}for
all }l\in\mathcal{K},$

hence by linearity

$\partial_{Z_{2}}f(Z).lZ_{2}+\ldots+\partial_{Z_{N}}f(Z).lZ_{N}=l.f(Z)\qquad\mbox{~{}for
all }l\in\mathcal{K}.$ (2.13)

This equation has the same structure of (2.9) with respect to the variables
$Z_{2},\ldots,Z_{N}$ and with Lie algebra $\mathcal{K}$. Then, results of
Section 2.3 can be translated to this context.

######  Corollary 20.

Time reparametrization in Euclidean and pseudo-Euclidean spaces described in
Corollary 17 hold if $G$ is an affine (pseudo-)isometry too.

######  Corollary 21.

Let $G=K\ltimes\mathbb{R}^{n}$ being the semi-direct product of a connected
Lie group $K$ by the Euclidean space $M=\mathbb{R}^{n}$, acting as follows:
given $g=(M,b)$ and $x\in\mathbb{R}^{n}$, it holds $g.x:=Mx+b$. Let
$\mathcal{K}$ be the Lie algebra of $K$, $\mathfrak{C}$ be its centralizer.
Let $\mathcal{R}$ be the ring of $\hat{\mathcal{K}}$-jointly radial functions
of $N-1$ variables $Z_{2},\ldots,Z_{N}$, that is the Abelian ring of analytic
functions
$\varphi:D_{\varphi}\subset\mathbb{R}^{n(N-1)}\rightarrow\mathbb{R}$, with
open dense diagonal $K$-space $D_{\varphi}$ that are radial with respect to
the diagonal action of $K$. Equivalently, by Proposition 4, jointly radial
functions are solutions of

$L_{\widehat{lZ}}\varphi=0\mbox{~{}~{}for
all~{}~{}}\hat{l}\in\hat{\mathcal{K}}.$ (2.14)

Denote by $\SS$ the set of functions $\varphi:D_{\varphi}\to\mathbb{R}^{n}$
that are solutions of (2.13). Denote by $\mathcal{C}$ the set of solutions
given by the centralizer $\mathfrak{C}$, i.e. the $\mathcal{R}$-module
generated by functions

$\left\\{c.Z_{i}\mbox{~{}~{}such that~{}~{}}i\in\\{2,\ldots,N\\}\mbox{ and
}c\in\mathfrak{C}\right\\}.$

Denote by $\mathcal{E}$ the set of elementary solutions, i.e. the
$\mathcal{R}$-module generated by $\left\\{Z_{2},\ldots,Z_{N}\right\\}.$

Then:

  1. Item 1.

If the space $\\{c.Z,c\in\mathfrak{C}\\}$ has rank $n$ at a point
$Z\in\mathbb{R}^{n}$, then $\SS=\mathcal{C}$ and it is a free
$\mathcal{R}$-module; moreover, given $\\{c_{1},\ldots,c_{n}\\}$ independent
generators of $\mathfrak{C}$ and $i\in\\{2,\ldots,N\\}$ a fixed index, a basis
of $\SS$ is given by $\\{c_{1}Z_{i},\ldots,c_{n}Z_{i}\\}$.

  2. Item 2.

If $N-1\geq n$, then $\SS=\mathcal{E}$ and it is a free $\mathcal{R}$-module,
with basis $\\{Z_{i_{1}},\ldots,Z_{i_{n}}\\}$ with $i_{1},\ldots,i_{n}$ being
any set of $n$ distinct indexes in $\\{2,\ldots,N\\}$.

In both cases, proofs are direct translations of the corresponding results.

We also consider the case of Permutation-Equivariant Population Dynamics. We
have the following result.

######  Corollary 22 (Translation-equivariant PEPDs).

Let $G=K\ltimes\mathbb{R}^{n}$ be as in Corollary 21, with $\mathcal{K}$ the
Lie algebra of $K$ and $\mathfrak{C}$ its centralizer. Let the space
$\\{cX,c\in\mathfrak{C}\\}$ have rank $n$ at some point $X\in\mathbb{R}^{n}$,
with $c_{1}X,\ldots,c_{n}X$ being a basis. Define

$\check{Z}_{i}:=(X_{1}-X_{i},\ldots,X_{i-1}-X_{i},X_{i+1}-X_{i},\ldots,X_{N}-X_{i}).$

Then, the FPEPD is given by
$F=\sum_{i=1}^{N}H(\check{Z_{i}})\partial_{X_{i}}$, where

$H(V_{2},\ldots,V_{N})=c_{1}(V_{2}+\ldots+V_{N})\phi_{1}(V_{2},\ldots,V_{N})+\ldots+c_{n}(V_{2}+\ldots+V_{N})\phi_{n}(V_{2},\ldots,V_{N})$

with $\phi_{1},\ldots,\phi_{n}$ being analytic, jointly radial and permutation
invariant. The formula holds in the $G$-diagonal invariant set
$\mathbb{R}^{nN}\setminus(\cup_{i=1}^{N}\\{(X_{1}+\ldots+X_{N})-NX_{i}=0\\})$.

###### Proof.

If $\\{cX,c\in\mathfrak{C}\\}$ has rank $n$ in a point, then it has rank $n$
in an open dense set, as in the proof of Theorem 2. By removing the union of
hyperplanes $\cup_{i=1}^{N}\\{(X_{1}+\ldots+X_{N})-NX_{i}=0\\}$, we are left
again with an open dense set $\mathcal{O}\subset\mathbb{R}^{n}$. Outside this
set, for each $i=1,\ldots,N$ it holds that

$c_{1}(V_{2}+\ldots+V_{N}),\ldots,c_{n}(V_{2}+\ldots+V_{N})$

is a basis of $\mathbb{R}^{n}$ for any choice of
$\check{Z}_{i}=(V_{2},\ldots,V_{N})$. By writing
$F=\sum_{i=1}^{N}F_{i}(\check{Z_{i}})\partial_{X_{i}}$, it holds

$F_{1}(\check{Z}_{1})=c_{1}(Z_{2,1}+\ldots+Z_{N,1})\phi_{1}(\check{Z}_{1})+\ldots+c_{n}(Z_{2,1}+\ldots+Z_{N,1})\phi_{n}(\check{Z}_{1}),$

for some functions $\phi_{1},\ldots,\phi_{N}$, as in the proof of Theorem 2,
Item 1, where $Z_{j,k}:=X_{j}-X_{k}$. Since the term $Z_{2,1}+\ldots+Z_{N,1}$
is permutation invariant, then $F_{1}$ is permutation invariant when the
$\phi_{i}$ are permutation invariant too. This completely determines
$H(\check{Z}_{1}):=F_{1}(\check{Z}_{1})$. Then, $F_{2},\ldots,F_{N}$ are
completely determined by permutation equivariance, i.e.
$F_{i}(\check{Z}_{i})=H(\check{Z}_{i})$. ∎

###  2.5 Population Dynamics on Lie groups

In this section, we describe Population Dynamics in the following interesting
case: the Lie group $G$ is both the state space for each agent and the group
of equivariance. In this case, the manifold is then $M=G^{N}$, where $N$ is
the number of agents. We will provide an example in Section 5.1, in which the
group is $G=SO(2,\mathbb{R})$ the group of rotations of the plane, that is
identified with the circle $S^{1}$. The result will also play a role in
studying the Constrained Population Dynamics of Section 7.

We have the following description of Families of Population Dynamics.

######  Proposition 23 (FPDs on Lie groups).

Let $G=M$ be a Lie group, acting on itself by left translation: $G\times M\to
M$ maps $(g,h)$ to $gh$. Then, a function $\phi:G^{N}\to\mathbb{R}$ is jointly
radial if and only if it satisfies

$\phi(g_{1},\ldots,g_{N})=\psi(g_{1}^{-1}g_{2},\ldots,g_{1}^{-1}g_{N})$

for an analytic function $\psi:G^{N-1}\to\mathbb{R}$.

Consider the FPD associated to $(N,M,G)$. It is composed by vector fields
$F=\sum_{i=1}^{N}F_{i}$, where

$F_{i}(g_{1},\ldots,g_{N})=\sum_{j=1}^{n}v_{j}(g_{i})\phi_{ij}(g_{1},\ldots,g_{N}),$

where $v_{1},\ldots,v_{n}$ is a basis of the Lie algebra of left-invariant
vector fields of $G$ and functions $\phi_{ij}$ are jointly-radial.

###### Proof.

The first statement is a direct consequence of Proposition 4, written with
respect to the Lie Algebra $\hat{\mathfrak{L}}$ of the diagonal extension of
left-invariant vector fields.

The second statement is a particular case of Theorem 1, after proving that
each $F_{i}(g_{1},\ldots,g_{N}):=v_{j}(g_{i})$ for $j=1,\ldots,n$ is a
solution. Since $G$ acts by left translations, its Lie algebra is given by
right-invariant vector fields. Let $l$ be one of such right-invariant vector
fields and define its diagonal extension
$\hat{l}(g_{1},\ldots,g_{N})=\sum_{k}l(g_{k})$, where $l(g_{k})$ acts on the
$k$-th agent. By studying the $i$-th component of $[F,\hat{l}]$, it holds
$[v_{j}(g_{i}),l(g_{i})]=0$, since each left-invariant vector field $v_{j}$
commutes with right-invariant vector fields. Thus, $F_{i}$ is a solution.∎

###  2.6 Gradient flow Population Dynamics

In this section, we describe Population Dynamics that are moreover gradient
flows. This means that the dynamics $\dot{X}=F(X)$ is indeed of the form
$\dot{X}=\nabla\phi(X)$ for some potential function $\phi:M^{N}\to\mathbb{R}$.
It is clear that the standard setting for such dynamics is given by Riemannian
manifolds, where the natural definition of the gradient is available.

The main result of this section is the following.

######  Proposition 24.

Let $M$ be a Riemannian manifold and $G$ a connected Lie group of isometries
of $M$.

Let $\phi:M^{N}\to\mathbb{R}$ be a jointly-radial function. Then, its gradient
is a jointly-equivariant vector field.

###### Proof.

It is easy to observe that the result for $G$ acting diagonally on $M^{N}$ is
equivalent to the case of $G^{\prime}:=G^{N}$ acting on $M^{\prime}:=M^{N}$.
Then, from now on we only consider the case $N=1$, hence $\hat{l}=l$.

We need to prove that $[\nabla\phi,l]=0$ for all $l\in\mathfrak{L}$. We fix a
point $X\in M$ and define normal coordinates around it: each point in a small
neighborhood $\mathcal{N}_{X}$ of $X$ is written in coordinates
$x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ so that the Riemannian tensor $g$ at
$x$ is
$g_{ij}(x)=\delta_{ij}-\frac{1}{3}\sum_{kl}R_{ijkl}x_{k}x_{l}+o(\|x\|^{2})$.
In particular, we have that Christoffel symbols are zero. The inverse
Riemannian tensor has a similar structure: $g^{ij}(x)=\delta^{ij}+o(\|x\|)$.

Write $\nabla f=\sum_{i}\partial_{i}f(x)g^{ij}(x)\partial_{i}$ and
$l=\sum_{i}l_{i}(x)\partial_{i}$. Here, the symbol $\partial_{i}$ is the
derivative with respect to the variable $x_{i}$ in normal coordinates. Since
$f$ is radial, then $L_{l}f=0$, i.e. $g_{x}(\nabla f(x),l(x))=0$ for all $x\in
M$ due to the definition of gradient. By writing the identity in normal
coordinates, we have $\psi(x)=\sum_{i}(\partial_{i}f(x))l_{i}(x)+o(\|x\|)=0$.
Since $\psi(x)=0$ for all $x\in M$, then its differential is zero. By
computing it in coordinates, we have
$\sum_{i}(\partial^{2}_{ji}f(x))l_{i}(x)+(\partial_{i}f(x))(\partial_{j}l_{i}(x))+o(1)=0$.
By sending $x\to 0$, we have $o(1)=0$, thus for all indexes $j$ the following
identity in $X$ holds:

$[(\partial^{2}_{ij}f)l_{i}+(\partial_{i}f)(\partial_{j}l_{i})]_{|_{x=0}}=0.$
(2.15)

We now prove that the matrix $\partial_{j}l_{i}$ is skew-symmetric at $X$. We
write $l(x)=L_{0}+L_{1}x+o(\|x\|)$ in Taylor series, thus
$L_{1}=\partial_{j}l_{i}$. Its exponential is then
$\exp(tl(x))=L_{0}t+(\mathrm{Id}+L_{1}t)x+o(x,t)$, thus by differentiation in
$x$ we have $D\exp(tl(x))=(\mathrm{Id}+L_{1}t)+o(1,t)$. The fact that $G$ is a
group of isometry reads as

$g_{x}(v,w)=g_{x}(D\exp(tl(x))v,D\exp(tl(x))w).$

By differentiating the identity with respect to $t$ at $t=0$, we have

$0=g_{x}(L_{1}v,w)+g(x)(v,L_{1}w).$

By letting $x\to 0$, the Riemannian tensor becomes the Euclidean one, thus the
identity becomes

$w^{T}L_{1}v+v^{T}L_{1}w=0,$

i.e. $L_{1}=\partial_{j}l_{i}$ is skew-symmetric.

