[ beforeskip=.2em plus 1pt,pagenumberformat=]toclinesection [ beforeskip=.2em
plus 1pt,pagenumberformat=]toclinesection

# Classical Mechanics in Noncommutative Spaces:
Confinement and More

Vladislav Kupriyanov Centro de Matemática, Computação e Cognição, Universidade
Federal do ABC, Santo André, SP, Brazil Maxim Kurkov Dipartimento di Fisica
“E. Pancini”, Università di Napoli Federico II, Complesso Universitario di
Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy.  Alexey Sharapov
Physics Faculty, Tomsk State University, Lenin ave. 36, Tomsk 634050, Russia

###### Abstract

We consider a semi-classical approximation to the dynamics of a point particle
in a noncommutative space. In this approximation, the noncommutativity of
space coordinates is described by a Poisson bracket. For linear Poisson
brackets, the corresponding phase space is given by the cotangent bundle of a
Lie group, with the Lie group playing the role of a curved momentum space. We
show that the curvature of the momentum space may lead to rather unexpected
physical phenomena such as an upper bound on the velocity of a free
nonrelativistic particle, bounded motion for repulsive central force, and no-
fall-into-the-centre for attractive Coulomb potential. We also consider a
superintegrable Hamiltonian for the Kepler problem in $3$-space with
$\mathfrak{su}(2)$ noncommutativity. The leading correction to the equations
of motion due to noncommutativity is shown to be described by an effective
monopole potential.

##  1 Introduction

The idea of noncommutative spacetime, being quite natural, goes back to the
early days of quantum theory. Indeed, if position and momentum coordinates
become noncommuting operators when quantized, why not assume the same for
position coordinates alone? Then the generalized uncertainty principle $\Delta
x^{a}\Delta x^{b}\geq\ell^{2}$ would bring about a fundamental length $\ell$,
setting the absolute limit for locating a spacetime point. Intuitively,
abandoning local interactions of particles could solve, or at least alleviate,
the problem of ultraviolet divergences. This line of reasoning was first spelt
out by Snyder [1]. Further developments in quantum field theory, particularly
the success of the renormalization program in quantum electrodynamics,
completely eclipsed these ideas for many decades. Renewed interest in
noncommutative spaces was sparked by the work of Seiberg and Witten [2] on the
eve of the millennium. They considered the quantization of open strings ending
on $D$-branes in the presence of a constant $B$-field. It was shown that the
low-energy effective field theory on a single $D$-brane has a consistent
deformation to a noncommutative gauge theory. The subsequent explosion of
papers on noncommutative field theory (see [3, 4, 5, 6] for a comprehensive
review) was also supported by parallel advances in mathematics, most notably
the theory of deformation quantization, quantum groups, and noncommutative
geometry.

Another old proposal for eliminating ultraviolet divergences is based on the
hypothesis of curved momentum space. This was first put forward by Born [7, 8]
and then considered by many authors (see e.g. [9, 10, 11, 12, 13, 14, 15, 16,
17, 18] and references therein). If the particle’s momentum space is ‘curved
enough’ to have a finite volume, then we get an effective cut-off that makes
all loop integrals finite at the upper limit. In particular, one can consider
a compact momentum space, in which case there exists an upper bound on the
uncertainty of momenta, $\Delta p\leq M$. According to Heisenberg’s
uncertainty relation $\Delta x\Delta p\geq\hbar/2$, this yields immediately a
lower bound for the uncertainty of positions $\Delta x\geq\hbar/M$. Thus, both
the hypotheses – noncommutativity and curvature – introduce a fundamental
length scale $\ell=\hbar/M$, leading to the granular structure of spacetime.
This also indicates a certain connection between the curvature of momentum
space and the noncommutativity of spacetime.

Since Born’s pioneering paper [7], the curvature of the momentum space has
usually been treated within the framework of Riemannian geometry. In our
opinion, this choice of geometric structure was motivated mainly by a formal
analogy with general relativity, without any deep physical or mathematical
reasons. It is also clear that there is no point in considering momentum space
separately from configuration one since only together do they constitute the
phase space of a scalar particle. If the configuration space turns out to be
noncommutative, this should affect the corresponding momentum space and vice
versa. In the recent paper [19], this interplay between the curvature of
momentum space and the noncommutativity of spacetime was formalized
mathematically in the framework of symplectic groupoids. The idea was to
consider semi-classical approximation wherein the noncommutativity of
spacetime coordinates is controlled by their Poisson brackets. Upon (strict,
deformation, geometric, canonical, etc.) quantization, these Poisson brackets
are expected to give rise to a noncommutative algebra of quantum observables
generated by the operators of spacetime coordinates. The whole phase space of
a scalar particle is then identified with a symplectic groupoid integrating
the underlying Poisson bracket. Such integration is not always possible, but
when it is possible, it is essentially unique. For example, the Lie-algebraic
noncommutativity defines and is defined by linear Poisson brackets of
coordinates, $\\{x^{a},x^{b}\\}=f^{ab}_{c}x^{c}$. In this case, the conjugated
momentum space is given by a Lie group integrating the corresponding Lie
algebra, and the total phase space is the cotangent bundle of the Lie group
endowed with the canonical symplectic structure. As is well known, there is a
unique simply connected Lie group with a given Lie algebra. If the Lie algebra
happens to be compact, then so is the corresponding Lie group. This provides a
large number of examples of mechanical systems with compact momentum spaces.
For a simple Lie algebra, the corresponding Lie group enjoys a metric that is
invariant under the right and left translations. This metric, being unique up
to a constant factor, allows us to speak about the momentum space as a
(pseudo-)Riemannian manifold. It should be emphasized that no canonical
metrics exist on momentum spaces associated with general (linear or nonlinear)
Poisson brackets. From this perspective, the existence of (pseudo-)Riemannian
metrics for simple Lie groups and algebras is an accident.

Not only does the geometry of symplectic groupoids underlay the kinematics of
point particles, but it also provides a geometric framework for formulating
Poisson electrodynamics [20, 21, 22], which is defined as a semi-classical
limit of noncommutative gauge theories. As explained in [19], the gauge fields
enter this picture as bisections of a symplectic groupoid. The bisections form
an infinite-dimensional Lie group and describe the low-energy limit of the
gauge vector field in noncommutative spacetime. Unlike Maxwell’s
electrodynamics, neither the target space nor the field equations of Poisson
electrodynamics are linear. This nonlinearity reflects the noncommutativity of
spacetime and disappears in the commutative limit. The conventional $U(1)$
gauge group is replaced here by the subgroup of Lagrangian bisections, which
act on the gauge fields by left (right) translations defining finite gauge
transformations***A bit later the authors of [23] obtained finite Poisson
gauge transformations integrating the sections of the corresponding Lie
algebroid.. Using the reach geometry of symplectic groupoids, it is also
possible to define gauge-covariant and gauge-invariant strength tensors of the
electromagnetic field as well as to introduce a minimal electromagnetic
coupling to point particles and matter fields [24]. All in all, the concept of
symplectic groupoids looks very promising for studying noncommutative dynamics
of fields and particles in semi-classical approximation.

It should be noted that the construction of a symplectic groupoid that
integrates a given Poisson structure is a difficult mathematical problem,
whose explicit solution is known only in some special cases. The first
nontrivial and best-studied case is that of linear Poisson brackets mentioned
above. Therefore, in this paper, we focus on the classical dynamics of point
particles whose position ‘coordinates’ form a Lie algebra. In semi-classical
approximation, this gives rise to Hamiltonian dynamics on the cotangent bundle
$T^{\ast}G$ of the corresponding Lie group $G$. A typical example of such a
Hamiltonian system is the motion of a rigid body about a fixed point. Here,
the configuration space is given by the rotation group $SO(3)$, whose
cotangent bundle – the phase space of the system – is diffeomorphic to
$SO(3)\times\mathbb{R}^{3}$. This means that, formally, a nonrelativistic
particle moving in $3$-space with $\mathfrak{so}(3)$ noncommutativity and a
rigid body with a stationary point inside the body have the same phase space.
The fundamental difference, however, lies in the physical interpretation of
the phase space variables: what is the configuration space for the rigid body
becomes the momentum space for the particle and vice versa. The compactness of
the rotation group $SO(3)$, considered as momentum space†††In Sec. 4, we
replace $SO(3)$ with its universal covering $SU(2)$, which is also compact.,
has far-reaching consequences. For example, the $3$-velocity of a free
(nonrelativistic!) particle turns out to be bounded from above, provided the
kinetic energy is a smooth function of momenta. In Sec. 3, we consider the
motion of such a particle in a central potential and show that it cannot fall
to the centre, whatever attractive force. For any repulsive central force, we
find a domain of bounded motion. All these observations contradict the usual
physical intuition. Sec. 4 is devoted to a detailed study of Kepler’s problem
for a particle with $\mathfrak{so}(3)\simeq\mathfrak{su}(2)$ noncommutativity
of space coordinates. As momentum space, we take the group manifold
$SU(2)\sim\mathbb{S}^{3}$, so that the total phase space is given by the
product $\mathbb{R}^{3}\times\mathbb{S}^{3}$. We show that all space
trajectories are plane curves provided the kinetic energy is invariant under
rotations. However, unlike the commutative case, the planes do not pass
through the centre of force. Finally, we present a Hamiltonian for which all
spatial trajectories are conical curves, as in the commutative Kepler problem.

##  2 Noncommutative mechanics in semi-classical approximation

Noncommutative geometry studies noncommutative algebras as if they were
algebras of functions on spaces, like, e.g., the commutative algebras
associated with smooth manifolds. Such geometric models for noncommutative
algebras provide a natural mathematical framework for classical and quantum
field theory in noncommutative spacetime. The correspondence principle between
commutative and noncommutative physics implies that appropriate noncommutative
algebras must be close to the usual commutative algebras of smooth functions,
at least starting from some fundamental scale $\ell$. This, in turn, suggests
the existence of a one-parameter family of algebras $\mathfrak{A}_{\ell}$ such
that $\mathfrak{A}_{0}$ is isomorphic to the algebra of smooth functions on a
given spacetime manifold $X$. In other words, $\mathfrak{A}_{\ell}$ can be
considered as a deformation of the commutative algebra $C^{\infty}(X)$, with
the deformation parameter $\ell\geq 0$ controlling the degree of non-
commutativity. The product in $\mathfrak{A}_{\ell}$, denoted by $\star$,
should then have the form

$\displaystyle f\star g=f\cdot
g+\frac{\ell}{2}\\{f,g\\}+\mathcal{O}(\ell^{2})\qquad\forall f,g\in
C^{\infty}(X)\,.$ (2.1)

If we further assume that the $\star$-product algebra is associative for all
$\ell$, then the bilinear operator determining the first-order deformation
must be equivalent to some Poisson bracket on $X$. In terms of local
coordinates $x^{a}$, each Poisson bracket on $X$ defines and is defined by a
Poisson bivector

$\Theta^{ab}(x)=\\{x^{a},x^{b}\\}\,.$ (2.2)

The celebrated Kontsevich formality theorem [25] ensures the existence of
higher-order terms in (2.1), complementing the first-order deformation to an
associative product. Using the terminology of quantum mechanics, we will refer
to the Poisson manifold $(X,\Theta)$ as a semi-classical approximation to
noncommutative spacetime (= the algebra $\mathfrak{A}_{\ell}$). It is this
approximation that we will deal with in this paper.

Regarding now the Poisson manifold $X$ as a configuration space of a point
scalar particle, we can ask about the corresponding phase space, that is, the
space of states of the particle. With the correspondence principle in mind, we
should expect this to be given by a symplectic manifold $\mathcal{G}$ whose
dimension is equal to twice the dimension of the configuration space $X$. The
compatibility between phase space and configuration space dynamics implies
that the Poisson brackets on $\mathcal{G}$, associated with the symplectic
structure, should reproduce the initial Poisson brackets (2.2) on functions
that depend exclusively on the spacetime coordinates $x^{a}$. Among other
things, this means that the symplectic manifold $\mathcal{G}$ is a fibre
bundle over the Poisson manifold $X$ whose projection ${\sf
s}:\mathcal{G}\rightarrow X$ is a Poisson map. This is known as a symplectic
realization of a Poisson manifold $(X,\Theta)$.

In general, finding a symplectic realization for a given symplectic manifold
is a difficult problem. Moreover, the problem may have different solutions in
different dimensions [26, 27], which makes the definition of a particle’s
phase space quite ambiguous. In [19], it was suggested to narrow the ambiguity
by requiring the phase space to be a symplectic groupoid integrating the
Poisson bracket on $X$. Here we will refrain from discussing symplectic
groupoids in any detail, referring the interested reader to the books [28, 29,
30]. In a nutshell, a symplectic groupoid $\mathcal{G}$ over a Poisson
manifold $X$ is a symplectic manifold of $\dim\mathcal{G}=2\dim X$ endowed
with a compatible groupoid structure. As part of this structure, there exists
a pair of canonical projections ${\sf s},{\sf t}:\mathcal{G}\rightarrow X$,
called the source and target, each of which makes $\mathcal{G}$ into a fibre
bundle over $X$. It is also required that the source map ${\sf s}$ be Poisson
and the target map ${\sf t}$ be anti-Poisson:

$\\{{\sf s}^{\ast}f,{\sf
s}^{\ast}g\\}_{\mathcal{G}}=\\{f,g\\}_{X}\,,\qquad\\{{\sf t}^{\ast}f,{\sf
t}^{\ast}g\\}_{\mathcal{G}}=-\\{f,g\\}_{X}\qquad\forall f,g\in
C^{\infty}(X)\,.$ (2.3)

The existence of two canonical projections is stressed in the shorthand
notation $\mathcal{G}\rightrightarrows X$ for a groupoid $\mathcal{G}$ over
$X$. One more ingredient of the groupoid structure is an involutive map ${\sf
i}:\mathcal{G}\rightarrow\mathcal{G}$, called inversion, that interchanges
source and target projection: ${\sf s}\circ{\sf i}={\sf t}$ and ${\sf
t}\circ{\sf i}={\sf s}$. From the viewpoint of classical mechanics, the fibre
${\sf s}^{-1}(x)$ is the space of (generalized) momenta at $x\in X$.

A trivial example of symplectic groupoids is provided by the cotangent bundle
$T^{\ast}X$ endowed with the canonical symplectic structure. This groupoid
integrates the zero Poisson bracket on $X$ and describes the phase space of an
ordinary mechanical system. Here the source and target maps coincide with each
other and with the canonical projection ${p}:T^{\ast}X\rightarrow X$. The
inversion acts by inverting the direction of canonical momenta.

Notice that whatever configuration space $X$, the generalised momenta of an
ordinary mechanical system always form a linear space, the cotangent space
$T^{\ast}_{x}X$ at $x\in X$. For a general symplectic groupoid
$\mathcal{G}\rightrightarrows X$ the role of a momentum space at $x\in X$ is
played by the fibre ${\sf s}^{-1}(x)\subset\mathcal{G}$, which may well be a
manifold with nontrivial topology. The possibility of a curved momentum space
sharply distinguishes noncommutative classical mechanics from its commutative
counterpart and has far-reaching dynamical implications as we will see soon.

In general, not every Poisson manifold $(X,\Theta)$ can be integrated into a
smooth symplectic groupoid $\mathcal{G}\rightrightarrows X$. For such
integration to be possible, the Poisson manifold should meet some
integrability conditions [30]. When the conditions are satisfied, the
corresponding Poisson structure is called integrable. Significantly, every
integrable Poisson structure gives rise to a symplectic groupoid, which is
unique up to covering.

Given a symplectic groupoid $\mathcal{G}\rightrightarrows X$, one defines
dynamics by specifying a Hamiltonian $H\in C^{\infty}(\mathcal{G})$. The time
evolution of any classical observable $f\in C^{\infty}(\mathcal{G})$ is
determined by the Hamiltonian equation of motion

$\dot{f}=\\{f,H\\}_{\mathcal{G}}\,.$ (2.4)

Equivalently, one can describe the phase-space trajectories
$\gamma:\mathbb{R}\rightarrow\mathcal{G}$ as stationary points of the action
functional

$S=-\int_{\gamma}\left[\theta+H{\rm d}t\right]\,,$ (2.5)

where $\theta$ is a symplectic potential for the symplectic form $\omega={\rm
d}\theta$ on $\mathcal{G}$.

##  3 Symplectic groupoids and dynamics for Lie–Poisson structures

Let us specify the above considerations for the case of linear Poisson bracket

$\\{x^{a},x^{b}\\}=f^{ab}_{c}x^{c}\,.$ (3.6)

Here, $x^{a}$ are Cartesian coordinates on $X=\mathbb{R}^{n}$. It follows
immediately from the Jacobi identity that $f_{c}^{ab}$ are structure constants
of some $n$-dimensional Lie algebra $\mathfrak{g}$. This allows us to identify
$\mathbb{R}^{n}$ with the space dual to $\mathfrak{g}$. The linear Poisson
bracket (3.6) on $\mathfrak{g}^{\ast}$ is known as the Lie–Poisson structure.

To any Lie algebra $\mathfrak{g}$ there corresponds a Lie group $G$, which is
unique up to covering. As we will see in a moment, the group $G$ plays the
role of the space of momenta conjugate to the position coordinates $x^{a}$. To
stress this interpretation, we will denote local coordinates on $G$ by
$p_{i}$. We define the basis $\bar{\gamma}^{a}$ of left-invariant $1$-forms on
$G$ by the standard relations

$g^{-1}{\rm
d}g=\bar{\gamma}_{a}t^{a}\,,\qquad\bar{\gamma}_{a}=\bar{\gamma}_{a}^{i}(p){\rm
d}p_{i}\,,$ (3.7)

where $g=g(p)\in G$ and $t^{a}$ is a basis in $\mathfrak{g}$ with commutation
relations $[t^{a},t^{b}]=f^{ab}_{c}t^{c}$. The dual to the basis of left-
invariant $1$-forms $\\{\bar{\gamma}_{a}\\}$ is the basis $\\{\gamma^{a}\\}$
of left-invariant vector fields on $G$. These satisfy the relations

$\bar{\gamma}_{a}(\gamma^{b})=\delta^{b}_{a}\,,\qquad{\rm
d}\bar{\gamma}_{a}=-\frac{1}{2}f^{bc}_{a}\bar{\gamma}_{b}\wedge\bar{\gamma}_{c}\,,\qquad[\gamma^{a},\gamma^{b}]=f^{ab}_{c}\gamma^{c}\,.$
(3.8)

Now we are ready to define a symplectic groupoid that integrates the linear
Poisson bracket (3.6). As a smooth manifold, it is given by the Cartesian
product $\mathcal{G}=\mathfrak{g}^{\ast}\times G$ and the symplectic structure
is determined by the exact $2$-form

$\omega={\rm d}\theta\,,\qquad\theta=\langle x,g^{-1}{\rm
d}g\rangle=x^{a}\bar{\gamma}_{a}^{i}(p){\rm d}p_{i}$ (3.9)

or, more explicitly,

$\omega={\rm
d}x^{a}\wedge\bar{\gamma}_{a}-\frac{1}{2}x^{c}f^{ab}_{c}\bar{\gamma}_{a}\wedge\bar{\gamma}_{b}\,.$
(3.10)

The bivector dual to $\omega$ defines the Poisson brackets

$\\{x^{a},x^{b}\\}=f^{ab}_{c}x^{c}\,,\qquad\\{x^{a},p_{i}\\}=\gamma^{a}_{i}(p)\,,\qquad\\{p_{i},p_{j}\\}=0\,.$
(3.11)

The source, target, and inversion maps are given by

${\sf s}(x,g)=x\,,\qquad{\sf t}(x,g)=\mathrm{Ad}^{\ast}_{g}(x)\,,\qquad{\sf
i}(x,g)=(\mathrm{Ad}^{\ast}_{g}(x),g^{-1})\,,$ (3.12)

where $(x,g)\in\mathfrak{g}^{\ast}\times G$ and $\mathrm{Ad}^{\ast}$ stands
for the coadjoint representation of the group $G$.

The action functional (2.5) for a nonrelativistic particle in noncommutative
space $\mathfrak{g}^{\ast}$ assumes now the form

$S[x,g]=-\int\big{[}\langle x,g^{-1}\dot{g}\rangle+H(x,g)\big{]}dt\,.$ (3.13)

The least action principle gives the Hamiltonian equations of motion

$\dot{x}^{a}=\\{x^{a},H\\}=\gamma^{a}_{i}(p)\frac{\partial H}{\partial
p_{i}}+x^{c}\,f^{ab}_{c}\frac{\partial H}{\partial
x^{b}}\,,\qquad\dot{p}_{i}=\\{p_{i},H\\}=-\gamma_{i}^{a}(p)\frac{\partial
H}{\partial x^{a}}\,.$ (3.14)

The direct product structure of the phase space $\mathfrak{g}^{\ast}\times G$
allows one to consider Hamiltonians that are given by a sum of kinetic and
potential energy, $H=T(g)+U(x)$. Such mechanical systems are usually called
natural. For a free particle, $U(x)=0$ and Eqs. (3.14) assume very simple form

$\dot{x}^{a}=\gamma^{a}_{i}(p)\frac{\partial T}{\partial
p_{i}}\,,\qquad\dot{p}_{i}=0\,.$ (3.15)

The first equation expresses the particle’s velocity $\dot{x}^{a}$ as a
function of its momentum $p_{i}$. The second equation says that the momentum
of a free particle is conserved. By the Noether theorem, the Hamiltonian
action of the integrals of motion $p_{i}$ generates the global symmetry of the
free action:

$x^{a}\rightarrow x^{a}+v^{i}\gamma^{a}_{i}(p)\,,\qquad\forall v^{i}=const\,.$
(3.16)

The general solution to (3.15) reads

$x^{a}(t)=x^{a}(0)+\dot{x}^{a}(0)t\,.$ (3.17)

Whatever the group $G$, we get a uniform linear motion in space, as for
ordinary free particles. Therefore, it seems impossible to distinguish between
commutative and noncommutative spaces by probing them with free particles. The
only exception is the case of compact $G$: because of the first equation in
(3.15) the components of the velocity vector $\dot{x}^{a}$ appear to be
bounded by absolute value (like in Special Relativity) provided the kinetic
energy $T(p)$ has continuous first derivatives.