We now use the fact that $\partial_{j}l_{i}$ is skew-symmetric in (2.15). It
then holds

$[(\partial^{2}_{ij}f)l_{i}-(\partial_{i}l_{j})(\partial_{i}f)]_{|_{x=0}}=0,$

for all indexes $j$. This is the formula for $[\nabla f,l]$ in coordinates at
$X$. It then holds $[\nabla f,l]=0$ at $X$. Since the choice of $X$ was
arbitrary, the identity holds for all $X\in M$. ∎

######  Corollary 25.

Let $M$ be a Riemannian manifold of dimension $n=2$ and $N\geq 1$. Let
$F=(F_{1},\ldots,F_{N})=\nabla\phi$ be the gradient of a jointly radial
function. Define $F^{\perp}:=((F_{1})^{\perp},\ldots,(F_{N})^{\perp})$, where
$(F_{i})^{\perp}$ is a continuous choice of perpendicular vectors to $F_{i}$
at each point of $M$. Then, $F^{\perp}$ is well defined and is a PD.

###### Proof.

Assume that $F$ has a connected (open dense) domain. As in the previous proof,
we use normal coordinates at each point of $(X_{1},\ldots,X_{N})\in M^{N}$,
that coincide with the normal coordinates on each component. Write
$X_{i}=(x_{i},y_{i})$ in normal coordinates, then
$F(X)=\sum_{i=1}^{N}f_{i}\partial_{x_{i}}+g_{i}\partial_{y_{i}}$. We then
define
$F(X)^{\perp}=\sum_{i=1}^{N}-g_{i}\partial_{x_{i}}+f_{i}\partial_{y_{i}}+o(1)$,
where we used the fact that the Riemannian metric in normal coordinates
coincides with the Euclidean one up to order 2. In other terms, for each
$F_{i}$ we choose $(F_{i})^{\perp}=JF_{i}$ with $J$ being the symplectic
matrix

$J=\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right).$ (2.16)

We then write the condition $[F^{\perp},\hat{l}]=0$ for each coordinate, that
reads as

$(\partial_{X_{1}}JF_{i})l(X_{1})+\ldots+(\partial_{X_{N}}JF_{i})l(X_{N})-(\partial_{X_{i}}l)JF_{i}=0.$
(2.17)

By recalling that $\partial_{X_{i}}l=L_{1}$ with $L_{1}$ skew-symmetric (as
proved in Proposition 24), we have that it commutes with $J$, due to the fact
that we deal with 2-dimensional skew-symmetric matrices; $L_{1}$ is indeed a
multiple of $J$. We can then rewrite (2.17) as
$J\left([F,\hat{l}]\right)_{i}=0$, that is true due to equivariance of $F$.

More generally, for each connected component of the domain of $F$ and each
coordinate $F_{i}$, we can choose $(F_{i})^{\perp}$ as either $JF_{i}$ or
$-JF_{i}$. The proof is identical, since the condition (2.17) holds by
replacing $J$ with $-J$. ∎

We will use these results throughout the remaining of the article, to identify
some Population Dynamics that are gradient flows. It is remarkable to observe
that it might happen that other Population Dynamics are gradient flows, even
though they are not gradient of jointly radial functions. We also have the
following useful corollary.

######  Corollary 26.

Let $M$ be a Riemannian manifold of dimension $n$ and $G$ a connected Lie
group of isometries of $M$. If $N\geq n+1$, a basis of the space of solutions
$\SS$ is given by the component of the vector fields $\nabla
d(x_{1},x_{2}),\ldots,\nabla d(x_{1},x_{n+1})$ for the variable $x_{1}$, where
$d$ is the Riemannian distance.

###### Proof.

It is clear that the vector fields are independent. Since they are gradient
flows, they are jointly equivariant. Then, their first components are both
independent functions and solutions. Since we have $n$ independent functions
for the $n$-dimensional manifold, it is a basis due to Theorem 1. ∎

##  3 Population Dynamics on the Euclidean plane

In this section, we describe Population Dynamics on the Euclidean plane, being
jointly equivariant with respect to rototraslations of the plane. We already
stated in the introduction that this setting is often used for several
important applications. We first describe the general FPD, by showing the role
of each functional parameter. We also describe gradient flows and Permutation-
Equivariant PDs.

The setting of this section is then the following: the manifold is
$M=\mathbb{R}^{2}$ and the group is the special Euclidean group of
rototranslations $G=SO(2,\mathbb{R})\ltimes\mathbb{R}^{2}$. We use standard
coordinates on $\mathbb{R}^{2N}$, that we denote by $X_{i}=(x_{i},y_{i})$ for
$i=1,\ldots,N$, and denote the standard Euclidean norm by $\|\cdot\|$.

The case $N=1$ is easy but already interesting.

######  Proposition 27.

The FPD associated to
$(N=1,\mathbb{R}^{2},SO(2,\mathbb{R})\ltimes\mathbb{R}^{2})$ contains the null
vector field only.

###### Proof.

By Proposition 19, PDs for $N=1$ are contained in the set of constant vector
fields $f(x)=v\in\mathbb{R}^{2}$. By adding equivariance with respect to
$SO(2)$, we impose $[Jx,v]=0$. This implies $v=0$. ∎

We then consider $N>1$ from now on. We introduce difference variables with
respect to $X_{1}$, that we also write in polar coordinates, i.e.

$Z_{i}:=X_{i}-X_{1}=\rho_{i}(\cos(\theta_{i}),\sin(\theta_{i})),$ (3.1)

where $\rho_{i}:=\|X_{i}-X_{1}\|$ and $\theta_{i}\in[0,2\pi)$ is the argument
of $Z_{i}$.

We have the following result.

######  Proposition 28.

Consider the FPD associated to
$(N,\mathbb{R}^{2},SO(2,\mathbb{R})\ltimes\mathbb{R}^{2})$ with $N>1$. Assume
$(x_{1}-x_{2},y_{1}-y_{2})\neq(0,0)$ for simplicity of notation. Then, the FPD
is composed of all vector fields of the form

$F=F_{1}+\ldots+F_{N}\qquad\mbox{~{}~{}with~{}~{}}\qquad
F_{i}=f_{i,1}\partial_{x_{i}}+f_{i,2}\partial_{y_{i}},$

and

$\left(\begin{array}[c]{c}f_{i,1}\\\
f_{i,2}\end{array}\right)=\left(\begin{array}[c]{cc}x_{2}-x_{1}&y_{1}-y_{2}\\\
y_{2}-y_{1}&x_{2}-x_{1}\end{array}\right)\binom{\lambda_{i}(\rho_{2},\ldots,\rho_{n},\theta_{3}-\theta_{2},\ldots,\theta_{N}-\theta_{2})}{\mu_{i}(\rho_{2},\ldots,\rho_{n},\theta_{3}-\theta_{2},\ldots,\theta_{N}-\theta_{2})},$
(3.2)

where functions $\lambda_{i},\mu_{i},$ are arbitrary analytic functions with a
domain being a diagonal $SO(2,\mathbb{R})$-space.

###### Proof.

The centralizer $\mathfrak{C}$ of $SO(2,\mathbb{R})$ is
$\mathfrak{C}=\mathbb{R}.\mathrm{Id}+\mathbb{R}.J$, where $J$ is the
symplectic matrix (2.16).

Since $\mathfrak{C}$ has dimension 2, one can apply Corollary 21, Item 1. This
gives the result. ∎

######  Remark 29.

The number $2N$ of functional parameters here is minimal, as each
$F_{i}\in\SS$ is independent and $\SS=\mathcal{C}$ has dimension $2$.
Moreover, it depends on $2N-3$ variables, that is the minimal number of
variables too.

In the formula above, it is clear that both $X_{1}$ and $X_{2}$ play specific
(but different) roles. We then now give examples of dynamics for a system of
$N=2$ particles only. In this case, we can describe the physical meaning of
the functional parameters: $\lambda_{i}$ plays the role of the radial
component of the interaction, while $\mu_{i}$ is the rotational component. In
particular:

  * 1.

$\lambda_{1}>0$ promotes convergence, and similarly for $\lambda_{2}<0$.
Opposite signs promote distancing.

  * 2.

$\mu_{1}>0$ promotes clockwise rotation, and similarly for $\mu_{2}<0$.
Opposite signs promote counter-clockwise rotation.

This is made clear from the following examples. Trajectories of these examples
can be found in Figure 1. Agent 1 is depicted in green, Agent 2 is depicted in
blue. The initial state is always $X_{1}=(0,0)$, $X_{2}=(1,1)$.

Example A.1

Set $\mu_{1}=\mu_{2}=0$. For $\lambda_{1}=1$ and $\lambda_{2}=-1$ we have
convergence to a common state.

Example A.2

Again with $\mu_{1}=\mu_{2}=0$, for $\lambda_{1}=1$ and $\lambda_{2}=0.5$,
agent 1 aims to converge towards agent 2, that in turn aims to distance itself
(with a smaller velocity).

Example A.3

Again with $\mu_{1}=\mu_{2}=0$, for $\lambda_{1}=-1+\rho_{2}$ and
$\lambda_{2}=1-\rho_{2}$, agents aim to converge towards a configuration with
distance 1.

Example A.4

Again with $\mu_{1}=\mu_{2}=0$, for $\lambda_{1}=-1+\rho_{2}$ and
$\lambda_{2}=0.5-\rho_{2}$, agents aim to converge towards a configuration
with different distances (1 and 0,5 respectively). Thus, agent 1 starts
converging, then distances itself after reaching distance 1.

Example A.5

Set $\lambda_{1}=\lambda_{2}=0$. For $\mu_{1}=1$ and $\mu_{2}=0$ we have
clockwise rotation of agent 1 around agent 2.

Example A.6

Set $\lambda_{1}=\lambda_{2}=0$. For $\mu_{1}=1$ and $\mu_{2}=1$ we have
parallel displacement, as agent 1 moves clockwise and agent 2 moves counter-
clockwise.

Example A.7

Set $\lambda_{1}=\lambda_{2}=0$. For $\mu_{1}=1$ and $\mu_{2}=-1$ we have
clockwise movement.

Example A.8

For $\lambda_{1}=1$, $\lambda_{2}=-1$, $\mu_{1}=1$ and $\mu_{2}=-1$ we have
clockwise movement, together with convergence.

Example A.9

For $\lambda_{1}=1$, $\lambda_{2}=0$, $\mu_{1}=0$ and $\mu_{2}=1$ we have
clockwise movement of agent 2 with respect to agent 1, together with
convergence of agent 1 towards agent 2 via a radial movement.

Example A.10

For $\lambda_{1}=-1+\rho_{2}$ and $\lambda_{2}=1-\rho_{2}$, with
$\mu_{1}=1,\mu_{2}=-1$ agents aim to converge towards a configuration with
distance 1, together with clockwise rotation.

Example A.11

For $\lambda_{1}=-1+\rho_{2}$ and $\lambda_{2}=0.5-\rho_{2}$, with
$\mu_{1}=1,\mu_{2}=-1$ agents aim to converge towards a configuration with
different distances, together with clockwise rotation. The result is a
convergence to a limit clockwise trajectory, with distance being the average
of the desired distances.

Example A.12

For $\lambda_{1}=-0.5+\rho_{2}$ and $\lambda_{2}=0.5-\rho_{2}$, with
$\mu_{1}=1-\rho_{2},\mu_{2}=-1+\rho_{2}$ agents aim to converge towards a
configuration with distance 0.5. This first induces counter-clockwise
rotation, that then turns into clockwise rotation.

(a) Example A.1: $\lambda_{1}=1,\lambda_{2}=-1$,
$\mu_{1}=\mu_{2}=0$.

(b) Example A.2: $\lambda_{1}=1,\lambda_{2}=0.5$,
$\mu_{1}=\mu_{2}=0$.

(c) Ex. A.3: $\lambda_{1}=-1+\rho_{2}$,
$\lambda_{2}=1-\rho_{2},\mu_{1}=\mu_{2}=0$.