Now consider a particle moving in response to an external force. In this case,
the equations of motion take the form

$\dot{x}^{a}=\gamma^{a}_{i}(p)\frac{\partial T}{\partial
p_{i}}+x^{c}\,f^{ab}_{c}\frac{\partial U}{\partial
x^{b}}\,,\qquad\dot{p}_{i}=-\gamma_{i}^{a}(p)\frac{\partial U}{\partial
x^{a}}\,.$ (3.18)

(Notice that due to the external force, the velocity $\dot{x}$ may now be
unbounded even for a compact $G$.) The co-adjoint action of $G$ on
$\mathfrak{g}^{\ast}$ stratifies $\mathfrak{g}^{\ast}$ into co-adjoint orbits.
Similarly, the action of $G$ on itself by conjugation stratifies $G$ into the
classes of conjugate elements. Let us suppose that the functions $T$ and $U$
are invariant under these actions, that is, $U$ is constant on each co-adjoint
orbit, while $T$ is constant on each conjugacy class:

$U(x)=U(\mathrm{Ad}^{\ast}_{h}(x))\,,\qquad T(g)=T(hgh^{-1})\,,\qquad\forall
h\in G\,.$ (3.19)

In mathematics, such functions $T$ are called class functions. Then one can
readily see that the action

$S[x,g]=-\int\big{(}\langle x,g^{-1}\dot{g}\rangle+T(g)+U(x)\big{)}dt$ (3.20)

is invariant under the transformations

$x\rightarrow\mathrm{Ad}^{\ast}_{h}x\,,\qquad g\rightarrow
hgh^{-1}\,,\qquad\forall h\in G\,.$ (3.21)

Again, by the Noether theorem, these symmetry transformations give rise to
conservation laws. To write the corresponding conserved quantities in an
explicit form we need to introduce the right-invariant $1$-forms and vector
fields on $G$:

$dgg^{-1}=t^{a}\rho^{i}_{a}(p)dp_{i}\,,\qquad\bar{\rho}^{a}=\bar{\rho}^{a}_{i}(p)\frac{\partial}{\partial
p_{i}}\,,\qquad\bar{\rho}^{a}(\rho_{b})=\delta_{b}^{a}\,.$ (3.22)

They satisfy the identities

$d\rho_{c}=\frac{1}{2}f_{c}^{ab}\rho_{a}\wedge\rho_{b}\,,\qquad[\bar{\rho}^{a},\bar{\rho}^{b}]=-f^{ab}_{c}\bar{\rho}^{c}\,,\qquad[\bar{\rho}^{a},\gamma^{b}]=0\,.$
(3.23)

We also introduce the matrix
$\Delta^{a}_{b}(p)=\bar{\gamma}_{b}(\bar{\rho}^{a})$, which obeys the equation

$\gamma^{c}\Delta^{a}_{b}=-f^{cd}_{b}\Delta_{d}^{a}\,.$ (3.24)

Conditions (3.19) are equivalent to

$x^{b}f_{b}^{ac}\frac{\partial U}{\partial
x^{c}}=0\,,\qquad(\gamma^{a}-\bar{\rho}^{a})T=0\,.$ (3.25)

We claim that, under the conditions above, the following functions are
conserved:

$\mu^{a}=x^{b}\bar{\gamma}_{b}^{i}(\gamma^{a}_{i}-\bar{\rho}_{i}^{a})=x^{a}-x^{b}\Delta_{b}^{a}(p)\,.$
(3.26)

Indeed,

$\begin{array}[]{c}\dot{\mu}^{a}=\dot{x}^{b}\bar{\gamma}_{b}^{i}(\gamma^{a}_{i}-\bar{\rho}_{i}^{a})-\dot{p}_{i}\partial^{i}_{p}\Delta_{b}^{a}(p)x^{b}\\\\[14.22636pt]
\displaystyle=\frac{\partial T}{\partial
p_{i}}(\gamma^{a}_{i}-\bar{\rho}_{i}^{a})+\frac{\partial U}{\partial
x^{c}}(\gamma^{c}\Delta^{a}_{b})x^{b}=\frac{\partial T}{\partial
p_{i}}(\gamma^{a}_{i}-\bar{\rho}_{i}^{a})-\frac{\partial U}{\partial
x^{c}}f^{cd}_{b}x^{b}\Delta_{d}^{a}=0\,.\end{array}$ (3.27)

The conserved quantities $\mu^{a}$ have the following Poisson brackets:

$\\{\mu^{a},\mu^{b}\\}=f^{ab}_{c}\mu^{c}\,.$ (3.28)

If $G$ is a semi-simple Lie group, then the Killing metric
$g^{ab}=f^{ac}_{d}f^{db}_{c}$ is nondegenerate and we can define the conserved
quantity $Q=g_{ab}\mu^{a}\mu^{b}$, which satisfies $\\{Q,\mu^{a}\\}=0$.

As mentioned above, for a compact $G$, the velocity of a free non-relativistic
particle turns out to be bounded above. However, this property may not survive
upon switching on interaction. We conclude this section by discussing two more
interesting phenomena related to the compactness of the momentum space $G$.
The phase-space dynamics develop on the isoenergetic surface

$T(g)+U(x)=E$ (3.29)

defined by some energy level $E$. For a compact momentum space, the kinetic
energy necessarily reaches its maximum and minimum values:

$T_{min}\leq T(g)\leq T_{max}\,.$ (3.30)

It is the existence of an upper bound on the kinetic energy that sharply
differs the situation under consideration from ordinary classical mechanics.
On substituting (3.29), we get

$E-T_{max}\leq U(x)\leq E-T_{min}\,.$ (3.31)

Depending on $U$, these inequalities can severely bound the domain of motion.
Suppose, for example, that the potential energy of a particle drops to zero at
infinity, i.e.,

$U(x)\rightarrow 0\,,\qquad x\rightarrow\infty\,.$ (3.32)

Then the particle has to move in a compact region of space

$\mathcal{C}_{E}=\big{\\{}x\in\mathfrak{g}^{\ast}\;\big{|}\;U(x)\geq
E-T_{max}\big{\\}}$ (3.33)

whenever its total energy $E>T_{max}$. The last inequality is automatically
fulfilled for all points of the bounded region‡‡‡$\mathcal{C}$ stands for
compact or confinement.

$\mathcal{C}=\big{\\{}x\in\mathfrak{g^{\ast}}\;\big{|}\;U(x)>T_{max}-T_{min}\big{\\}}\,.$
(3.34)

(The case $\mathcal{C}=\varnothing$ is not excluded.) Therefore, if a
trajectory starts at some point $x\in\mathcal{C}$, it will be fully confined
in a bounded region of $\mathfrak{g}^{\ast}$. Remarkably, the confinement may
take place even for repulsive central force vanishing at infinity! In Sec.
5.2, we will illustrate this phenomenon with the Coulomb potential.

Similarly, we can consider an attractive central force that becomes infinite
at some point $a$, i.e.,

$U(x)\rightarrow-\infty\,,\qquad x\rightarrow a\,.$ (3.35)

Normally, we would expect some trajectories to lead to the singular point
$x=a$ (falling to the centre). As the particle approaches the centre, the
decrease in potential energy should be balanced by the increase in kinetic
energy. The domain available for motion is still described by Eq. (3.33). Now
it means that the particle cannot approach the singular point $a$ arbitrarily
close; whatever total energy $E$, the region where

$U(x)\leq E-T_{max}$ (3.36)

is forbidden for motion. This inequality does not exclude the physical
situation where an accelerating charge loses its energy due to electromagnetic
radiation and spirals towards the centre. In this paper, however, we do not
consider the radiation effects.

In conclusion, we note that the abnormal behavior above owes its existence
entirely to the compactness of momentum space and can occur even in a
commutative situation. As shown in [31], for one, the toric compactification
of the momentum space associated with the commutative configuration space
$\mathbb{R}^{n}$ eliminates the distinction between attractive and repulsive
potentials.

##  4 Example: a scalar particle in $\mathfrak{su}(2)$ noncommutative space

Let us illustrate the above formulas by a nonrelativistic particle moving in
the noncommutative $3$-space $\mathfrak{su}(2)^{\ast}$. To our knowledge, this
mechanical model was first studied in [32, 33, 34]. Note that
$\mathfrak{su}(2)$ is a unique compact simple Lie algebra of dimension three.
As the space of momenta of the particle, we take the simply-connected Lie
group $SU(2)$, so that the whole phase space is given by the direct product
$\mathcal{G}=\mathfrak{su}(2)^{\ast}\times SU(2)$. To write the Hamiltonian
equations of motion on $\mathcal{G}$ it is convenient to use the language of
quaternions.

Recall that the quaternion algebra $\mathbb{H}$ is a unique $4$-dimensional
associative division algebra over the reals. As a vector space $\mathbb{H}$ is
spanned by the multiplicative identity element $1$ together with the three
‘imaginary units’ $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ subject to the
relations

$\displaystyle\mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=\mathbf{i}\,\mathbf{j}\,\mathbf{k}=-1\
.$

Hence, the general element of $\mathbb{H}$ is

$q=q_{0}+q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k}\,,\qquad
q_{0},q_{1},q_{2},q_{3}\in\mathbb{R}\,.$ (4.37)

The conjugation, norm, and trace on $\mathbb{H}$ are defined by the usual
formulas:

$\begin{array}[]{l}\bar{q}=q_{0}-q_{1}\mathbf{i}-q_{2}\mathbf{j}-q_{3}\mathbf{k}\,,\\\\[8.53581pt]
|q|^{2}=q\bar{q}=q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}\,,\\\\[8.53581pt]
\mathrm{Tr}(q)=q+\bar{q}=2q_{0}\,.\end{array}$ (4.38)

These have the standard properties

$\overline{qq^{\prime}}=\bar{q}^{\prime}\bar{q}\,,\qquad|qq^{\prime}|=|q||q^{\prime}|\,,\qquad\mathrm{Tr}(qq^{\prime})=\mathrm{Tr}(q^{\prime}q)\,.$
(4.39)

In terms of quaternions, the Lie algebra $\mathfrak{su}(2)$ is identified with
the commutator algebra of purely imaginary quaternions, while the Lie group
$SU(2)$ can be described as the multiplicative group of quaternions of norm
$1$. Using the Eucledian structure on the space of imaginary quaternions, we
can also identify $\mathfrak{su}(2)$ and $\mathfrak{su}(2)^{\ast}$. Then the
coadjoint action of $SU(2)$ on the physical $3$-space
$\mathfrak{su}(2)^{\ast}$ is given simply by conjugation

$\mathsf{Ad}^{\ast}_{u}(\mathbf{x})=u\mathbf{x}\bar{u},$ (4.40)

where $\mathbf{x}+\bar{\mathbf{x}}=0$ and $u\bar{u}=1$. In the following, we
will denote pure imaginary quaternions in bold.