(d) Example A.4: $\lambda_{1}=-1+\rho_{2}$,
$\lambda_{2}=0.5-\rho_{2},\mu_{1}=\mu_{2}=0$.

(e) Example A.5: $\lambda_{1}=\lambda_{2}=0$,
$\mu_{1}=1,\mu_{2}=0$.

(f) Example A.6: $\lambda_{1}=\lambda_{2}=0$,
$\mu_{1}=\mu_{2}=1$.

(g) Example A.7: $\lambda_{1}=\lambda_{2}=0$,
$\mu_{1}=1,\mu_{2}=-1$.

(h) Example A.8: $\lambda_{1}=1,\lambda_{2}=-1$,
$\mu_{1}=1,\mu_{2}=-1$.

(i) Example A.9: $\lambda_{1}=1,\lambda_{2}=0$,
$\mu_{1}=0,\mu_{2}=1$.

(j) Ex. A.10: $\lambda_{1}=-1+\rho_{2}$,
$\lambda_{2}=1-\rho_{2},\mu_{1}=1,\mu_{2}=-1$.

(k) Ex. A.11: $\lambda_{1}=-1+\rho_{2}$,
$\lambda_{2}=0.5-\rho_{2},\mu_{1}=1,\mu_{2}=-1$.

(l) Ex. A.12: $\lambda_{1}=-0.5+\rho_{2}=-\lambda_{2}$,
$\mu_{1}=1-\rho_{2}=-\mu_{2}$.

Figure 1: Examples A (1 to 12).

######  Remark 30.

The case of $M=\mathbb{R}^{n}$ and $G=SO(n,\mathbb{R})\ltimes\mathbb{R}^{n}$
for $n>2$ is very different. Indeed, in this case, one has
$\mathfrak{C}=\mathbb{R}.\mathrm{Id}$, hence hypotheses of Corollary 21, Item
1, are never satisfied. Then, one can resort to Corollary 21, Item 2 if
$N-1\geq n$ to describe FPD. As it has been already stated in the
introduction, our results do not cover “intermediate” number of agents.

######  Remark 31.

Contrarily to the case described in Remark 30, there is another interesting
case in which assumptions of Corollary 21, Item 1 hold: this is the group of
unit quaternions $\mathcal{H}$ (or the semi-direct product
$\mathcal{H}\ltimes\mathbb{R}^{4}$) acting on $\mathbb{R}^{4}$. Indeed, the
Lie algebra $\mathcal{L}$ is generated by the 3 quaternion matrices $i,j,k.$

$i=\left(\begin{array}[c]{cccc}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{array}\right),\qquad j=\left(\begin{array}[c]{cccc}0&0&-1&0\\\
0&0&0&1\\\ 1&0&0&0\\\ 0&-1&0&0\end{array}\right),\qquad
k=\left(\begin{array}[c]{cccc}0&0&0&-1\\\ 0&0&-1&0\\\ 0&1&0&0\\\
1&0&0&0\end{array}\right).$

The centralizer $\mathfrak{C}$ is the full Lie algebra $\mathcal{H}^{\ast}$of
skew quaternions,
$\mathcal{H}^{\ast}=\\{a.Id+b.\hat{\imath}+c.\hat{\jmath}+d.\hat{k}\\}$ with

$\hat{\imath}=\left(\begin{array}[c]{cccc}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{array}\right),\qquad\hat{\jmath}=\left(\begin{array}[c]{cccc}0&0&1&0\\\
0&0&0&1\\\ -1&0&0&0\\\
0&-1&0&0\end{array}\right),\qquad\hat{k}=\left(\begin{array}[c]{cccc}0&0&0&-1\\\
0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\end{array}\right).$

It is easy to check that the four vectors
$x,\hat{\imath}.x,\hat{\jmath}.x,\hat{k}.x$ are independent for all
$x\in\mathbb{R}^{4}\setminus\\{0\\}$. Thus, Theorem 2, Item 1 or Corollary 21,
Item 1 can be used to characterize the FPD.

###  3.1 Gradient flows for 2 particles

In the case of $N=2$ particles, we are able to completely characterize the
gradient flow PDs.

######  Proposition 32.

Let $F$ be a PD in the FPD associated to
$(N=2,\mathbb{R}^{2},SO(2,\mathbb{R})\ltimes\mathbb{R}^{2})$, i.e. of the form
(3.2). Then, it is a gradient flow if and only if $\lambda_{1}=-\lambda_{2}$
and $\mu_{1}=\mu_{2}=0$.

###### Proof.

It is easy to prove that both conditions $\lambda_{1}=-\lambda_{2}$ and
$\mu_{1}=\mu_{2}=0$ imply that $F$ is a gradient flow. Indeed, define

$\phi(x_{1},y_{1},x_{2},y_{2}):=\Phi(\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}})\mbox{~{}~{}with~{}~{}}\Phi(\rho)=\int_{0}^{\rho}\sqrt{s}\lambda_{2}(s)\,ds$

and check that $F=\nabla\phi$.

We now prove the converse result. Let $\phi(x_{1},x_{2},x_{2},y_{2})$ be such
that $\nabla\phi$ is a PD, i.e. of the form (3.2). A direct computation shows
that, by imposing that $f_{1,1}=\partial_{x_{1}}\phi$ and
$f_{2,1}=\partial_{x_{2}}\phi$, the condition
$\partial_{x_{2}}f_{1,1}=\partial_{x_{1}}f_{2,1}$ ensures
$\lambda_{1}=-\lambda_{2}$ and $\mu_{1}=-\mu_{2}$. We already know that the
dynamics given by $\lambda_{1}=-\lambda_{2}$ with $\mu_{1}=\mu_{2}=0$ is a
gradient flow, as proved above. By recalling that gradient flows form a vector
space, we can then focus on the case $\lambda_{1}=\lambda_{2}=0$ and
$\mu_{1}=-\mu_{2}$.

We now impose the condition $\partial_{y_{2}}f_{2,1}=\partial_{x_{2}}f_{2,2}$,
that reads as

$\displaystyle-\mu_{2}(\rho_{2})+(y_{1}-y_{2})\partial_{\rho_{2}}\mu_{2}(\rho_{2})\frac{y_{2}-y_{1}}{\rho_{2}}=\mu_{2}(\rho_{2})+(x_{2}-x_{1})\partial_{\rho_{2}}\mu_{2}(\rho_{2})\frac{x_{2}-x_{1}}{\rho_{2}},$

hence $2\mu_{2}(\rho_{2})+\rho_{2}\partial_{\rho_{2}}\mu_{2}(\rho_{2})=0$.
This implies $\mu_{2}(\rho_{2})=\frac{k}{\rho_{2}^{2}}$. Remark that the
domain is clearly given by $\rho_{2}\neq 0$, that is a $G$-invariant set.
Again by recalling that gradient flows form a vector space, we see that we
need only to check wether the following vector field is a gradient flow:

$\dot{x}_{1}=\frac{y_{2}-y_{1}}{\rho_{2}^{2}},\qquad\dot{y}_{1}=\frac{x_{1}-x_{2}}{\rho_{2}^{2}},\qquad\dot{x}_{2}=\frac{y_{1}-y_{2}}{\rho_{2}^{2}},\qquad\dot{y}_{2}=\frac{x_{2}-x_{1}}{\rho_{2}^{2}}.$
(3.3)

By duality in $\mathbb{R}^{4}$, it is a gradient flow if and only if the dual
1-form

$\omega=\frac{(y_{1}-y_{2})d(x_{2}-x_{1})+(x_{2}-x_{1})d(y_{2}-y_{1})}{\rho_{2}^{2}}$

is exact. By writing it in coordinates $(\rho_{2},\theta_{2},x_{2},y_{2})$
with $\rho_{2}(\cos(\theta_{2}),\sin(\theta_{2}))=(x_{2}-x_{1},y_{2}-y_{1})$,
we have $\omega=d\theta_{2}$. It is clear that this form is not exact. Thus,
the vector field (3.3) is not a gradient flow. This in turn implies that a
gradient flow that is a PD necessarily satisfies $\mu_{1}=\mu_{2}=0$. ∎

######  Remark 33.

The proof above shows that a complete characterization of gradient flow PDs is
difficult, since one needs to check wether forms are exact on domains that are
not simply connected. For this reason, we have no complete characterization
for the case $N>2$, for which we only have the partial results given by
Proposition 24.

###  3.2 Permutation-equivariant population dynamics

We now add permutation equivariance to the previous setting, by studying FPEPD
introduced in Definition 10. Since we already observed that the centralizer
$\mathfrak{C}$ of $SO(2,\mathbb{R})$ has dimension $n=2$, we can apply
Corollary 22 to compute the FPEPD in this case. We have the following result.

######  Proposition 34.

Consider the FPEPD associated to
$(N,\mathbb{R}^{2},SO(2,\mathbb{R})\ltimes\mathbb{R}^{2})$ with $N>1$.

Consider the dynamics taking place on the open dense set

$\displaystyle\mathcal{O}$ $\displaystyle=$
$\displaystyle\Big{\\{}(X_{1},\ldots,X_{N})\in\mathbb{R}^{2N}\mbox{~{}~{}such
that~{}~{}}\sum_{j\neq i}(X_{j}-X_{i})\neq(0,0)\mbox{~{}for
all~{}}i=1,\ldots,N\Big{\\}}.$

Write in polar coordinates
$X_{j}-X_{i}=\rho_{j,i}(\cos(\theta_{j,i}),\sin(\theta_{j,i}))$ and define
$\tau_{j,i}:=\theta_{j,i}-\theta_{i+1,i}$, with the convention that $i=N+1$ is
replaced by $i=1$ and that the difference of angles is mod-$2\pi$.

Then, the FPEPD is composed of all vector fields of the form

$F=F_{1}+\ldots+F_{N}\qquad\mbox{~{}~{}with~{}~{}}\qquad
F_{i}=f_{i,1}\partial_{x_{i}}+f_{i,2}\partial_{y_{i}},$

and

$\left(\begin{array}[c]{c}f_{i,1}\\\
f_{i,2}\end{array}\right)=\left(\begin{array}[c]{cc}\sum_{j\neq
i}(x_{j}-x_{i})&-\sum_{j\neq i}(y_{j}-y_{i})\\\ \sum_{j\neq
i}(y_{j}-y_{i})&\sum_{j\neq
i}(x_{j}-x_{i})\end{array}\right)\binom{\lambda(\check{\rho}_{i},\check{\tau_{i}})}{\mu(\check{\rho}_{i},\check{\tau_{i}})},$

where functions $\lambda,\mu$ are arbitrary analytic functions that are
invariant with respect to permutations of both the first $N-1$ coordinates

$\check{\rho}_{i}:=(\rho_{1,i},\ldots,\rho_{i-1,i},\rho_{i+1,i},\ldots,\rho_{N,i})$

and the remaining $N-2$ coordinates

$\check{\tau}_{i}:=(\tau_{1,i},\ldots,\tau_{i-1,i},\tau_{i+2,i},\ldots,\tau_{N,i}).$

###### Proof.

The proof is a direct application of Corollary 22. The only remarkable detail
is that all functions that are jointly radial (i.e. invariant by
rototranslation) and permutation invariant are of the form $\lambda,\mu$ given
in the statement; this fact has already been proved for Proposition 28. ∎

######  Remark 35.

This result, combined with Proposition 32, shows that the only possible PDs
for $N=2$ that are gradient flows are also PEPDs.

We do not provide additional examples. For $N=2$ particles, we already had
PEPD in Examples A.(1-3-7-8-10-12) in Figure 1. Examples for $N>2$ particles
are similar, again with $\lambda>0$ promoting convergence and $\mu>0$
promoting clockwise rotation.

##  4 Population Dynamics on the relativistic spaces

In this section, we focus on Population Dynamics on the relativistic spaces.

###  4.1 Population Dynamics on the relativistic line

In this section, we deal with PDs on the relativistic line: the state space is
the pseudo-Euclidean plane $\mathbb{R}^{2}$ endowed with the quadratic form
with signature $(1,1)$. We denote by $c>0$ the speed of light and consider the
following scalar product:

$(T_{1},x_{1})\cdot(T_{2},x_{2})=c^{2}T_{1}T_{2}-x_{1}x_{2}.$

The group is the semidirect product $G=SO(1,1)\ltimes\mathbb{R}^{2}$ that
preserves the associated quadratic form. We have the following result.