The presymplectic potential (3.9) on $\mathfrak{su}(2)^{\ast}\times SU(2)$
takes now the form

$\theta=\mathrm{Tr}(x\bar{u}{\rm d}u)$ (4.41)

and for the symplectic $2$-form we get

$\omega={\rm d}\theta=\mathrm{Tr}\big{(}{\rm d}x\wedge\bar{u}{\rm
d}u-x\bar{u}{\rm d}u\wedge\bar{u}{\rm d}u\big{)}\,.$ (4.42)

Instead of working with constrained momenta $u$, we can add the constraint
$u\bar{u}-1=0$ to the Hamiltonian $H(u,x)$ with the Lagrange multiplier
$\lambda$. Then the full action (3.13) with unconstrained $u$ is given by

$S[\mathbf{x},u,\lambda]=\int\Big{(}-\frac{1}{2}\mathrm{Tr}(\mathbf{x}\bar{u}\dot{u})-H(\mathbf{x},u)-\lambda(u\bar{u}-1)\Big{)}{\rm
d}t\,.$ (4.43)

Let us take the Hamiltonian in the form ${H}=T(u)+U(\mathbf{x})$ and introduce
the pair of quaternions

$V_{0}+\mathbf{V}=\frac{\partial T}{\partial u_{0}}+\frac{\partial T}{\partial
u_{1}}\mathbf{i}+\frac{\partial T}{\partial u_{2}}\mathbf{j}+\frac{\partial
T}{\partial u_{3}}\mathbf{k}\,,\qquad\mathbf{F}=-\frac{\partial U}{\partial
x_{1}}\mathbf{i}-\frac{\partial U}{\partial x_{2}}\mathbf{j}-\frac{\partial
U}{\partial x_{3}}\mathbf{k}\,.$ (4.44)

The variation of (4.43) results in the equations of motion

$\dot{{u}}=u\mathbf{F}\,,\qquad\dot{\mathbf{x}}=u_{0}\mathbf{V}-V_{0}\mathbf{u}+\frac{1}{2}[\mathbf{V},\mathbf{u}]+[\mathbf{x},\mathbf{F}]\,,\qquad|u|^{2}=1\,.$
(4.45)

Here, the square brackets stand for the commutator of quaternions. Notice that
the algebraic constraint $|u|^{2}=1$ is conserved over time.

The equations of motion (4.45) are considerably simplified if we assume that
$U$ is a central potential of the form $U=f(|\mathbf{x}|^{2})$ and take
$T=-u_{0}/\ell m$. In this case,

$\dot{u}_{0}=-\mathrm{Tr}(\mathbf{u}\mathbf{x})f^{\prime}\,,\qquad\dot{\mathbf{u}}=\big{(}[\mathbf{x},\mathbf{u}]-2u_{0}\mathbf{x}\big{)}f^{\prime}\,,\qquad\ell
m\dot{\mathbf{x}}=\mathbf{u}\,,\qquad|u|^{2}=1\,.$ (4.46)

We see that the direction of the radial acceleration depends on the sign of
$u_{0}$, regardless of whether the central potential is attractive or
repulsive; there is also an additional force proportional to the angular
momentum of the particle. Excluding $\mathbf{u}$, we obtain the system of
first- and second-order equations

$\ddot{\mathbf{x}}=(2T\mathbf{x}+[\mathbf{x},\dot{\mathbf{x}}])f^{\prime}\,,\qquad\dot{T}=-\mathrm{Tr}(\dot{\mathbf{x}}\mathbf{x})f^{\prime}$
(4.47)

with initial data lying on the constraint surface
$T^{2}+|\dot{\mathbf{x}}|^{2}=(\ell m)^{-2}$. Since the constraint is
conserved over time, the absolute values of the kinetic energy and velocity
appear to be bounded above, $|T|,|\dot{\mathbf{x}}|\leq(\ell m)^{-1}$. For
small velocities, when $|\dot{\mathbf{x}}|<<(\ell m)^{-1}$ and $T\approx-(\ell
m)^{-1}$, the first equation in (4.47) turns to the standard Newton equation
for a particle of mass $m$ moving in the external potential $U$. As for the
second equation, it represents the law of energy conservation $T+U=E$, which
allows us to rewrite the system (4.47) in the form

$\ddot{\mathbf{x}}=\big{(}2(E-f)\mathbf{x}+[\mathbf{x},\dot{\mathbf{x}}]\big{)}f^{\prime}\,,$
(4.48)

where the constant $E$ is determined by the initial data through the equation

$\big{(}E-f(|\mathbf{x}|^{2})\big{)}^{2}+|\dot{\mathbf{x}}|^{2}=(\ell
m)^{-2}\,.$

One more integral of motion is given by the ‘angular momentum’ (3.26). In
terms of quaternions it can be written as

${\mu}=\mathbf{x}-u\mathbf{x}\bar{u}\,.$ (4.49)

######  Remark 4.50.

We see that a free particle with $SU(2)$ as momentum space and a class
function $T(u_{0})$ as Hamiltonian has the same number of fundamental (time-
independent) conservation lows as the ordinary nonrelativistic particle. These
are given by the three independent components of the momentum $u$, the angular
momentum vector $\mu$, and the kinetic energy $T(u_{0})$. All these quantities
generate and come from symmetry transformations. Furthermore, free motion is
uniform and linear. What about the remaining symmetry transformations of
Newton’s mechanics, namely, the Galilean boosts: $x\rightarrow x+tv$? One can
see that under mild assumptions, they are implicitly present in the free model
(3.20) for any Lie group $G$. Let us suppose that the kinetic energy $T(g)$
has a nondegenerate local minimum at the identity element $e\in G$, i.e.,
$\partial_{i}T(e)=0$ and the matrix
$\big{(}\partial_{i}\partial_{j}T(e)\big{)}$ is positive-definite. By the
Morse lemma, one can find new coordinates $P^{i}=P^{i}(g)$ in a vicinity of
$e$ such that $T=T(e)+\frac{1}{2}\delta_{ij}P^{i}P^{j}$. In these coordinates,
the free Lagrangian (3.20) takes the form

$L=-x_{a}\bar{\gamma}^{a}_{i}(P)\dot{P}^{i}-\frac{1}{2}\delta_{ij}P^{i}P^{j}\,.$

(The constant $T(e)$ can be omitted.) Introducing the new position coordinates
$X_{i}=x_{a}\bar{\gamma}_{i}^{a}(P)$, we can bring the Lagrangian into the
standard form:

$L=\dot{X}_{i}P^{i}-\frac{1}{2}\delta_{ij}P^{i}P^{j}\,.$

The last Lagrangian is invariant (up to a total derivative) under the Galilean
boosts: $X\rightarrow X+Vt$ and $P\rightarrow P+V$. By making the inverse
change of variables $(X,P)\rightarrow(x,g)$, we get nonlinear symmetry
transformations for the free action (3.20). One can regard $(X,P)$ as Darboux
coordinates on $T^{\ast}G$ adapted to the Hamiltonian $T$.

To further elucidate the connection with ordinary Hamiltonian mechanics, it is
convenient to introduce dimensional momenta. If $p$ is the momentum
canonically conjugated to a position coordinate $x$, then $[x][p]=[\hbar]$. In
the unit system where $\hbar=1$, we have $[p]=[x]^{-1}$. Let $\ell$ be a
characteristic length scale, $[\ell]=[x]$. Introduce the dimensional momenta
$p$ as $u=\ell p\in\mathbb{H}$. For the action functional $S$ to have the
dimension of $\hbar$ we also need to multiply the first term in (4.43) by
$\ell^{-1}$. The Hamiltonian for a nonrelativistic particle of mass $m$ can be
written as

$\mathcal{H}=\frac{k(u_{0})}{m\ell^{2}}+U(\mathbf{x})\,,$ (4.51)

where $k(u_{0})$ is a dimensionless quantity. Due to the constraint
$|p|^{2}=\ell^{-2}$, we can parameterize the southern hemisphere of
$\mathbb{S}^{3}$ by the vector part of the quaternion $p=p_{0}+{\mathbf{p}}$.
In these coordinates, the first-order Lagrangian assumes the form

$\begin{array}[]{c}\displaystyle
L=-\frac{\ell}{2}\mathrm{Tr}(\mathbf{x}\bar{p}\dot{p})-\mathcal{H}\\\\[14.22636pt]
\displaystyle=\frac{1}{2}\mathrm{Tr}({\mathbf{p}}\dot{\mathbf{x}})\sqrt{1-\ell^{2}|\mathbf{p}|^{2}}+\frac{\ell}{2}\mathrm{Tr}\big{(}\mathbf{x}\mathbf{p}\dot{\mathbf{p}}\big{)}+\frac{\ell^{2}}{2}\frac{\mathrm{Tr}(\mathbf{x}\mathbf{p})\mathrm{Tr}({\mathbf{p}}\dot{\mathbf{p}})}{\sqrt{1-\ell^{2}|\mathbf{p}|^{2}}}\\\\[14.22636pt]
\displaystyle-\frac{k(\sqrt{1-\ell^{2}|\mathbf{p}|^{2}})}{m\ell^{2}}-U(\mathbf{x})+\text{tot.
der.}\end{array}$ (4.52)

The terms with time derivatives determine the symplectic potential, which
leads to the following Poisson brackets:

$\\{x^{i},x^{j}\\}=2\,\ell\,\varepsilon^{ijk}x^{k}\,,\qquad\\{x^{i},p_{j}\\}=\sqrt{1-\ell^{2}|\mathbf{p}|^{2}}\,\delta^{i}_{j}+\ell\,\varepsilon^{ijk}p_{k}\,,\qquad\\{p_{i},p_{j}\\}=0\,.$
(4.53)

We see that the commutative limit $\ell\rightarrow 0$ for the space
coordinates is simultaneously the decompactification limit for the momentum
$3$-sphere $\mathbb{S}^{3}$. In this limit, the compact nonabelian group
$SU(2)$ turns into the additive group of the vector space $\mathbb{R}^{3}$.
Expanding the Lagrangian (4.52) in powers of $\ell$, we find

$L=\frac{1}{2}\mathrm{Tr}(\mathbf{p}\dot{\mathbf{x}})+\frac{\ell}{2}\mathrm{Tr}({\mathbf{x}}\mathbf{p}\dot{\mathbf{p}})-\frac{k(1)}{m\ell^{2}}+k^{\prime}(1)\frac{|\mathbf{p}|^{2}}{2m}-U(\mathbf{x})+\mathcal{O}(\ell^{2})\,.$
(4.54)

We can drop the constant term as it does not affect dynamics. Then (4.54) goes
into the standard Lagrangian of a nonrelativistic particle as $\ell\rightarrow
0$ provided that $k^{\prime}(1)=-1$. For example, we can set as above
$k=-u_{0}$. For $\ell\neq 0$, the corrections due to noncommutativity become
negligible whenever the energy-momentum $p$ of the particle lies in a small
vicinity of the identity element of $SU(2)$, that is, $p\approx\ell^{-1}$. One
can use the Lagrangian (4.54) to derive the leading corrections to ordinary
nonrelativistic dynamics due to space noncommutativity. We find

$m\dot{\mathbf{x}}=\mathbf{p}-m\ell[\mathbf{x},\dot{\mathbf{p}}]+\mathcal{O}(\ell^{2})\,,\qquad\dot{\mathbf{p}}=\mathbf{F}+\frac{\ell}{2}[\mathbf{p},\dot{\mathbf{p}}]+\mathcal{O}(\ell^{2})$
(4.55)

or

$m\ddot{\mathbf{x}}=\mathbf{F}+\frac{m\ell}{2}[\dot{\mathbf{x}},\mathbf{F}]-m\ell\frac{d}{dt}[\mathbf{x},\mathbf{F}]+\mathcal{O}(\ell^{2})\,.$
(4.56)

The last term vanishes for central potentials. For instance, if
$U=\alpha/|\mathbf{x}|$ is the Coulomb potential, then the equation becomes

$m\ddot{\mathbf{x}}=\alpha\frac{\mathbf{x}}{\;|\mathbf{x}|^{3}}+\frac{\alpha
m\ell}{2}\frac{[\dot{\mathbf{x}},\mathbf{x}]}{\;|\mathbf{x}|^{3}}+\mathcal{O}(\ell^{2})\,.$
(4.57)

Notice the term proportional to the angular momentum of the particle. This
looks like a force exerted on the charge by a magnetic monopole located at the
origin. A nontrivial point is that the magnitude of this additional force is
proportional to the mass of the charged particle. As with the case of gravity,
this indicates that the force is of geometric origin. Expression (4.49) gives
the following integral of motion:

$\mu=m[\mathbf{x},\dot{\mathbf{x}}]-2\alpha
m\ell\frac{\mathbf{x}}{|\mathbf{x}|}+\mathcal{O}(\ell^{2})\,.$ (4.58)

It is noteworthy that the vector $\mu$ is exactly conserved if one restricts
to the leading corrections in (4.57) and (4.58). In this approximation, $\mu$
coinsides with the Poincaré vector, which allows one to find exact
trajectories of an electric charge moving in the dyon potential, see e.g.
[35].

##  5 Noncommutative Kepler problem in semi-classical limit

In this section, we consider in some detail the motion of a point charge $q$
in the Coulomb potential centred at the origin of the $\mathfrak{su}(2)$
noncommutative space. It may be shown [36, 37] that the standard Coulomb
potential, i.e., the field configuration $\vec{A}=0$ and
$A_{0}=C/|\mathbf{x}|$, solves the field equations of Poisson electrodynamics
with $\mathfrak{su}(2)$ non-commutativity in the whole space except the
origin. Applying the recipe of minimal interaction proposed in Ref. [19]
allows one to couple a charged particle on $\mathfrak{su}(2)$ noncommutative
space to the Coulomb field. In the electrostatic limit, we can restrict
ourselves to non-relativistic dynamics, which are defined by the Hamiltonian

$\mathcal{H}=\frac{p_{0}}{\ell
m}+\frac{1}{\ell^{2}m}+\frac{\alpha}{|\mathbf{x}|}\,,\qquad\alpha=qC\,,$
(5.59)

and the Poisson brackets

$\\{x^{i},x^{j}\\}=2\ell\,\varepsilon^{ijk}x^{k},\qquad\\{x^{i},p_{j}\\}=\ell
p_{0}\,\delta^{i}_{j}-\ell\,\varepsilon^{ikj}\,p_{k}\,,\qquad\\{p_{i},p_{j}\\}=0\,.$
(5.60)

The dynamical variables are assumed to be subject to the constraint
$p_{0}^{2}+\mathbf{p}^{2}=1/\ell^{2}$. In the standard notation of vector
algebra, the Hamiltonian equations of motion take the form

$\dot{p}_{0}=-\alpha\ell\,\frac{\mathbf{p}\cdot\mathbf{x}}{\;|\mathbf{x}|^{3}}\,,\qquad\dot{\mathbf{p}}=\alpha\ell\,p_{0}\,\frac{\mathbf{x}}{\;|\mathbf{x}|^{3}}+\alpha\ell\,\frac{\mathbf{p}\times\mathbf{x}}{\;|\mathbf{x}|^{3}}\,,\qquad
m\dot{\mathbf{x}}=\mathbf{p}\,.$ (5.61)

In the commutative limit when $\ell\rightarrow 0$ and $\ell p_{0}\rightarrow
1$, we arrive at the familiar equations of the Kepler problem. In different
coordinates, the Hamiltonian equations (5.61) have been integrated in [33,
34].

###  5.1 Conservation laws

As explained in the previous section, the Hamiltonian equations (5.61) enjoy a
number of conservation lows that make them integrable. First of all, the total
energy defined by the Hamiltonian (5.59) is conserved. For the repulsive
Coulomb potential ($\alpha>0$), the energy is always non-negative,
$\mathcal{H}\geq 0$. In the case of attraction ($\alpha<0$), the energy is
bounded above by the ‘critical energy’ $E_{c}=2/\ell^{2}m$, which coincides
with the maximal value of the kinetic energy of the particle. Geometrically,
it corresponds to the northern pole of the momentum $3$-sphere. The upper
bound $\mathcal{H}\leq E_{c}$ obviously disappears in the commutative limit
$\ell\rightarrow 0$.

The rotational invariance of the system gives rise to one more conserved
quantity, namely, the deformed angular momentum vector

$\mathbf{L}=\ell
p_{0}(\mathbf{x}\times\mathbf{p})\,+\ell\,\mathbf{p}\times(\mathbf{x}\times\mathbf{p})\,.$
(5.62)

By construction,

$\dot{\mathbf{L}}=\\{\mathbf{L},\mathcal{H}\\}=0\qquad\mbox{and}\qquad\lim_{\ell\to
0}\mathbf{L}=\mathbf{x}\times\mathbf{p}\,.$ (5.63)

Interestingly enough the square of the deformed angular momentum does not
depend on the deformation parameter,

$L^{2}:=|\mathbf{L}|^{2}=|\mathbf{x}\times\mathbf{p}|^{2}.$ (5.64)

Besides the angular momentum, the system admits an additional conserved vector
of the form [33]

$\mathbf{A}=\mathbf{p}\times\mathbf{L}+\frac{m\alpha}{|\mathbf{x}|}\,\big{(}\mathbf{x}-\ell\,\mathbf{L}\big{)}\,.$
(5.65)

One can check that $\dot{\mathbf{A}}=\\{\mathbf{A},\mathcal{H}\\}=0$. Clearly,
$\mathbf{A}$ goes into the standard Laplace–Runge–Lenz vector as $\ell\to 0$.
The square of the deformed Laplace–Runge–Lenz vector $\mathbf{A}$ is given by

$A^{2}=\alpha^{2}m^{2}+2m\,L^{2}\,\mathcal{H}\big{(}1-\mathcal{H}/E_{c}\big{)}\,.$
(5.66)

Since $A^{2}\geq 0$, we get a nontrivial inequality

$L^{2}\leq\frac{\alpha^{2}m}{2|\mathcal{H}\big{(}1-\mathcal{H}/E_{c}\big{)}|}$
(5.67)

provided that $\alpha>0$ and $\mathcal{H}>E_{c}$ or $\alpha<0$ and
$\mathcal{H}<0$. It sets an upper bound on possible values of the angular
momentum.

The conserved vector quantities close the following algebra w.r.t. the Poisson
brackets [33]:

$\displaystyle\\{L_{i},L_{j}\\}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijk}L_{k}\,,\qquad\\{A_{i},L_{j}\\}=\varepsilon_{ijk}A_{k}\,,$
$\displaystyle\qquad\\{A_{i},A_{j}\\}$ $\displaystyle=$
$\displaystyle-2m\mathcal{H}\big{(}1-\mathcal{H}/E_{c}\big{)}\,\varepsilon_{ijk}L_{k}\,.$
(5.68)

In the commutative limit $E_{c}\rightarrow\infty$ and we arrive at the
familiar Poisson bracket relations of the Kepler problem.

###  5.2 Space trajectories

With all integrals of motion above we can examine particle’s trajectories for
various initial data. Multiplying (5.62) by $\mathbf{x}$, we obtain

$\mathbf{L}\cdot\mathbf{x}=\ell L^{2}\,.$ (5.69)

Therefore, all space trajectories are plane curves. The corresponding planes
are orthogonal to the deformed angular momentum $\mathbf{L}$ and do not pass
through the origin unless $\mathbf{L}=\mathbf{0}$. The distance $d$ between
the origin and the trajectory plane is equal to

$d=\ell\,L.$ (5.70)

It is convenient to introduce the shifted position vector

$\mathbf{R}=\mathbf{x}-\ell\,\mathbf{L}.$ (5.71)

As in the commutative case, both the deformed Laplace–Runge–Lenz vector (5.65)
and $\mathbf{R}$ lie in the orbit plane,

$\mathbf{L}\cdot\mathbf{A}=0\,,\qquad\mathbf{L}\cdot\mathbf{R}=0\,.$ (5.72)

Therefore we can parametrize the orbit by the length $R=|\mathbf{R}|$ and the
angle $\theta$ between the vectors $\mathbf{A}$ and $\mathbf{R}$. It follows
from the definition (5.65) that

$\mathbf{A}\cdot\mathbf{R}=\mathbf{R}\cdot(\mathbf{p}\times\mathbf{L})+\frac{\alpha
m\,R^{2}}{|\mathbf{x}|}.$ (5.73)

Making a circular shift in the scalar triple product and using the identity

$\mathbf{R}\times\mathbf{p}=\ell p_{0}\,\mathbf{L}\,,$ (5.74)

we can write

$\mathbf{R}\cdot(\mathbf{p}\times\mathbf{L})=\mathbf{L}\cdot(\mathbf{R}\times\mathbf{p})\,,=\ell
p_{0}\,L^{2},$ (5.75)

whence

$A\,R\,\cos\,\theta=\ell p_{0}\,L^{2}+\frac{\alpha m\,R^{2}}{|\mathbf{x}|}.$
(5.76)

Since the energy is conserved, $\mathcal{H}=E=\mathrm{const}$, the expression
(5.59) for the Hamiltonian implies that

$\ell p_{0}=1-E\,\ell^{2}m+\frac{\alpha\,\ell^{2}m}{|\mathbf{x}|}\,.$ (5.77)

Rel. (5.76) can now be rewritten as

$A\,R\,\cos{\,\theta}=\big{(}1-2E/E_{c}\big{)}\,L^{2}+\frac{m\,\alpha}{|\mathbf{x}|}\,\big{(}R^{2}+\ell^{2}L^{2}\big{)}.$
(5.78)

Finally, noticing that $|\mathbf{x}|^{2}={R^{2}+\ell^{2}L^{2}}$, we arrive at
the orbit equation

$A\,R\,\cos{\,\theta}=\big{(}1-2E/E_{c}\big{)}\,L^{2}+m\,\alpha\,\sqrt{R^{2}+\ell^{2}L^{2}},$
(5.79)

where the constant $A$ is expressed through the integrals of motion $L$ and
$E$ as

$A=\sqrt{\alpha^{2}m^{2}+2m\,L^{2}\,E\big{(}1-E/E_{c}\big{)}}\,,$ (5.80)

c.f. Eq. (5.66).