######  Proposition 36.

Consider the FPD associated to
$(N,\mathbb{R}^{2},SO(1,1)\ltimes\mathbb{R}^{2})$. Then:

  * 1.

for $N=1$, the FPD contains the zero vector field only;

  * 2.

for $N=2$, all PDs are given by $F=F_{1}+F_{2}$ with
$F_{i}=f_{i,1}\partial_{T_{i}}+f_{i,2}\partial_{x_{i}}$, where

$\left(\begin{array}[c]{c}f_{i,1}\\\
f_{i,2}\end{array}\right)=\left(\begin{array}[c]{cc}T_{2}-T_{1}&c^{-1}(x_{2}-x_{1})\\\
x_{2}-x_{1}&c(T_{2}-T_{1})\end{array}\right)\left(\begin{array}[]{c}\phi_{i}(r)\\\
\psi_{i}(r)\end{array}\right)$

with $\phi_{i}(r),\psi_{i}(r)$ analytic functions of
$r=\sqrt{c^{2}(T_{2}-T_{1})^{2}-(x_{2}-x_{1})^{2}}$;

  * 3.

for $N\geq 3$, and choosing any two distinct indexes
$k_{1},k_{2}\in\\{2,\ldots,N\\}$, all PDs are of the form

$F(X)=\sum_{i=1}^{N}f_{i}(Z)\partial_{X_{i}}\qquad\mbox{~{}~{} with
~{}~{}}\qquad f_{i}=\psi_{i,1}(Z)Z_{k_{1}}+\psi_{i,2}(Z)Z_{k_{2}},$

where $\psi_{i,k}(Z)$ are $\hat{\mathfrak{L}}$-jointly radial functions of
$Z=(Z_{2},\ldots,Z_{N})$ with $Z_{i}:=(T_{i}-T_{1},x_{i}-x_{1})$.

###### Proof.

For $N=1$, we have that Proposition 19 ensures that PDs are contained in the
set of constant vector fields. By adding equivariance with respect to
$SO(1,1)$, we have that the only PD is the zero vector field.

The case $N\geq 3$ is a direct application of Corollary 21, Item 2. One can
also apply Corollary 21, Item 1.

The case $N=2$ is a direct application of Corollary 21, Item 1. Indeed, the
centralizer of $so(1,1)=\mathbb{R}.K$ with

$K=\left(\begin{array}[]{cc}0&c^{-1}\\\ c&0\end{array}\right)$

is $\mathfrak{C}=\mathbb{R}.\mathrm{Id}+\mathbb{R}.K$, that has dimension
$n=2$. Then, solutions are of the form $\phi(Z)Z+\psi(Z)KZ$, where
$Z=(T_{2}-T_{1},x_{2}-x_{1})$. Moreover, $\phi(Z),\psi(Z)$ are
$so(1,1)$-radial, thus depending on
$r=\|Z\|=\sqrt{c^{2}(T_{2}-T_{1})^{2}-(x_{2}-x_{1})^{2}}$ only.

∎

######  Remark 37.

The case $N=2$ is an interesting example in which it holds
$\mathcal{E}\subsetneq\SS$, i.e. in which elementary solutions do not provide
all solutions. We can anyway provide a complete classification, due to the
equality $\mathcal{C}=\SS$, i.e. observing that all solutions are solutions
given by the centralizer.

######  Remark 38.

A direct computation for $N=2$ shows that
$\dot{r}=r(\phi_{1}(r)-\phi_{2}(r))$. In particular, this implies that terms
$\psi_{1},\psi_{2}$ do not contribute to the dynamics of $r$ (i.e. the
dynamics on the quotient space).

We now provide some simple examples for $N=2$. In Examples B.1-2-3, we
consider the case of $\phi_{1}(r)=r$, $\phi_{2}(r)=1$ and
$\psi_{1}=\psi_{2}=0$ with increasing values of $c\in\\{1,10,100\\}$. By
Remark 38, it holds $\partial_{t}r(t)=r(1-r)$, hence the variable $r$
converges to the value 1. The limit dynamics is then given by
$\phi_{1}=\phi_{2}=1$, i.e. translations with constant velocity 1 in all
variables $T_{1},x_{1},T_{2},x_{2}$. This corresponds to the fact that the
relativistic dynamics converges to the so-called relative equilibria of the
classical dynamics, i.e. classical dynamics with constant $x_{1}-x_{2}$. The
key remark here is that the convergence of the relativistic dynamics to the
classical one is faster when values of $c$ increase, as expected.

In Examples B.4-5-6, we investigate the role of $\psi_{1},\psi_{2}$. They do
not contribute to the dynamics of the variable $r$, but we show that they play
a strong role in the dynamics of the state variable. In particular, we show
that the dynamics is qualitatively different for different values of the
parameter $c\in\\{1,10,100\\}$. We choose again $\phi_{1}(r)=r$,
$\phi_{2}(r)=1$, ensuring convergence of $r$ to 1. By defining
$\psi_{1}=100/c=-\psi_{2}$, we find that:

  * 1.

for $c=1$ the distances $T_{2}-T_{1},x_{2}-x_{1}$ explode;

  * 2.

for $c=10$ the distance $T_{2}-T_{1}$ converges to 0, while $x_{2}-x_{1}$
converges to a constant (i.e. to a classical relative equilibrium);

  * 3.

for $c=100$ the dynamics is similar to the case $c=10$, but convergence is
faster for both variables.

(a) Example B.1 with $c=1$.

(b) Example B.2 with $c=10$.

(c) Example B.3 with $c=100$.

(d) Example B.4 with $c=1$.

(e) Example B.5 with $c=10$.

(f) Example B.6 with $c=100$.

Figure 2: Examples B (1 to 6). Variables $T$ above, $x$ below; Variables 1
(blue) and 2 (orange).

###  4.2 Population Dynamics on the relativistic plane

In this section, we deal with PDs on the relativistic plane: the state space
is the pseudo-Euclidean space $\mathbb{R}^{3}$ endowed with the quadratic form
with signature $(1,2)$. Given two states $(T_{1},x_{1},y_{1})$,
$(T_{2},x_{2},y_{2})$, the scalar product is

$(T_{1},x_{1},y_{1})\cdot(T_{2},x_{2},y_{2})=T_{1}T_{2}-x_{1}x_{2}-y_{1}y_{2}.$

The group is then the semidirect product $G=SO(1,2)\ltimes\mathbb{R}^{3}$. We
treat the cases $N=1$ and $N\geq 4$ with the general theory. We are then left
with two special dimensions, that are $N=2,3$. For $N=2$, we prove that PDs
are determined by elementary solutions via a direct computation. Instead, for
$N=3$ we have no general statement. The results are summarized as follows.

######  Proposition 39.

Consider the FPD associated to
$(N,\mathbb{R}^{3},SO(1,2)\ltimes\mathbb{R}^{3})$. Then:

  * 1.

for $N=1$, the FPD is composed of the zero vector field only;

  * 2.

for $N=2$, all PDs are of the form

$\displaystyle F(X_{1},X_{2})$ $\displaystyle=$
$\displaystyle\psi_{1}(r)\left((T_{2}-T_{1})\partial_{T_{1}}+(x_{2}-x_{1})\partial_{x_{1}}+(y_{2}-y_{1})\partial_{y_{1}}\right)+$
$\displaystyle\psi_{2}(r)\left((T_{2}-T_{1})\partial_{T_{2}}+(x_{2}-x_{1})\partial_{x_{2}}+(y_{2}-y_{1})\partial_{y_{2}}\right),$

with $r=\sqrt{(T_{2}-T_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}}$;

  * 3.

for $N=3$, the FPD contains vector fields of the form

$\displaystyle F(X_{1},X_{2},X_{3})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\phi_{i}(Z_{2},Z_{3})\left((T_{2}-T_{1})\partial_{T_{i}}+(x_{2}-x_{1})\partial_{x_{i}}+(y_{2}-y_{1})\partial_{y_{i}}\right)+$
$\displaystyle\psi_{i}(Z_{2},Z_{3})\left((T_{3}-T_{1})\partial_{T_{i}}+(x_{3}-x_{1})\partial_{x_{i}}+(y_{3}-y_{1})\partial_{y_{i}}\right),$

with

$Z_{2}:=(T_{2}-T_{1},x_{2}-x_{1},y_{2}-y_{1}),\qquad
Z_{3}:=(T_{3}-T_{1},x_{3}-x_{1},y_{3}-y_{1})$

and $\phi_{i},\psi_{i}$ analytic and $\hat{\mathfrak{L}}$-jointly radial with
respect to the action of $SO(1,2)$;

  * 4.

for $N\geq 4$, and choosing any three distinct indexes
$k_{1},k_{2},k_{3}\in\\{1,\ldots,N\\}$, all PDs are of the form

$F(X)=\sum_{i=1}^{N}f_{i}(Z)\partial_{X_{i}}\qquad\mbox{~{}~{} with
~{}~{}}\qquad
f_{i}=\psi_{i,1}(Z)Z_{k_{1}}+\psi_{i,2}(Z)Z_{k_{2}}+\psi_{i,3}(Z)Z_{k_{3}},$

where $\psi_{i,k}(Z)$ are $\hat{\mathfrak{L}}$-jointly radial functions of
$Z=(Z_{2},\ldots,Z_{N})$ with $Z_{i}:=(T_{i}-T_{1},x_{i}-x_{1},y_{i}-y_{1})$.

###### Proof.

The cases $N=1$ and $N\geq 4$ are identical to Proposition 36.

We now study $N=2$, for which we introduce the difference variable

$Z=(T,x,y):=(T_{2}-T_{1},x_{2}-x_{1},y_{2}-y_{1})$

and consider vector fields of a single particle of coordinate $Z$ that are
equivariant under the action of $SO(1,2)$, i.e. solutions of (2.13). For this
reason, we introduce polar coordinates in the $(x,y)$ difference variables,
i.e.

$(x_{2}-x_{1},y_{2}-y_{1})=\rho(\cos(\theta),\sin(\theta))\mbox{~{}~{}~{}with~{}}\rho\geq
0,\theta\in[0,2\pi).$

It is easy to prove that the Lie algebra $so(1,2)$ is generated by

$V_{1}=x\partial_{T}+T\partial_{x},\qquad
V_{2}=y\partial_{T}+T\partial_{y},\qquad V_{3}=-y\partial_{x}+x\partial_{y}.$

By rewriting them in cylindrical coordinates $(T,\rho,\theta)$, it holds

$V_{1}=\rho\cos(\theta)\partial_{T}+T\cos(\theta)\partial_{\rho}-T\frac{\sin(\theta)}{\rho}\partial_{\theta},\qquad
V_{2}=\rho\sin(\theta)\partial_{T}+T\sin(\theta)\partial_{\rho}+T\frac{\cos(\theta)}{\rho}\partial_{\theta},\qquad
V_{3}=\partial_{\theta}.$

Let

$F=f(T,\rho,\theta)\partial_{T}+g(T,\rho,\theta)\partial_{\rho}+h(T,\rho,\theta)\partial_{\theta}$

be a PD. The condition $[F,V_{3}]=0$ is equivalent to require that $f,g,h$ do
not depend on $\theta$. For simplicity of notation, we now drop the dependence
of $f,g,h$ on the remaining variables $T,\rho$. Then, we are left with
imposing $[F,{V_{1}}]=[F,{V_{2}}]=0$, i.e.

$\displaystyle[F,V_{1}]$ $\displaystyle=$ $\displaystyle
f\left(\cos(\theta)\partial_{\rho}-\frac{\sin(\theta)}{\rho}\partial_{\theta}\right)+g\left(\cos(\theta)\partial_{T}+T\frac{\sin(\theta)}{\rho^{2}}\partial_{\theta}\right)-hV_{2}$
$\displaystyle-\rho\cos(\theta)((\partial_{T}f)\partial_{T}+(\partial_{T}g)\partial_{\rho}+(\partial_{T}h)\partial_{\theta})-T\cos(\theta)((\partial_{\rho}f)\partial_{T}+(\partial_{\rho}g)\partial_{\rho}+(\partial_{\rho}h)\partial_{\theta})=0,$
$\displaystyle[F,V_{2}]$ $\displaystyle=$ $\displaystyle
f\left(\sin(\theta)\partial_{\rho}+\frac{\cos(\theta)}{\rho}\partial_{\theta}\right)+g\left(\sin(\theta)\partial_{T}-T\frac{\cos(\theta)}{\rho^{2}}\partial_{\theta}\right)+hV_{1}$
$\displaystyle-\rho\sin(\theta)((\partial_{T}f)\partial_{T}+(\partial_{T}g)\partial_{\rho}+(\partial_{T}h)\partial_{\theta})-T\sin(\theta)((\partial_{\rho}f)\partial_{T}+(\partial_{\rho}g)\partial_{\rho}+(\partial_{\rho}h)\partial_{\theta})=0.$

It is clear that this condition is equivalent to

$\cos(\theta)[F,{V_{1}}]+\sin(\theta)[F,{V_{2}}]=0,\qquad-\sin(\theta)[F,{V_{1}}]+\cos(\theta)[F,{V_{2}}]=0,$

i.e.