Let us take a closer look at the particle’s trajectories. By introducing the
Cartesian coordinates in the plane of the orbit,

$X=R\,\cos{\theta}\,,\qquad Y=R\,\sin{\theta},$ (5.81)

we can rewrite Eq. (5.79) as follows:

$A\,X-\big{(}1-2E/E_{c}\big{)}\,L^{2}=m\,\alpha\,\sqrt{(X^{2}+Y^{2})+\ell^{2}L^{2}}\,.$
(5.82)

#### The case of $E\neq E_{c}$ and $E\neq 0$.

First, consider the situation where $L\neq 0$. By squaring both sides of Eq.
(5.82), after a few simplifications which involve Eq. (5.66), we get the
equation

$\omega\,(X-X_{0})^{2}+\xi\,Y^{2}=1\,,$ (5.83)

where

$\omega=\frac{4E^{2}(1-E/E_{c})^{2}}{\alpha^{2}}\,,\qquad\xi=-\frac{2m\,E\,(1-E/E_{c})}{L^{2}}\,,\qquad
X_{0}=\frac{(1-2E/E_{c})A}{2m\,E\,(1-E/E_{c})}\,.$ (5.84)

Therefore we have elliptic motion iff $\xi>0$, that is,

$E<0\quad\mbox{or}\quad E>E_{c}\,.$ (5.85)

The motion is hyperbolic iff $\xi<0$ or, what is the same,

$0<E<E_{c}\,.$ (5.86)

For elliptic motion, the semi-axes of the ellipse are equal to
$1/\sqrt{\omega}$ and $1/\sqrt{\xi}$, respectively. In the limit $L\to 0$, the
minor semi-axis $1/\sqrt{\xi}$ vanishes and the motion becomes radial along
the line segment

$|X-X_{0}|\leq 1/\sqrt{\omega}\,,\qquad Y=0\,.$ (5.87)

The first novelty compared to the commutative case is the possibility of
bounded (elliptic or radial) motion for the _repulsive_ potential
($\alpha>0$). In this case, the energy region $E>E_{c}$ is perfectly
accessible when the particle is sufficiently close to the centre. Applying
Eqs. (3.33) and (3.34) with $T_{max}=E_{c}$ and $T_{min}=0$, we arrive at the
following conclusion: Any trajectory that passes through the ‘trapping region’

$\mathcal{C}=\\{\mathbf{x}\in\mathbb{R}^{3}\;|\;|\mathbf{x}|<\alpha/E_{c}\\}\,,$
(5.88)

remains confined in the compact region

$\mathcal{C}_{E}=\big{\\{}\mathbf{x}\in\mathbb{R}^{3}\;\big{|}\;|\;\mathbf{x}|\leq{\alpha}/{(E-E_{c})}\big{\\}}\,.$
(5.89)

This effect is illustrated in Fig. 1.

Figure 1: Confinement due to noncommutativity for the repulsive potential with
$\alpha=-1$. The blue curve is a hyperbolic trajectory for $\ell=0$. The red
curve is an elliptic trajectory for $\ell=3/2$. In both cases $m=1$ and the
initial conditions are chosen as follows: $x^{1}(0)=1$, $x^{2}(0)=x^{3}(0)=0$,
$p_{1}(0)=p_{3}(0)=0$, $p_{2}(0)=1/2$. In the noncommutative case
($\ell=3/2$), the initial value of $p_{0}$ is chosen in the southern
hemisphere.

When the motion is hyperbolic, the semi-axes of the hyperbola are equal to
$1/\sqrt{\omega}$ and $1/\sqrt{|\xi|}$. Two branches of the hyperbola are
located in the regions

$X\leq\frac{A}{m\,(E-E_{c})}+\frac{E_{c}\,(m|\alpha|-A)}{2\,m\,E\,(E-E_{c})}$
(5.90)

and

$X\geq\frac{A}{m\,E}-\frac{E_{c}\,(m|\alpha|-A)}{2\,m\,E\,(E-E_{c})}\,,$
(5.91)

where the boundary values of $X$ correspond to the intersection points with
the $X$-axis. These two branches correspond to $\alpha<0$ and $\alpha>0$
respectively§§§This statement is obvious at $\ell=0$. For $\ell>0$, it holds
by continuity. .

In the limit $L\to 0$, the trajectories ‘shrink’ to the rays

$Y=0,\qquad
X\leq\frac{|\alpha|}{m\,(E-E_{c})}<0\qquad\mbox{for}\qquad\alpha<0$ (5.92)

and

$Y=0,\qquad X\geq\frac{|\alpha|}{m\,E}>0\qquad\mbox{for}\qquad\alpha>0\,.$
(5.93)

Eq. (5.92) describes another important novelty: even though the potential is
attractive, the particle cannot fall to the centre! Indeed, the region
$|X|<|\alpha|/(E_{c}-E)$ is forbidden, in agreement with Eq. (3.36). A
numerical solution that illustrates this effect is presented in Fig. 2.

Figure 2: Radial motion along the axis $x^{1}$ for the attractive potential
with $\alpha=1$.  The blue line represents a solution to the equations of
motion with $\ell=0$. In a finite time $t\approx 1.1$, the particle falls on
the centre. The red line represents a solution for $\ell=3/2$. Though the
potential is attractive, the particle cannot reach the centre, oscillating
along the $x^{1}$ axis. In both cases $m=1$ and the initial conditions are
chosen as follows: $x^{1}(0)=1$, $x^{2}(0)=x^{3}(0)=0$,
$p_{1}(0)=p_{2}(0)=p_{3}(0)=0$. In the noncommutative case ($\ell=3/2$), the
initial value of $p_{0}$ is chosen in the southern hemisphere.

#### The case of $E=0$.

One can easily see from Eq. (5.79) that, whenever $E=0$ and $L\neq 0$, the
motion is parabolic. The trajectories are given by the plane curves

$Y^{2}=\nu\,(X-\zeta)$ (5.94)

with

$\nu=\frac{2\,L^{2}}{\alpha\,m},\qquad\zeta=\frac{\alpha^{2}\ell^{2}m^{2}-L^{2}}{2\,\alpha\,m}\,.$
(5.95)

The equality $E=0$ is only possible for the _attractive_ potential, i.e.,
$\alpha<0$. Therefore, the parabola is located in the region $X\leq\zeta$. In
the limit $L\to 0$, the parabola ‘shrinks’ to the ray

$X\leq\alpha/E_{c}<0\,,\qquad Y=0$ (5.96)

and the motion becomes radial. Of course, this formula can also be obtained
from Eq. (5.92) by passing to the limit $E\to 0$. Despite the fact that the
potential is attractive, the particle with $L=0$ cannot fall to the centre:
the region $|x|<|\alpha|/E_{c}$ is forbidden.

#### The case of $E=E_{c}$.

For $E=E_{c}$ and $L\neq 0$, the motion is also parabolic. Indeed, relation
(5.79) yields equations (5.94) and (5.95). However, the equality $E=E_{c}$ is
only possible for $\alpha>0$. Therefore, in contrast to the previous case, our
parabola is located in the region $X\geq\zeta$. The possibility of having the
parabolic motion for the repulsive potential is the third nontrivial property
of the model under consideration. As $L$ goes to zero this parabola ‘shrinks’
to the ray

$X\geq{\alpha}/{E_{c}},$ (5.97)

so the motion becomes radial. This result perfectly matches our previous
conclusion (5.93) when $E\to E_{c}$.

To summarize, all space trajectories resulting from the Hamiltonian equations
(5.61) are given by conic sections, as in the ordinary Kepler problem.
Moreover, for a given kinetic energy, the Coulomb potential is proved to be
the only central power-law potential with such a property [34].

###  5.3 Other Hamiltonians

Figure 3: A typical trajectory for the alternative choice (5.98) of the
kinetic energy. The parameters are chosen as follows: $\alpha=-1$, $\ell=1/5$,
$m=1$. The initial conditions are given by $x^{1}(0)=1$,
$x^{2}(0)=x^{3}(0)=0$, $p_{1}(0)=p_{3}(0)=0$, $p_{2}(0)=1/2$. For these
parameters, the whole trajectory projects onto the southern hemisphere of the
momenta $3$-sphere.

At this point, the reader may wonder what the advantage is of the Hamiltonian
(5.59) over other options. A partial answer to this question is that the above
choice for the kinetic energy function results in a superintegrable system. It
is the presence of an additional integral of motion (5.65) that significantly
simplifies the analysis of trajectories. An arbitrary choice of kinetic energy
will most likely not provide an additional conserved quantity. For example,
the simplest candidate consistent with the commutative limit would be

$\mathcal{H}=\frac{|\mathbf{p}|^{2}}{2m}+\frac{\alpha}{|\mathbf{x}|}\,,\qquad\alpha=qC\,.$
(5.98)

All our conclusions regarding the confinement for the repulsive potential and
the impossibility to fall into the center for the attractive potential remain
valid. Moreover, since the rotational invariance is still there, the
(deformed) angular momentum is conserved, and the system is integrable. The
numerical analysis of the Hamiltonian equations

$\dot{x}^{i}=\\{x^{i},\mathcal{H}\\},\qquad\dot{p}_{i}=\\{p_{i},\mathcal{H}\\}$
(5.99)

shows that the orbits, still being flat curves, exhibit a precession as in
Fig. 3. Therefore, we do not expect to find any reasonable deformation of the
Laplace–Runge–Lenz vector which would yield superintegrability. A detailed
analysis of this Hamiltonian is beyond the scope of this paper.

## Acknowledgments

V.G.K. acknowledges support from the National Council for Scientific and
Technological Development (CNPq) Grant 304130/2021-4.