$\displaystyle
f\partial_{\rho}+g\partial_{T}-h\frac{T}{\rho}\partial_{\theta}-\rho((\partial_{T}f)\partial_{T}+(\partial_{T}g)\partial_{\rho}+(\partial_{T}h)\partial_{\theta})-T((\partial_{\rho}f)\partial_{T}+(\partial_{\rho}g)\partial_{\rho}+(\partial_{\rho}h)\partial_{\theta})=0,$
(4.1) $\displaystyle
f\frac{1}{\rho}\partial_{\theta}-g\frac{T}{\rho^{2}}\partial_{\theta}+h(\rho\partial_{T}+T\partial_{\rho})=0.$
(4.2)

Equation (4.2) is equivalent to both

$h=0,\qquad\rho f=Tg.$ (4.3)

By plugging these conditions into (4.1), we have

$f\partial_{\rho}+g\partial_{T}-(\partial_{T}(Tg))\partial_{T}-\rho(\partial_{T}g)\partial_{\rho}-T(\partial_{\rho}f)\partial_{T}-(\partial_{\rho}(\rho
f))\partial_{\rho}=0.$

A direct computation shows that this condition is equivalent to

$\partial_{T}g+\partial_{\rho}f=0.$ (4.4)

By using (4.3), we write $f=T\psi,g=\rho\psi$ for some $\psi=\psi(T,\rho)$ to
be found. Equation (4.4) then reads as
$\rho\partial_{T}\psi+T\partial_{\rho}\psi=0$, that is equivalent to state
that $\psi$ only depends on the variable $r:=\sqrt{T^{2}-\rho^{2}}$. Summing
up, it holds

$F(Z)=\psi(r)(T\partial_{T}+\rho\partial_{\rho})$ (4.5)

for some $\psi(r)$. Going back to cartesian coordinates and to the original
dynamics for $N=2$ particles, we have the result.

For $N=3$, we write the difference variables $Z_{2},Z_{3}$ as in the
statement. We then observe that elementary solutions $f$ of (2.13) are of the
form $\phi(Z_{2},Z_{3})Z_{2}+\psi(Z_{2},Z_{3})Z_{3}$ with $\phi,\psi$ being
$\hat{\mathfrak{L}}$-jointly radial with respect to the action of $SO(1,2)$.
The statement then follows from the fact that elementary solutions are
solution, see Proposition 14. ∎

######  Remark 40.

The case $N=2$ is interesting, since the following inclusions of solutions
holds $\mathcal{E}\subsetneq\mathcal{C}=\SS$, as already discussed in Remark
37. In the case case $N=3$, we are unable to completely describe the FPD,
since we only have $\mathcal{E}=\mathcal{C}\subset\SS$, but it is unclear
wether the last inclusion is proper or not.

A very similar case is given by $SO(3,\mathbb{R})\ltimes\mathbb{R}^{3}$ acting
on $\mathbb{R}^{3}$, already discussed in Remark 30. There, the centralizer is
too small to describe all solutions via Corollary 21, Item 1. Anyway,
computations for $N=1$ or $N=2$ can be carried out exactly as in the proof of
Proposition 39. This is one more example of the fact that one can characterize
PDs either for small $N$ by computations, or for large $N$ by Corollary 21,
Item 2. The intermediate values of $N$ are not completely solved by the theory
developed here.

We do not provide examples, since the case $N=1$ is trivial and $N\geq 2$ is
hard to picture, since one needs a plot in dimension 3 or larger. If one
considers PDs only depending on $T_{i}$ and
$\rho_{i}=\|(x_{i}-x_{1},y_{i}-y_{1})\|$, i.e. not depending on the argument
$\arg(x_{i}-x_{1},y_{i}-y_{1})$, observe that the dynamics of the
$T_{i},\rho_{i}$ variables is given by PDs on the relativistic line, studied
in Section 4.1.

##  5 Population Dynamics on spheres

In this section, we study Population Dynamics on the spheres of dimension 1
and 2.

###  5.1 Population dynamics on the circle $S^{1}$

In this section, we study Population Dynamics on the circle
$S^{1}\subset\mathbb{R}^{2}$. The natural group of invariance is
$SO(2,\mathbb{R})$, the rotations of the plane $\mathbb{R}^{2}$ around the
origin, that is isomorphic to $S^{1}$. We are then in the case in which the
Lie group $G$ and the manifold $M$ on which agents evolve coincide. We treated
this case in Section 2.5.

######  Proposition 41.

Consider the FPD associated to $(N,S^{1},SO(2,\mathbb{R}))$. Denote by
$\theta_{i}$ the angular position of agent $i\in\\{1,\ldots,N\\}$. Then,

  * 1.

for $N=1$, all PDs are given by constant vector fields;

  * 2.

for $N\geq 2$ all PDs are given by $F(\theta_{1},\ldots,\theta_{N})=\sum
f_{i}\partial_{\theta_{i}}$, where

$f_{i}(\theta_{1},\ldots,\theta_{N})=\phi_{i}(\theta_{2}-\theta_{1},\theta_{3}-\theta_{1},\ldots,\theta_{N}-\theta_{1})$
(5.1)

for some $\phi_{i}(\alpha_{2},\ldots,\alpha_{N})$ analytic function.

###### Proof.

The proof is a direct application of Proposition 23. Recall that $\alpha\in
SO(2)$ acts on $\theta\in S^{1}$ by addition $\theta+\alpha$, thus Proposition
23, Statement 1 ensures that jointly equivariant functions $f_{i}$ are of the
form given in (5.1); in particular, for $N=1$ they are constant. Since the
group operation is commutative, the basis of both right- and left-invariant
vector fields is simply $\partial_{\theta}$. By Proposition 23, Statement 2,
we have the result. ∎

We provide some examples for $N=2$. In Figure 3, we show the time evolution of
the states, that are in the interval $[0,2\pi]$ with identification $0=2\pi$.
Remark that one needs to have functions $\phi_{i}$ that are $2\pi$-periodic
with respect to their argument, that is always $\theta_{2}-\theta_{1}$.

Example C.1

The initial state is $\theta_{1}(0)=\pi-0.1$ and $\theta_{2}(0)=0.1$. By
choosing $\phi_{1}(a)=-\sin(a)+0.1$, $\phi_{2}(a)=-\sin(a)+0.1$, we have
convergence of both particles towards a trajectory of constant velocity 0.1.

Example C.2

The initial state is $\theta_{1}(0)=-0.1$ and $\theta_{2}(0)=0.1$. By choosing
$\phi_{1}(a)=|\sin(a/2)|$, $\phi_{2}(a)=-\phi_{1}(a)$, we have initial
divergence of particles. Yet, particles then converge to the same point
$\pi=-\pi$.

Example C.3

The initial state is again $\theta_{1}(0)=-0.1$ and $\theta_{2}(0)=0.1$. By
choosing $\phi_{1}(a)=\sin(a)$, $\phi_{2}(a)=-\phi_{1}(a)$, we have divergence
of particles. Compactness of the manifold implies that the distance between
particles cannot go to infinity, but to its maximum value $\pi$.

Example C.4

The initial state is $\theta_{1}(0)=0$ and $\theta_{2}(0)=\pi$. By choosing
$\phi_{1}(a)=\sin(2a)+2$, $\phi_{2}(a)=\sin(a)$, in particular with different
frequencies, we have a more complex behavior.

(a) Example C.1: $\phi_{1}(a)=\phi_{2}(a)=-\sin(a)+0.1$.

(b) Example C.2: $\phi_{1}(a)=|\sin(a/2)|$, $\phi_{2}(a)=-\phi_{1}(a)$

(c) Example C.3: $\phi_{1}(a)=\sin(a)$, $\phi_{2}(a)=-\phi_{1}(a)$

(d) Example C.4: $\phi_{1}(a)=\sin(2a)+2$, $\phi_{2}(a)=\sin(a)$

Figure 3: Examples C (1 to 4).

###  5.2 Action of $SO(3,\mathbb{R})$ on the sphere $S^{2}$

In this section, we describe population dynamics on the two-dimensional sphere
$S^{2}\subset\mathbb{R}^{3}$. From now on, we consider a sphere $S^{2}$ of
radius 1. We use polar coordinates on the $(x,y)$ plane and mainly consider
$z>0$ for the moment, i.e. we write

$x=\rho\cos(\theta),\qquad y=\rho\sin(\theta),\qquad z=\sqrt{1-\rho^{2}}.$

We have two different groups that naturally act on the sphere. On one side, we
consider the action of $SO(3,\mathbb{R})$, that represents all rotations of
the sphere. In the next Section 5.3, we will discuss the action of
$SO(2,\mathbb{R})$.

The Lie algebra $so(3,\mathbb{R})$ of $SO(3,\mathbb{R})$, that are all
rotations of $\mathbb{R}^{3}$ around the zero, are as follows in the
$\rho,\theta$-coordinates:

$A=\sqrt{1-\rho^{2}}\left(-\sin(\theta)\partial_{\rho}-\frac{\cos(\theta)}{\rho}\partial_{\theta}\right),\qquad
B=\sqrt{1-\rho^{2}}\left(\cos(\theta)\partial_{\rho}-\frac{\sin(\theta)}{\rho}\partial_{\theta}\right),\qquad
C=\partial_{\theta}.$

It holds $[A,B]=-C$, $[B,C]=-A$, $[C,A]=-B$.

We now study the FPD in this case.

######  Proposition 42.

Consider the FPD associated to $(N,S^{2},SO(3,\mathbb{R}))$. Then:

  * 1.

for $N=1$, the FPD is reduced to the null vector field;

  * 2.

for $N=2$, all PDs are of the form

$F(X)=\sum_{i=1}^{N}f_{i}(X)\partial_{X_{i}}\qquad\mbox{~{}~{} with
~{}~{}}\qquad
f_{i}=\phi_{i,1}(X)h(X_{1},X_{2})+\phi_{i,2}(X)h^{\perp}(X_{1},X_{2}),$

where $h(X_{1},X_{2})$ is the component of the gradient of the Riemannian
distance $d(X_{1},X_{2})$ on the variable $X_{1}$ and $h^{\perp}$ is its
orthogonal on $S^{2}$;

  * 3.

for $N\geq 3$, by choosing 3 distinct indexes $k_{1},k_{2},k_{3}$, all PDs are
of the form

$F(X)=\sum_{i=1}^{N}f_{i}(X)\partial_{X_{i}}\qquad\mbox{~{}~{} with
~{}~{}}\qquad
f_{i}=\phi_{i,1}(X)h(X_{k_{1}},X_{k_{2}})+\phi_{i,2}(X)h(X_{k_{1}},X_{k_{3}}),$

where $\phi_{i,k}(X)$ are $\hat{\mathfrak{L}}$-jointly radial functions of
$X=(X_{1},\ldots,X_{N})$ and $h(X_{a},X_{b})$ is the component of the gradient
of the Riemannian distance $d(X_{a},X_{b})$ on the variable $X_{a}$.

###### Proof.

We first consider the case $N=1$. Let $F$ be a PD, whose domain is required to
be open, dense and $G$-invariant: since $SO(3)$ acts transitively on $S^{2}$,
its domain needs to be the whole $S^{2}$. Since $S^{2}$ has Euler
characteristic 2, then $F$ has at least one zero, i.e. $F(\bar{X})=0$ for some
$\bar{X}\in S^{2}$. By equivariance of the vector field $F$, all points in the
orbit of $\bar{X}$ satisfy the same condition, i.e. $F(g\bar{X})=0$ for all
$g\in SO(3)$. Since the action is transitive, we have $F(X)=0$ for all $X\in
S^{2}$.

We now consider the case $N=2$. We have that $h$ is a solution, by recalling
that the distance is radial and applying Proposition 24. It is easy to prove
that the domain of $h$ is $\\{X_{1}\neq\pm X_{2}\\}$, that is $G$-invariant
and connected. Then, the definition of $h^{\perp}$ is unique, up to a single
change of sign. It is a solution, due to Corollary 25. It is easy to prove
that it is independent on $h$, thus $h,h^{\perp}$ provide a basis of solutions
for the 2-dimensional manifold. By Theorem 1, we have the result.

The case $N\geq 3$ is a direct application of Corollary 26.

∎

###  5.3 Action of $SO(2,\mathbb{R})$ on the sphere $S^{2}$

In this section, we consider PDs on $S^{2}$, where the group action is
$SO(2,\mathbb{R})$, that is the Lie group of all rotations of $\mathbb{R}^{3}$
around the $z$-axis. In this example, the group of rotations preserves a
specific axis: this is particularly interesting for models of swarms moving on
the Earth that take into account the magnetic field, i.e. that preserve the
Earth’s rotational axis passing through poles.

From now on, we consider the stereographic projection
$(x,y,z)\mapsto\left(\frac{x}{1-z},\frac{y}{1-z}\right)$, that is a
diffeomorphism between the sphere without poles $S^{2}\setminus\\{(0,0,\pm
1)\\}$ and the plane without the origin $\mathbb{R}^{2}\setminus\\{0\\}$.
Moreover, with this change of coordinates, the action of $SO(2,\mathbb{R})$ on
$S^{2}\setminus\\{(0,0,\pm 1)\\}$ is identical to the classical one (rotations
around the origin) on $\mathbb{R}^{2}\setminus\\{0\\}$. We then represent
equivariant trajectories on $S^{2}$ by their stereographic projection, for
simplicity of visualization.

######  Proposition 43.

Consider the FPD associated to
$(N,\mathbb{R}^{2}\setminus\\{0\\},SO(2,\mathbb{R}))$ with $N\geq 1$, that is
equivalent to the FPD associated to $(N,S^{2}\setminus\\{(0,0,\pm
1)\\},SO(2,\mathbb{R}))$ by stereographic projection. Write
$X_{i}=(x_{i},y_{i})=\rho_{i}(\cos(\theta_{i}),\sin(\theta_{i}))$ on
$\mathbb{R}^{2}\setminus\\{0\\}$ in Cartesian and polar coordinates.

PDs are of the form
$F={\displaystyle\sum\limits_{i=1}^{N}}f_{i,1}\partial_{x_{i}}+f_{i,2}\partial_{y_{i}}$,
with

$\left(\begin{array}[]{c}f_{i,1}\\\
f_{i,2}\end{array}\right)=\left(\begin{array}[c]{cc}x_{i}&-y_{i}\\\
y_{i}&x_{i}\end{array}\right)\left(\begin{array}[]{c}\phi_{i}(\rho_{1},\ldots,\rho_{N},\theta_{1}-\theta_{2},\ldots,\theta_{1}-\theta_{N})\\\
\psi_{i}(\rho_{1},\ldots,\rho_{N},\theta_{1}-\theta_{2},\ldots,\theta_{1}-\theta_{N})\end{array}\right).$

###### Proof.

Recall that the centralizer $\mathfrak{C}$ of $so(2,\mathbb{R})$ is
$\mathbb{R}.\mathrm{Id}+\mathbb{R}.J$ with $J$ given by (2.16). Since
$\mathfrak{C}$ has dimension 2, thus one can apply Theorem 2, Item 1. The
space of solutions is then given by $\phi(X)X+\psi(X)JX$ with
$\phi(X),\psi(X)$ being invariant with respect to the action of
$SO(2,\mathbb{R})$. By writing in polar coordinates
$X_{i}=\rho_{i}(\cos(\theta_{i}),\sin(\theta_{i}))$, one can apply a rotation
of angle $-\theta_{1}$ and introduce the variables
$Y_{i}=\rho_{i}(\cos(\theta_{i}-\theta_{1}),\sin(\theta_{i}-\theta_{1}))$.
Remark that, contrarily to the Euclidean case studied in Section 3, we still
have the non-zero variable $Y_{1}=\rho_{1}(1,0)$. This implies that
$\phi,\psi$ depend on variables
$\rho_{1},\ldots,\rho_{N},\theta_{2}-\theta_{1},\ldots,\theta_{N}-\theta_{1}$.
∎

######  Remark 44.

For $N=1$, we observe that the space of elementary solutions is strictly
contained in the space of solutions, i.e. $\mathcal{E}\subsetneq\SS$. Indeed,
it is reduced to solutions of the form $\phi(X)X$. We are anyway able to
completely characterize solutions, since $\mathcal{C}=\SS$, i.e. solutions
given by the centralizer are all solutions.

For $N\geq 2$, one can find alternative (equivalent) expressions for the PDs,
by applying Theorem 2, Item 2.

We now show some examples of PDs associated to
$(N,\mathbb{R}^{2}\setminus\\{0\\},SO(2,\mathbb{R}))$ in the simple cases of
$N=1,2$.

For $N=1$, in Figure 4-Left, we show four different examples of PDs, all
starting from $(x,y)=(1,1)$. We show that $\phi>0$ corresponds to move away
from the origin, while $\psi>0$ corresponds to rotate counter-clockwise.
Opposite signs mean opposite behavior. In particular, in Examples D.1, D.2 we
have positive $\phi$, with zero and negative $\psi$, respectively. In Example
D.3, we have $\phi\equiv 0$ and $\psi=10$, providing a periodic circular
trajectory with constant angular velocity. In Example D.4, we have
$\phi(r)=2-4r$, that provides stabilization of the radius to $0.5$; at the
same time, $\psi=50>0$ provides rotation with constant angular velocity.
Example D.4, that has a nature similar to Example A.3 above, shows that one
can stabilize the system towards a given radius (i.e. a given parallel on
$S^{2}$). Instead, stabilization of the angular variable (or both radial and
angular) is not achievable with such PDs, due to the invariance with respect
to rotations.

For $N=2$, in Figure 4-Right we show three different examples of PDs, in which
agent 1 is marked in green and agent 2 is marked in blue. Similarly to the
previous case, we can have convergence of both the radius and angular
variables (Example E.1), eventually choosing the radius (Example E.2, the
target radius being $r=2$). One can also converge to a chosen angular
difference (Example E.3, angular difference $\pi/8$), but cannot send agents
to a chosen common angle. We explicitly have the following choices of
functional parameters:

Example E.1

$\phi_{1}=\rho_{2}-\rho_{1}=-\phi_{2}$,
$\psi_{1}=\sin(\theta_{2}-\theta_{1})=-\psi_{2}$.

Example E.2

$\phi_{1}=2-\rho_{1}$, $\phi_{2}=2-\rho_{2}$,
$\psi_{1}=\sin(\theta_{2}-\theta_{1})=-\psi_{2}$.

Example E.3

$\phi_{1}=0.1$, $\phi_{2}=\rho_{1}-\rho_{2}$,
$\psi_{1}=\sin(\theta_{2}-\theta_{1}-\pi/8)=-\psi_{2}$.

(a) Examples D (1 to 4).

(b) Examples E (1 to 3).

Figure 4: Examples on $S^{2}$ with $N=1$ and $N=2$.

##  6 Volume preserving Population Dynamics on the plane

In this section, we consider Population Dynamics on the punctured plane
$\mathbb{R}^{2}\setminus\\{0\\}$ under the action of $SL(2)$. The case is
interesting, since it is the only one, among our examples, in which the group
is not a group of isometries (or pseudo-isometries).

The resulting dynamics on the punctured plane $\mathbb{R}^{2}\setminus\\{0\\}$
preserve volumes: given two points $X_{1}=(x_{1},y_{1})$ and
$X_{2}=(x_{2},y_{2})$, we denote by

$X_{1}\wedge X_{2}:=\left|\begin{array}[]{cc}x_{1}&x_{2}\\\
y_{1}&y_{2}\end{array}\right|=x_{1}y_{2}-y_{1}x_{2}$

the (oriented) area of the parallelogram with vertices
$0,X_{1},X_{1}+X_{2},X_{2}$. It is easy to prove that volumes are preserved by
the action of the special linear group $SL(2,\mathbb{R})$ composed of matrices
with determinant 1.

We now describe jointly radial functions and PDs for the action of $SL(2)$.

######  Proposition 45.

Let $M=\mathbb{R}^{2}\setminus\\{0\\}$. Consider the set
$\mathcal{O}=\\{X_{1}\wedge X_{2}\neq 0\\}\subset M^{N}$, that it is open
dense, $G$-invariant.

The jointly radial functions on $M^{N}$ defined on $\mathcal{O}$ are

$\psi(X_{1},\ldots,X_{N})=\phi(X_{1}\wedge
X_{2},[X_{1}|X_{2}]^{-1}X_{3},\ldots,[X_{1}|X_{2}]^{-1}X_{N}),$ (6.1)

where $[X_{1}|X_{2}]$ is the $2\times 2$-matrix with columns given by
$X_{1},X_{2}$ and $\phi$ is an analytic function of its $N-2$ variables in
$\mathbb{R}^{2}$. Consider now the FPD associated to $(N,,SL(2,\mathbb{R}))$.

  * 1.

For $N=1$, all PDs are of the form
$F=\alpha\left(x_{1}\partial_{x_{1}}+y_{1}\partial_{y_{1}}\right)$ for
$\alpha\in\mathbb{R}$.

  * 2.

For $N\geq 2$, choose two distinct indexes $j_{1},j_{2}\in\\{1,\ldots,N\\}$.
Then, all PDs are of the form

$F=\sum_{i=1}^{N}\phi_{1,i}\left(x_{j_{1}}\partial_{x_{i}}+y_{j_{1}}\partial_{y_{i}}\right)+\phi_{2,i}\left(x_{j_{2}}\partial_{x_{i}}+y_{j_{2}}\partial_{y_{i}}\right)$

where $\phi_{i,1},\phi_{i,2}$ are arbitrary jointly-radial functions, i.e. of
the form (6.1).

###### Proof.

We first prove that jointly radial functions are of the form (6.1). It is easy
to prove that $\psi$ of such form is jointly equivariant: the diagonal action
of $g\in SL(2)$ reads as

$\displaystyle\psi(gX_{1},\ldots,gX_{N})$ $\displaystyle=$
$\displaystyle\phi((gX_{1})\wedge(gX_{2}),[gX_{1}|gX_{2}]^{-1}(gX_{3}),\ldots,[gX_{1}|gX_{2}]^{-1}gX_{N})$
$\displaystyle=$ $\displaystyle\phi(X_{1}\wedge
X_{2},[X_{1}|X_{2}]^{-1}X_{3},\ldots,[X_{1}|X_{2}]^{-1}X_{N})=\psi(X_{1},\ldots,X_{N}).$

Here we used that the determinant is invariant under the action of $g$ and
that

$[gX_{1}|gX_{2}]^{-1}gX_{i}=[X_{1}|X_{2}]^{-1}g^{-1}gX_{i}=[X_{1}|X_{2}]^{-1}X_{i}.$

It is clear that all jointly radial functions are of this form, since the
variables in (6.1) play the role of coordinates in the quotient
$\mathcal{O}/G$.

We now study the case $N=1$. Let $F$ be a PD, which domain is a $G$-space.
Since $SL(2,\mathbb{R})$ acts transitively on $M$, its domain is the whole
$M$. Write $F(X)=f(x,y)\partial_{x}+g(x,y)\partial_{y}$ and write $[F,l]=0$
for $l$ in the basis of the Lie algebra $so(2,\mathbb{R})$, that is

$l_{1}=y\partial_{x},\qquad l_{2}=x\partial_{y},\qquad
l_{3}=x\partial_{x}-y\partial_{y}.$

A direct computation shows:

$[F,l_{1}]=g\partial_{x}-y(\partial_{x}f)\partial_{x}-y(\partial_{x}g)\partial_{y},\qquad[F,l_{2}]=f\partial_{y}-x(\partial_{y}f)\partial_{x}-x(\partial_{y}g)\partial_{y}.$

This implies that $g=g(y)$ and $f=f(x)$. Since $g(y)=y\partial_{x}f$, then
$g=ky$ for some $k\in\mathbb{R}$. Similarly, $f=kx$ for the same $k$. By
linearity, we are now left to prove that $F(x)=x\partial_{x}+y\partial_{y}$
satisfies

$[F,l_{3}]=(x\partial_{x}-y\partial_{y})-(x\partial_{x}-y\partial_{y})=0.$

This proves the result.

The case $N\geq 2$ is a direct application of Theorem 2, Item 2. ∎

##  7 Controlled Population Dynamics: the unicycle

In this section, we describe an interesting problem of Controlled Population
Dynamics as given in Definition 11, i.e. a PD in which the dynamics is
constrained to belong to a subset $\Omega$ of the Lie algebra of vector
fields. In the terminology of mechanical systems, constraints on the velocity
are known as non-holonomic.