## References

  * [1] H. S. Snyder, “Quantized space-time,” Phys. Rev. 71 (1947) 38–41.
  * [2] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP 09 (1999) 032, arXiv:hep-th/9908142.
  * [3] M. R. Douglas and N. A. Nekrasov, “Noncommutative field theory,” Rev. Mod. Phys. 73 (2001) 977–1029, arXiv:hep-th/0106048.
  * [4] R. J. Szabo, “Quantum field theory on noncommutative spaces,” Phys. Rept. 378 (2003) 207–299, arXiv:hep-th/0109162.
  * [5] P. Aschieri and L. Castellani, “Noncommutative gauge and gravity theories and geometric Seiberg–Witten map,” The European Physical Journal Special Topics (June, 2023) .
  * [6] K. Hersent, P. Mathieu, and J.-C. Wallet, “Gauge theories on quantum spaces,” Phys. Rept. 1014 (2023) 1–83, arXiv:2210.11890 [hep-th].
  * [7] M. Born, “A suggestion for unifying quantum theory and relativity,” Proc. Roy. Soc. London A 165 no. 921, (1938) 291–303.
  * [8] M. Born, “Reciprocity Theory of Elementary Particles,” Rev. Mod. Phys. 21 no. 3, (1949) 463–473.
  * [9] Y. A. Golfand, “Quantum field theory in constant curvature p-space,” Zh. Eksp. Teor. Fiz. 43 no. 1, (1962) 256–267.
  * [10] I. E. Tamm, “On curved momentum space,” in Proceedings of the International Conference on Elementary Particles, p. 314.  Kyoto, Kyoto University, 1965.
  * [11] V. G. Kadyshevsky, M. D. Mateev, R. M. Mir-Kasimov, and I. P. Volobuev, “Equations of Motion for the Scalar and the Spinor Fields in Four-Dimensional Noneuclidean Momentum Space,” Theor. Math. Phys. 40 (1979) 800–807.
  * [12] L. Freidel and E. R. Livine, “3D Quantum Gravity and Effective Noncommutative Quantum Field Theory,” Phys. Rev. Lett. 96 (2006) 221301, arXiv:hep-th/0512113.
  * [13] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, and L. Smolin, “The principle of relative locality,” Phys. Rev. D 84 (2011) 084010, arXiv:1101.0931 [hep-th].
  * [14] J. Kowalski-Glikman, “Living in Curved Momentum Space,” Int. J. Mod. Phys. A 28 (2013) 1330014, arXiv:1303.0195 [hep-th].
  * [15] C. Guedes, D. Oriti, and M. Raasakka, “Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups,” J. Math. Phys. 54 (2013) 083508, arXiv:1301.7750 [math-ph].
  * [16] A. Smilga, “Comments on noncommutative quantum mechanical systems associated with Lie algebras,” J. Geom. Phys. 180 (2022) 104628, arXiv:2204.08705 [hep-th].
  * [17] S. Franchino-Vinas, S. Mignemi, and J. J. Relancio, “The beauty of curved momentum space,” Proc. Sci. 436 (2023) 340, arXiv:2303.08220 [hep-th].
  * [18] Komaie-Moghaddam, H., Fatollahi, A., and Khorrami, M. “Field theory amplitudes in a space with SU(2) fuzziness,” Eur. Phys. J. C 53 (2008) 679–688,  arXiv:0712.1670 [hep-th].
  * [19] V. G. Kupriyanov, A. A. Sharapov and R. J. Szabo, “Symplectic groupoids and Poisson electrodynamics,” JHEP 03 (2024), 039 [arXiv:2308.07406 [hep-th]].
  * [20] V. G. Kupriyanov and R. J. Szabo, “Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry,” J. Phys. A 55 no. 3, (2022) 035201, arXiv:2101.12618 [hep-th].
  * [21] V. G. Kupriyanov, “Poisson gauge theory,” JHEP 09 (2021) 016, arXiv:2105.14965 [hep-th].
  * [22] V. G. Kupriyanov, M. A. Kurkov, and P. Vitale, “Poisson gauge models and Seiberg–Witten map,” JHEP 11 (2022) 062, arXiv:2209.13044 [hep-th].
  * [23] F. Di Cosmo, A. Ibort, G. Marmo, and P. Vitale, “Symplectic realizations and Lie groupoids in Poisson Electrodynamics,” arXiv:2312.16308 [hep-th].
  * [24] A. A. Sharapov, “On Poisson electrodynamics with charged fields,” arXiv preprint arXiv:2402.14439 (2024) .
  * [25] M. Kontsevich, “Deformation quantization of Poisson manifolds. 1.,” Lett. Math. Phys. 66 (2003) 157–216, arXiv:q-alg/9709040 [q-alg].
  * [26] A. Weinstein, “The local structure of poisson manifolds,” J. Diff. Geom. 18 (1983) 523.
  * [27] J. M. Gracia-Bondia, F. Lizzi, G. Marmo, and P. Vitale, “Infinitely many star products to play with,” JHEP 04 (2002) 026, arXiv:hep-th/0112092.
  * [28] M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization.  Translations of Mathematical Monographs, Vol. 119. American Mathematical Society, 1993.
  * [29] I. Vaisman, Lectures on the Geometry of Poisson Manifolds.  Progress in Mathematics. Birkhäuser Basel, 2012.
  * [30] M. Crainic, R. L. Fernandes, and I. Mărcuţ, Lectures on Poisson Geometry, vol. 217 of Graduate Studies in Mathematics.  American Mathematical Society, 2021.
  * [31] Marjan-S. Mirahmadi, Amir H. Fatollahi, and Mohammad Khorrami, “The similarity of attractive and repulsive forces on a lattice,” Mod. Phys. Lett. A 30 no. 23 (2015) 1550112,  arXiv:1405.4199 [quant-ph].
  * [32] Mohammad Khorrami, Amir H. Fatollahi, and Ahmad Shariati, “Poisson structure on a space with linear SU(2) fuzziness,” J. Math. Phys. 50 no. 7 (2009) 072902, arXiv:0806.1808 [hep-th].
  * [33] Amir H. Fatollahi, Ahmad Shariati, and Mohammad Khorrami, ‘‘Closedness of orbits in a space with SU(2) Poisson structure,’’ Int. J. Mod. Phys. A 29 no. 17 (2014) 1450081,  arXiv:1412.4614 [physics.class-ph].
  * [34] Andalib, T., Fatollahi, A.H. ‘‘Central force problem in space with SU(2) Poisson structure,’’ J. High Energ. Phys. 07 (2013) 022,  arXiv:1312.2115 [hep-th].
  * [35] J. Sivardière, ‘‘On the classical motion of a charge in the field of a magnetic monopole,’’ European Journal of Physics 21 no. 2, (Mar., 2000) 183--190.
  * [36] M. Kurkov and P. Vitale, ‘‘Four-dimensional noncommutative deformations of $U(1)$ gauge theory and $L_{\infty}$-bootstrap,’’ JHEP 01 (2022) 032, arXiv:2108.04856 [hep-th].
  * [37] V. G. Kupriyanov, M. A. Kurkov, and P. Vitale, ‘‘Lie--Poisson gauge theories and $\kappa$-Minkowski electrodynamics,’’ JHEP 11 (2023) 200, arXiv:2304.04857 [hep-th].


* [24] Liu, Y., Pan, S., Qian, Y.: An inexact $q$-order regularized proximal Newton method for nonconvex composite optimization. https://arxiv.org/abs/2311.06871.
  * [25] Luo, Z.-Q., Tseng, P.: Error bounds and convergence analysis of feasible descent methods: A general approach. Ann. Oper. Res. 46, 157–178 (1993)
  * [26] Mordukhovich, B.S.: Variational Analysis and Generalized differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)
  * [27] Mordukhovich, B.S.: Second-Order Variational Analysis in Optimization, Variational Stability, and Control: Theory, Algorithm, Applications. Springer, Cham (2024)
  * [28] Mordukhovich, B.S., Ouyang, W.: Higher-order metric subregularity and its applications. J. Global Optim. 63, 777–795 (2015)
  * [29] Mordukhovich, B.S., Sarabi, M.E.: Generalized Newton algorithms for tilt-stable minimizers in nonsmooth Optimization. SIAM J. Optim. 31, 1184–1214 (2021)
  * [30] Mordukhovich, B.S., Yuan, X., Zeng, S., Zhang, J.: A globally convergent proximal Newton-type method in nonsmooth convex optimization. Math. Program. 198, 899–936 (2023)
  * [31] Nabou, Y., Necoara, I.: Efficiency of higher-order algorithms for minimizing composite functions. Comput. Optim. Appl. 87, 441–473 (2024)
  * [32] Nesterov, Yu., Polyak, B.T.: Cubic regularization of Newton method and its global performance. Math. Program. 108, 177–205 (2006)
  * [33] Ochs, P.: Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano. SIAM J. Optim. 29(1), 541–570 (2019)
  * [34] Ouyang, W., Liu Y., Pong, T.K., Wang, H.: Kurdyka-Łojasiewicz exponent via Hadamard parametrization. https://arxiv.org/abs/2402.00377
  * [35] Poon, C., Peyré, G.: Smooth bilevel programming for sparse regularization. In: Proceedings of NeurIPS’21 (2021). https://arxiv.org/abs/2106.01429
  * [36] Poon, C., Peyré, G.: Smooth over-parameterized solvers for nonsmooth structured optimization. Math. Program. 201, 897–952 (2023)
  * [37] Rockafellar, R.T., Wets, R.J-B.: Variational Analysis. Springer, Berlin (1998)
  * [38] Yao, J.C., Zheng, X.Y., Zhu, J.: Stable minimizers of $\varphi$-regular functions. SIAM J. Optim. 27, 1150–1170 (2017)
  * [39] Yu, P., Li, G., Pong T.K.: Kurdyka-Łojasiewicz exponent via inf-projection. Found. Comput. Math. 22, 1171–1217 (2022)
  * [40] Yue, M.-C., Zhou, Z., Man-Cho So, A.: On the quadratic convergence of the cubic regularization method under a local error bound condition. SIAM J. Optim. 29, 904–932 (2019)
  * [41] Zhang, B., Ng, K.F., Zheng, X.Y., He, Q.: Hölder metric subregularity for multifunctions in $\mathcal{C}^{2}$ type Banach spaces. Optimization. 65, 1963–1982 (2016)
  * [42] Zheng, X.Y., Zhu, J.: Generalized metric subregularity and regularity with respect to an admissible function. SIAM J. Optim. 26, 535–563 (2016)
  * [43] Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

##  9 Proofs of Technical Lemmas

The main goal of this appendix is to give a detailed proof of Lemma 7.6. We
split this proof into several technical steps (also called lemmas), which are
of their own interest.

######  Lemma 9.1.

Under Assumption 7.1, for any $x\in\mathcal{F}$ we have

$\langle\nabla f(x),x-y_{\sigma}(x)\rangle\geq 0.$ (9.1)

If furthermore $\sigma\geq\frac{2L_{2}}{q+2}$ and $f(x)\leq f(x_{0})$, then
$y_{\sigma}(x)\in\mathcal{L}(f(x))\subset\mathcal{F}$.