###  7.1 The classical unicycle

The unicycle is a classical model of a vehicle on the plane $\mathbb{R}^{2}$
moving along straight lines or turning on itself, or combine these
displacements. The state space is then $\mathbb{R}^{2}\times S^{1}$, i.e. the
state is described by the position-orientation variables $(x,y,\theta)$. The
dynamics is given by

$\dot{x}=\cos(\theta)u(t),\qquad\dot{y}=\sin(\theta)u(t),\qquad\dot{\theta}=v(t).$
(7.1)

Here, $u(t),v(t)$ are the control variables, accounting for straight movements
and rotations, respectively. Remark that the constraint is linear, since
$\Omega$ is generated by
$F_{1}=\cos(\theta)\partial_{x}+\sin(\theta)\partial_{y},F_{2}=\partial_{\theta}$.
It is not a Lie algebra, since it holds
$[F_{1},F_{2}]=-\sin(\theta)\partial_{x}+\cos(\theta)\partial_{y}\not\in\Omega$.

One of the interests of this model in the context of PDs is that it is a
first-order system that allows to describe orientation of the displacements
(via the variable $\theta$). This is in sharp contrast with other models, such
as the celebrated Cucker-Smale model [14], in which orientation is encoded by
a second-order dynamics. Clearly, the advantage of dealing with first-order
systems is here mitigated by the non-holonomic constraint.

We now impose equivariance with respect to the action of the group $G=SE(2)$
of rototranslation of the plane. The action on $\mathbb{R}^{2}\times S^{1}$ is
the following: given an element

$g=\left(\begin{array}[]{ccc}\cos(\tau)&-\sin(\tau)&a\\\
\sin(\tau)&\cos(\tau)&b\\\ 0&0&1\end{array}\right)\in SE(2)$

and $(x,y,\theta)\in\mathbb{R}^{2}\times S^{1}$, it holds

$\Phi(g)\left(\begin{array}[]{c}x\\\ y\\\
\theta\end{array}\right)=\left(\begin{array}[]{ccc}\cos(\tau)&-\sin(\tau)&0\\\
\sin(\tau)&\cos(\tau)&0\\\ 0&0&1\end{array}\right)\left(\begin{array}[]{c}x\\\
y\\\ \theta\end{array}\right)+\left(\begin{array}[]{c}a\\\ b\\\
\tau\end{array}\right).$ (7.2)

The Lie algebra $\mathfrak{L}$ is then generated by the 3 vector fields
$\left\\{\partial_{x},\partial_{y},-y\partial_{x}+x\partial_{y}+\partial_{\theta}\right\\}$.
The equation (7.1) is the differential equation associated with left-invariant
vector fields, while (7.2) is associated with right-invariant vector field
(i.e. left translations).

We are now ready to describe the Controlled FPD for the unycicle.

######  Proposition 46.

Consider the Family of CPD associated to $(N,\mathbb{R}^{2}\times
S^{1},SO(2,\mathbb{R})\ltimes\mathbb{R}^{2},\Omega)$ with $N\geq 1$ and
$\Omega$ given by (7.1). Denote by $X_{i}=(x_{i},y_{i},\theta_{i})$ the state
of the $i$-th agent and $X:=(X_{1},\ldots,X_{N})$. Let

$\dot{X}_{i}=u_{i}(X)F_{i}+v_{i}(X)G_{i}\qquad\qquad\mbox{ with }\qquad
F_{i}=\cos(\theta_{i})\partial_{x_{i}}+\sin(\theta_{i})\partial_{y_{i}}\quad\mbox{~{}~{}
and ~{}~{}}\quad G_{i}=\partial_{\theta_{i}}.$

Then, the controlled FPD is characterized by the fact that $u_{i},v_{i}$ are
analytic functions of $\rho_{2},\ldots,\rho_{N}$,
$\alpha_{2}-\theta_{1},\ldots,\alpha_{N}-\theta_{1},\theta_{2}-\theta_{1},\ldots,\theta_{N}-\theta_{1}$
only, where
$Z_{i}:=(x_{i}-x_{1},y_{i}-y_{1})=\rho_{i}(\cos(\alpha_{i}),\sin(\alpha_{i}))$
for $i=2,\ldots,N$.

###### Proof.

We identify the unicycle variables $(x,y,\theta)$ with the element $X\in
SE(2)$ defined by

$X=\left(\begin{array}[]{ccc}\cos(\theta)&-\sin(\theta)&x\\\
\sin(\theta)&\cos(\theta)&y\\\ 0&0&1\end{array}\right).$

The action described in (7.2) can be rewritten as $\Phi(g)X=gX$, i.e. it is a
left translation on $SE(2)$. We can now study FPD on $SE(2)$ without the
constraint, that are completely characterized by Proposition 23. Indeed, the
FPD is given by

$\dot{X}_{i}=\sum_{j=1}^{3}\psi_{ij}(X_{1},\ldots,X_{N})(X_{i}\cdot l^{j}),$
(7.3)

where $l^{1},l^{2},l^{3}$ is a basis of the Lie algebra, hence $X\cdot
l^{1},X\cdot l^{2},X\cdot l^{3}$ is a basis of the left-invariant vector
fields. Such a basis can be chosen as

$l^{1}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
0&0&0\end{array}\right),\qquad l^{2}=\left(\begin{array}[]{ccc}0&-1&0\\\
1&0&0\\\ 0&0&0\end{array}\right),\qquad
l^{3}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\ 0&0&0\end{array}\right).$
(7.4)

Moreover, the $\psi_{ij}(X_{1},\ldots,X_{N})$ are jointly radial. Again by
Proposition 23, they are characterized by the fact that they are analytic
functions of $\rho_{2},\ldots,\rho_{N}$,
$\alpha_{2}-\theta_{1},\ldots,\alpha_{N}-\theta_{1},\theta_{2}-\theta_{1},\ldots,\theta_{N}-\theta_{1}$
only, where
$Z_{i}:=(x_{i}-x_{1},y_{i}-y_{1})=\rho_{i}(\cos(\alpha_{i}),\sin(\alpha_{i}))$
for $i=2,\ldots,N$.

We now identify the Controlled Population Dynamics, by simply verifying which
dynamics for the unicycle (7.1) are of the form (7.3). Observe that (7.1)
reads as $\dot{X}=u(t)F_{1}(X)+v(t)F_{2}(X)$ and that $F_{1}(X)=X\cdot l^{1}$,
$F_{2}(X)=X\cdot l^{2}$. As a consequence, solutions of the unicycle (7.1) are
of the form (7.3) if and only if $\psi_{i3}\equiv 0$. This proves the result.
∎

We now discuss examples. For $N=1$, controls $u,v$ need to be constant, hence
providing circular trajectories (that we do not present here). We provide a
single example for $N=2$ in Figure 5, where
$u_{1}=0.3\rho_{2},v_{1}=-.2\sin(\theta_{1}-\alpha_{2}),u_{2}=0.2\rho_{2},v_{2}=-0.1\rho_{2}$.
With such choice, agent 1 (green) aims to reach agent 2 (blue) with the same
angle. Moreover, the dynamics promotes convergence to an equilibrium, in which
$\rho_{2}=0$.

Figure 5: A Controlled Population Dynamics for the unicycle with $N=2$.

###  7.2 The relativistic unicycle

In this section, we consider a unicycle in space-time, by adding the time
variable $T$. The dynamics then takes place in $\mathbb{R}^{3}\times S^{1}$
and it can be written as follows:

$\dot{T}=f(t),\qquad\dot{x}=\cos(\theta)u(t),\qquad\dot{y}=\sin(\theta)u(t),\qquad\dot{\theta}=v(t).$
(7.5)

The natural group of equivariance for the variables $(T,x,y)$ only is the
group $SO(1,2)\ltimes\mathbb{R}^{3}$, that we already studied in Section 4.2.
No natural action of such group seems to take into account the non-holonomic
angular constraint for $(x,y,\theta)$ in (7.5).

A natural group of equivariance is $K=SO(1,1)\times SO(2,\mathbb{R})$, acting
as follows: by introducing polar coordinates for
$(x,y)=\rho(\cos(\alpha),\sin(\alpha))$, the action is
$(g,\omega).(T,\rho,\alpha,\theta)=(g.(T,\rho),\alpha+\omega,\theta+\omega)$,
where $g\in SO(1,1)$ is written as
$g=\left(\begin{array}[c]{cc}\cosh(\lambda)&\sinh(\lambda)\\\
\sinh(\lambda)&\cosh(\lambda)\end{array}\right)$. The action can be seen as
the natural “Lorentzian” group action, as it preserves the metric
$ds^{2}=dT^{2}-d\rho^{2}$. The rotation $\omega$ simultaneously acts on the
variables $\alpha$ and $\theta$. We have the following result.

######  Proposition 47.

Consider the controlled FPD associated to the relativistic unicycle
$(N,\mathbb{R}^{3}\times S^{1},SO(1,1)\times SO(2,\mathbb{R}))$.

For $N=1$, the controlled FPD is composed by vector fields

$T\cos(\theta-\alpha)\bar{u}(r,\theta-\alpha)\partial_{T}+\rho\cos(\theta-\alpha)\bar{u}(r,\theta-\alpha)\partial_{\rho}+\sin(\theta-\alpha)\bar{u}(r,\theta-\alpha)\partial_{\alpha}+v(r,\theta-\alpha)\partial_{\theta}$
(7.6)

with $\bar{u},v$ analytic functions and $r=\sqrt{T^{2}-\rho^{2}}$.

For $N\geq 2$, the FPD contains vector fields
$F=\sum_{i=1}^{N}F_{i}\partial_{X_{i}}$ with

$\displaystyle F_{i}$ $\displaystyle=$ $\displaystyle
T_{i}\cos(\theta_{i}-\alpha_{i})\bar{u}_{i}(X_{1},\ldots,X_{N})\partial_{T_{i}}+\rho_{i}\cos(\theta_{i}-\alpha_{i})\bar{u}_{i}(X_{1},\ldots,X_{N})\partial_{\rho_{i}}+$
$\displaystyle\sin(\theta_{i}-\alpha_{i})\bar{u}_{i}(X_{1},\ldots,X_{N})\partial_{\alpha_{i}}+\bar{v}_{i}(X_{1},\ldots,X_{N})\partial_{\theta_{i}},$

where $\bar{u}_{i},\bar{v}_{i}$ are jointly radial functions of their
variables.

###### Proof.

We first focus on $N=1$. We use polar coordinates for
$(x,y)=\rho(\cos(\alpha),\sin(\alpha))$. The system reads as

$\dot{T}=f(T,\rho,\alpha,\theta),\qquad\dot{\rho}=\cos(\theta-\alpha)u(T,\rho,\alpha,\theta),\qquad\dot{\alpha}=\frac{1}{\rho}\sin(\theta-\alpha)u(T,\rho,\alpha,\theta),\qquad\dot{\theta}=v(T,\rho,\alpha,\theta).$

The action of $SO(2,\mathbb{R})$ implies that the dynamics depends on the
variables $T,\rho,\theta-\alpha$ only, thus $f,u,v$ only depend on them.

The action of $SO(1,1)$ implies two conditions: first, that functions $f,u,v$
depend on the variables $r,\theta-\alpha$ only, with
$r=\sqrt{T^{2}-\rho^{2}}$. Second, it implies
$f(T,\rho,\theta-\alpha)=\psi(r,\theta-\alpha)T$ and
$\cos(\theta-\alpha)u(T,\rho,\theta-\alpha)=\psi(r,\theta-\alpha)\rho$ for an
analytic function $\psi(r,\theta-\alpha)$. Computations are identical to the
proof of Proposition 39 with $N=2$. We now write
$\psi(r,\theta-\alpha)=\cos(\theta-\alpha)\bar{u}(r,\theta-\alpha)$ for an
analytic function $\bar{u}(r,\theta-\alpha)=\frac{u(t)}{\rho}$ and find (7.6).

For $N\geq 2$, it is easy to prove that $F$ defined by (47) are PD, with the
same computations. ∎

##  8 Population Dynamics for quantum agents

In this section, we discuss the problem of joint equivariance of systems of
quantum particles. In the standard formalism of quantum theory (see e.g.
[33]), particles are described by (square roots of) probability densities. We
consider a finite-dimensional dynamics, i.e. $n\in\mathbb{N}$ possible states.
The Schrödinger equation describing the evolution of a single particle is
then:

$\dot{\Psi}=A\Psi,$ (8.1)

where $A$ is a skew-adjoint operator over $\mathbb{C}^{n}\cong
l^{2}(\\{1,\ldots,n\\},\mathbb{C})$. Here we have $<\Psi,\bar{\Psi}>=1$ and
the probability that the particle is in the state $k\in\\{1,\ldots,n\\}$ is
$p_{k}=\Psi_{k}\bar{\Psi}_{k}$.

We start by considering $N$ non-interacting particles, indexed by
$i\in\\{1,\ldots,N\\}$. The particle $i$ has $n_{i}$ possible states, the
quantum state is denoted by $\Psi_{i}$ and the dynamics is given by a matrix
$A_{i}$. It is clear that the state of the whole system is
$X=\Psi_{1}\otimes\Psi_{2}\otimes\ldots\otimes\Psi_{N}\in\mathbb{C}^{n_{1}}\otimes\ldots\otimes\mathbb{C}^{n_{N}}$,
since the probability of $X$ to be in the state $(k_{1},\ldots,k_{N})$ is just
$p_{k_{1}}p_{k_{2}}\ldots p_{k_{N}}$. Its evolution is indeed given by

$\displaystyle X(t)$ $\displaystyle=$
$\displaystyle\Psi_{1}(t)\otimes\Psi_{2}(t)\otimes\ldots\otimes\Psi_{N}(t)$
$\displaystyle=$ $\displaystyle(e^{tA_{1}}\otimes\ldots\otimes
e^{tA_{N}})(\Psi_{1}(0)\otimes\Psi_{2}(0)\otimes\ldots\otimes\Psi_{N}(0)).$

By computing the corresponding infinitesimal generator $\mathcal{A}$, we have

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle
A_{1}\otimes\mathrm{Id}_{n_{2}}\otimes\ldots\otimes\mathrm{Id}_{n_{N}}+\mathrm{Id}_{n_{1}}\otimes
A_{2}\otimes\ldots\otimes\mathrm{Id}_{n_{N}}+\mathrm{Id}_{n_{1}}\otimes\ldots\otimes\mathrm{Id}_{n_{N-1}}\otimes
A_{N},$

where $\mathrm{Id}_{n}$ is the identity matrix of dimension $n$. The key point
is then the following: while in classical kinematic systems the joint dynamics
occurs on the direct sum of the state spaces, here the natural context is the
tensor product of the individual state spaces.

Assume now that all particles lie in the same state space, that is then
$\mathbb{C}^{n}$ for a common dimension $n\in\mathbb{N}$. The state space for
the whole system is then $(\mathbb{C}^{n})^{\otimes
N}\simeq\mathbb{C}^{(n^{N})}$. Let $G$ be a symmetry group acting on
$\mathbb{C}^{n}$, and denote by $\mathfrak{L}$ its Lie algebra. Then, the
infinitesimal joint action on $(\mathbb{C}^{n})^{\otimes N}$ for
$l\in\mathfrak{L}$ is

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle
l\otimes\mathrm{Id}_{n}\otimes\ldots\otimes\mathrm{Id}_{n}+\mathrm{Id}_{n}\otimes
l\otimes\ldots\otimes\mathrm{Id}_{n}+\mathrm{Id}_{n}\otimes\ldots\otimes\mathrm{Id}_{n}\otimes
l.$

This general idea is now developed on a specific case.

###  8.1 Family of Population Dynamics for two quantum agents in
$\mathbb{C}^{2}$

In this section, we present a simple case for the general theory of PDs with
quantum symmetries. We focus on FPD of $N=2$ agents with state
$\Psi=\Psi_{1}\otimes\Psi_{2}$ evolving on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ under the $SU(2)$ symmetry. This is one
of the most important symmetries in quantum physics, the symmetry of quantum
spins. We require the dynamics to be equivariant with respect to its diagonal
action, i.e. with respect to the action

$g.(\Psi_{1}\otimes\Psi_{2})=g\Psi_{1}\otimes g\Psi_{2}.$ (8.2)

The Lie algebra is then $\mathfrak{L}=su(2)$. We want to characterize vector
fields $F$ satisfying

$[F(\Psi),(l\otimes\mathrm{Id}_{2})\Psi+(\mathrm{Id}_{2}\otimes
l)\Psi]=0\qquad\mbox{for all ~{}~{}~{}}l\in su(2).$ (8.3)

From now on, we use the following identification:
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}=\mathcal{M}_{2}(\mathbb{C})$, i.e. the
space of quadratic forms on $\mathbb{C}$. We first observe that the action
(8.2) now reads as

$g.X=gXg^{T},$ (8.4)

where $X\in\mathcal{M}_{2}(\mathbb{C})$ and $g^{T}$ is the transpose of $g\in
SU(2)$. Remark that the action (8.4) is reducible. Indeed, by using the
Clebsch-Gordan decomposition ([33]), the unitary diagonal action of $SU(2)$
over $\mathcal{M}_{2}(\mathbb{C})$ is (unitarily equivalent to) the direct sum
of two irreducible components: the (natural) action of $SU(2)$ on symmetric
and skew-symmetric matrices, respectively. This is the 4=3+1 of physicists.

######  Remark 48.

We stress here that we deal with (standard) transposition of matrices, and not
on the conjugate transposition. This explains the interest of the Takagi
decomposition used below.

The structure of skew-symmetric matrices in $\mathcal{M}_{2}(\mathbb{C})$ is
very simple, as they satisfy $S=aJ$ for some $a\in\mathbb{C}$ and $J$ is given
by (2.16). For symmetric matrices, we need the following useful decomposition.

######  Proposition 49 (Takagi decomposition [32]).

Any complex symmetric matrix $S$ of dimension $2\times 2$ satisfies the
following decomposition: there exists $U\in U(2)$ unitary matrix such that

$S=U\Delta
U^{T}\qquad\mbox{~{}~{}with~{}~{}}\qquad\Delta=\left(\begin{array}[c]{cc}\delta_{1}&0\\\
0&\delta_{2}\end{array}\right).$

Here, $\delta_{1},\delta_{2}\geq 0$ are the singular values of $S$.

As a consequence, there exists $\omega$ phase factor and $H\in SU(2)$ such
that

$S=e^{i\omega}H\Delta H^{T}.$ (8.5)

Moreover, for $\delta_{1}\neq\delta_{2}$, the parameters $(\Delta,\omega,H)$
are uniquely determined by $S$.

As a further consequence, define $\mathcal{D}_{S}$ the set of complex
symmetric matrices with distinct singular values, that is an open dense
$SU(2)$-space, i.e. invariant under the action of $SU(2)$. For all
$S\in\mathcal{D}_{S}$, the Takagi invariants $(\Delta,\omega)$ form a complete
set of invariants under the action (8.4). Moreover, these invariants smoothly
depend on $S$.

###### Proof.

See [32]. It is clear that $\delta_{1},\delta_{2}$ are the singular values,
since

$S^{*}S=(U^{T})^{*}\Delta U^{*}U\Delta U^{T}=(U^{T})^{*}\Delta^{2}U^{T}.$

The passage from $U\in U(2)$ to $H\in SU(2)$ is also direct, since any unitary
matrix satisfies $U=\exp(i\alpha)H$ with $\alpha\in\mathbb{R}/(2\pi)$. Thus,
equation (8.5) holds with $\omega=2\alpha$. ∎

We are now ready to describe jointly radial functions. The idea here is to use
the (unique) Takagi decomposition of the symmetric part of $X$ to define
invariants.

######  Proposition 50.

Let $\mathcal{D}$ be the set of matrices $X$ in $\mathcal{M}_{2}(\mathbb{C})$
such that the symmetric part $S$ of $X$ has distinct eigenvalues. The
following holds:

  * 1.

the set $\mathcal{D}$ is a $G$-space, i.e. open, dense and invariant under the
diagonal action of $SU(2)$;

  * 2.

by writing $X=S+A$ and applying the Takagi decomposition (8.5) to $S$, it
holds that the set of 5 real parameters
$(\omega(S),\delta_{1}(S),\delta_{2}(S),A)$ is a complete set of independent
invariants.

As a consequence, the ring $\mathcal{R}$ of jointly radial functions with
domain $\mathcal{D}$ is the space of functions of such parameters.

###### Proof.

The first result is a direct consequence of the fact that the diagonal action
of $SU(2)$ on $\mathcal{M}_{2}(\mathbb{C})$ is decomposed into the actions on
the symmetric and skew-symmetric part. Thus, if $S$ has distinct singular
values, it belongs to $\mathcal{D}_{S}$, that is $G$-invariant.

As for the second result, it is clear that
$\omega(S),\delta_{1}(S),\delta_{2}(S)$ are invariants, as a consequence of
the Takagi decomposition. Moreover, the matrix $H\in SU(2)$ is unique, thus we
can apply it to the matrix $X=S+A$, thus in particular to its skew-symmetric
part $A=aJ$. A direct computation shows that it holds $HAH^{T}=A$, thus the
parameter $a\in\mathbb{C}$ is invariant under the group action. By considering
the dimension over the space $\mathbb{R}$, we then have the 5-dimensional set
of parameters $(\omega(S),\delta_{1}(S),\delta_{2}(S),a)$ for the
8-dimensional space $\mathcal{M}_{2}(\mathbb{C})$, under the action of the
3-dimensional group $SU(2)$. Thus, the set of parameters is complete. ∎

We are now ready to describe some PDs for
$(N=2,\mathbb{C}^{2}\otimes\mathbb{C}^{2},SU(2))$. Due to the identification
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}=\mathcal{M}_{2}(\mathbb{C})$, we need to
rewrite the Lie algebra condition (8.3). By computing the infinitesimal
generator of (8.4), it reads as $l.X+X.l^{T}$ for some $l\in su(2)$, thus
condition (8.3) reads as

$[F(X),l.X+X.l^{T}]=0\qquad\mbox{for all ~{}~{}~{}}l\in su(2).$ (8.6)

We then aim to describe all vector fields $F$ satisfying condition (8.6). It
is clear that the dynamics on $\mathcal{M}_{2}(\mathbb{C})$ can also be
decomposed in the symmetric and skew-symmetric parts. Thus, the infinitesimal
generator of the group action $(l.X+X.l^{T})\partial_{X}$ now reads as
$(l.S+S.l^{T})\partial_{S}+(l.A-A.l^{T})\partial_{A}$. A general vector field
now reads as $F(S,A)=f_{1}(S,A)\partial_{S}+f_{2}(S,A)\partial_{A}$. Condition
(8.6) splits as follows:

$\displaystyle f_{1}(lS+Sl^{T})-(lS+Sl^{T})\partial_{S}f_{1}-(lA-
Al^{T})\partial_{A}f_{1}=0$ (8.7) $\displaystyle
f_{2}(lA+Al^{T})-(lS+Sl^{T})\partial_{S}f_{2}-(lA-Al^{T})\partial_{A}f_{2}=0$
(8.8)

These equations play the same role as (2.9) for linear actions on Euclidean
spaces. Even though the equations are different, they share a common important
feature: solutions of (8.7) are a module over the ring $\mathcal{R}$ of
jointly radial functions, and the same holds for solutions of (8.8). The proof
is identical to Theorem 1.

We are unable to provide all solutions to (8.7), while completely solving
(8.8) is easy. We are interested in the dynamics preserving the unit ball of
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Our result is stated here.

######  Proposition 51.

Consider the FPD associated to
$(N=2,\mathbb{C}^{2}\otimes\mathbb{C}^{2},SU(2))$, i.e. to
$(N=2,\mathcal{M}_{2}(\mathbb{C}),SU(2))$ preserving the unit ball of
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Decompose
$X\in\mathcal{M}_{2}(\mathbb{C})$ as $X=S+A$, its symmetric and skew-symmetric
part, respectively. Then, the FPD contains the following vector fields:

$F(X)=i\phi_{1}(S,A)S\partial_{S}+i\phi_{2}(S,A)A\partial_{A}$

for any real-valued jointly-radial functions
$\phi_{1},\phi_{2}\in\mathcal{R}$. Moreover, all solutions $f_{2}$ of (8.8)
are of the form $i\phi_{2}(S,A)A$.

###### Proof.

The first result is a direct consequence of the fact that $iS$ is a solution
to (8.7), while $iA$ is a solution to (8.8).

It is clear that solutions of (8.8) define vector fields on the manifold of
skew-symmetric matrices, that is 2-dimensional at each point. Since the space
of solutions of $i\phi_{2}(S,A)A$ spans the tangent space at each point, we
have all solutions.

Preservation of the unit ball is standard, since vector fields now read as
$i\phi_{1}S\partial_{S}+i\phi_{2}A\partial_{A}$ with $\phi_{1},\phi_{2}$ real
functions of variables $S,A$. ∎

######  Remark 52.

The (real) dimension of unit ball in the space
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ is 7. The space of vector fields given
in Proposition 51 is 2-dimensional (1 for the symmetric and 1 for the skew-
symmetric ones). Then, we are missing a 5-dimensional space of vector fields,
all related to the dynamics of the symmetric parts.

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