###### Proof.

It readily follows from (7.7) that

$\langle\nabla
f(x),y_{\sigma}(x)-x\rangle+\langle\nabla^{2}f(x)(y_{\sigma}(x)-x),y_{\sigma}(x)-x\rangle+\frac{\sigma}{q+1}\|y_{\sigma}(x)-x\|^{q+2}=0.$
(9.2)

Employing (7.8), we obtain

$\langle\nabla^{2}f(x)(y_{\sigma}(x)-x),y_{\sigma}(x)-x\rangle+\frac{\sigma}{q+1}\|y_{\sigma}(x)-x\|^{q+2}\geq
0,$

and hence (9.1) is a direct consequence of (9.2).

Let now $\sigma\geq\frac{2L_{2}}{q+2}$ and $f(x)\leq f(x_{0})$. To see the
conclusion, we argue by contradiction and suppose that
$y_{\sigma}(x)\notin\mathcal{F}$, which yields $\|x-y_{\sigma}(x)\|>0$. Denote

$y_{\alpha}:=x+\alpha(y_{\sigma}(x)-x),\quad\alpha\in[0,1],$ (9.3)

and deduce from $\mathcal{L}(f(x_{0})\subset{\rm int}(\mathcal{F})$ (by (2) in
Assumption 7.1) and $f(x)\leq f(x_{0})$ that $y_{0}=x\in{\rm
int}(\mathcal{F})$. So the number
$\overline{\alpha}:=\min\\{\alpha>0\;|\;y_{\alpha}\in{\rm bd}(\mathcal{F})\\}$
is well-defined. It follows that $\overline{\alpha}<1$ and that
$y_{\alpha}\in\mathcal{F}$ for all $\alpha\in[0,\overline{\alpha}]$.
Consequently, combining Lemma 7.3 with (9.1)–(9.3) gives us the estimates

$\displaystyle f(y_{\alpha})\leq$ $\displaystyle f(x)+\langle\nabla
f(x),y_{\alpha}-x\rangle+\frac{1}{2}\langle\nabla^{2}f(x)(y_{\alpha}-x),y_{\alpha}-x\rangle+\frac{L_{2}}{(q+1)(q+2)}\|y_{\alpha}-x\|^{q+2}$
$\displaystyle\leq$ $\displaystyle f(x)+\alpha\langle\nabla
f(x),y_{\sigma}(x)-x\rangle+\frac{\alpha^{2}}{2}\langle\nabla^{2}f(x)(y_{\sigma}(x)-x),y_{\sigma}(x)-x\rangle+\frac{\alpha^{q+2}\sigma}{2(q+1)}\|y_{\sigma}(x)-x\|^{q+2}$
$\displaystyle\leq$ $\displaystyle
f(x)+\left(\alpha-\frac{\alpha^{2}}{2}\right)\langle\nabla
f(x),y_{\sigma}(x)-x\rangle-\left(\frac{\sigma\alpha^{2}}{2(q+1)}-\frac{\alpha^{q+2}\sigma}{2(q+1)}\right)\|y_{\sigma}(x)-x\|^{q+2}$
$\displaystyle\leq$ $\displaystyle
f(x)-\frac{\sigma\alpha^{2}(1-\alpha^{q})}{2(q+1)}\|y_{\sigma}(x)-x\|^{q+2},$

which imply in turn that $f(y_{\overline{\alpha}})<f(x)$. Therefore,
$y_{\overline{\alpha}}\in{\rm int}(\mathcal{F})$, a contradiction ensuring
that $y_{\sigma}(x)\in\mathcal{F}$. In a similar way, we can check that
$f(y_{\sigma}(x))\leq f(x)$. ∎

######  Lemma 9.2.

Under Assumption 7.1, for any $x\in\mathcal{F}$ with
$y_{\sigma}(x)\in\mathcal{F}$ we have

$\|\nabla
f(y_{\sigma}(x))\|\leq\frac{L_{2}+\sigma}{q+1}\|y_{\sigma}(x)-x\|^{q+1}.$
(9.4)

###### Proof.

It follows from equation (7.7) that

$\|\nabla
f(x)+\nabla^{2}f(x)(y_{\sigma}(x)-x)\|=\frac{\sigma}{q+1}\|y_{\sigma}(x)-x\|^{q+1}.$

On the other hand, we deduce from (7.9) that

$\|\nabla f(y_{\sigma}(x))-\nabla
f(x)-\nabla^{2}f(x)(y_{\sigma}(x)-x)\|\leq\frac{L_{2}}{q+1}\|y_{\sigma}(x)-x\|^{q+1}.$

Combining these two relations verifies the desired inequality (9.4). ∎

######  Lemma 9.3.

Under Assumption 7.1, for any $x\in\mathcal{F}$ we have the estimates

$\overline{f}_{\sigma}(x)\leq\min_{y\in\mathcal{F}}\left[f(y)+\frac{L_{2}+\sigma}{(q+1)(q+2)}\|y-x\|^{q+2}\right],$
(9.5)
$f(x)-\overline{f}_{\sigma}(x)\geq\frac{q\sigma}{2(q+1)(q+2)}\|y_{\sigma}(x)-x\|^{q+2}.$
(9.6)

Moreover, if $\sigma\geq L_{2}$ and $f(x)\leq f(x_{0})$, then
$y_{\sigma}(x)\in\mathcal{F}$ and

$f(y_{\sigma}(x))\leq\overline{f}_{\sigma}(x).$ (9.7)

###### Proof.

Employing the lower bound in (7.10), for any $x,y\in\mathcal{F}$, we get

$f(x)+\langle\nabla
f(x),y-x\rangle+\frac{1}{2}\langle\nabla^{2}f(x)(y-x),y-x\rangle\leq
f(y)+\frac{L_{2}}{(q+1)(q+2)}\|y-x\|^{q+2},$

and so inequality (9.5) follows from the construction of
$\overline{f}_{\sigma}(x)$. Furthermore, we deduce from
$y_{\sigma}(x)\in\mathop{\arg\min}_{y\in\mathbb{R}^{m}}\overline{f}_{\sigma}(x)$,
(9.2), and Lemma 9.1 that

$\displaystyle f(x)-\overline{f}_{\sigma}(x)$ $\displaystyle=\langle\nabla
f(x),x-y_{\sigma}(x)\rangle-\frac{1}{2}\langle\nabla^{2}f(x)(y_{\sigma}(x)-x),y_{\sigma}(x)-x\rangle-\frac{\sigma}{(q+1)(q+2)}\|y_{\sigma}(x)-x\|^{q+2}$
$\displaystyle=\frac{1}{2}\langle\nabla
f(x),x-y_{\sigma}(x)\rangle+\frac{q\sigma}{2(q+1)(q+2)}\|y_{\sigma}(x)-x\|^{q+2}\geq\frac{q\sigma}{2(q+1)(q+2)}\|y_{\sigma}(x)-x\|^{q+2}.$

Finally, assuming $\sigma\geq L_{2}$ and $f(x)\leq f(x_{0})$ gives us by Lemma
9.1 that $y_{\sigma}(x)\in\mathcal{F}$, and thus (9.7) follows from the upper
bound in (7.10). ∎

######  Lemma 9.4.

Under Assumption 7.1, for any $x\in\mathcal{F}$ with
$y_{\sigma}(x)\in\mathcal{F}$ we have

$\lambda_{\min}\Big{(}\nabla^{2}f(y_{\sigma}(x))\Big{)}\geq-\left(\frac{\sigma}{q+1}+L_{2}\right)\|y_{\sigma}(x)-x\|^{q}.$
(9.8)

###### Proof.

The Hölder continuity of $\nabla^{2}f(x)$ and inequality (7.8) tell us that

$\displaystyle\nabla^{2}f(y_{\sigma}(x))\succeq\nabla^{2}f(x)-L_{2}\|y_{\sigma}(x)-x\|^{q}I_{m}\succeq-\left(\frac{\sigma}{q+1}+L_{2}\right)\|y_{\sigma}(x)-x\|^{q}I_{m},$

which clearly yields the estimate in (9.8). ∎

######  Lemma 9.5.

Let Assumption 7.1 hold, and let the sequences $\\{x_{k}\\}$ and
$\\{\widehat{x}_{k}\\}$ be taken from Algorithm 1 with
$\overline{\sigma}>\frac{2L_{2}}{q+2}$. Then for all $k\in\mathbb{N}$ we have
the estimates

$f(\widehat{x}_{k+1})-f(x_{k})\leq-\frac{(q+2)\overline{\sigma}-2L_{2}}{2(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2},$
(9.9)
$\sum_{i=0}^{k}\|\widehat{x}_{k+1}-x_{k}\|^{q+2}\leq\frac{2(q+1)(q+2)}{(q+2)\overline{\sigma}-2L_{2}}(f(x_{0})-f^{*}),$
(9.10) $\|\nabla
f(\widehat{x}_{k+1})\|\leq\frac{3L_{2}}{q+1}\|\widehat{x}_{k+1}-x_{k}\|^{q+1},$
(9.11)
$\lambda_{\min}(\nabla^{2}f(\widehat{x}_{k+1}))\geq-\frac{(q+3)L_{2}}{q+1}\|\widehat{x}_{k+1}-x_{k}\|^{q}.$
(9.12)

###### Proof.

First we verify (9.9). It follows from Lemma 7.3, (7.2), and (9.6) that

$\displaystyle f(\widehat{x}_{k+1})\leq f(x_{k})+\langle\nabla
f(x_{k}),\widehat{x}_{k+1}-x_{k}\rangle$
$\displaystyle+\frac{1}{2}\langle\nabla^{2}f(x_{k})(\widehat{x}_{k+1}-x_{k}),\widehat{x}_{k+1}-x_{k}\rangle+\frac{L_{2}}{(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2}$
$\displaystyle=f(x_{k})+\langle\nabla
f(x_{k}),\widehat{x}_{k+1}-x_{k}\rangle+\frac{1}{2}\langle\nabla^{2}f(x_{k})(\widehat{x}_{k+1}-x_{k}),\widehat{x}_{k+1}-x_{k}\rangle$
$\displaystyle+\frac{\sigma_{k}}{(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2}+\frac{L_{2}-\sigma_{k}}{(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2}$
$\displaystyle\leq
f(x_{k})-\frac{q\sigma_{k}}{2(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2}+\frac{L_{2}-\sigma_{k}}{(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2}$
$\displaystyle=f(x_{k})-\frac{(q+2)\sigma_{k}-2L_{2}}{2(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2}\leq
f(x_{k})-\frac{(q+2)\overline{\sigma}-2L_{2}}{2(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2},$

which justifies (9.9). To prove (9.10), we use (7.6) and (9.9) to get

$\displaystyle f(x_{0})-f^{*}\geq$
$\displaystyle\sum_{i=0}^{k}[f(x_{i})-f(x_{i+1})]\geq\sum_{i=0}^{k}[f(x_{i})-f(\widehat{x}_{i+1})]$
$\displaystyle\geq$
$\displaystyle\sum_{i=0}^{k}\frac{(q+2)\overline{\sigma}-2L_{2}}{2(q+1)(q+2)}\|\widehat{x}_{i+1}-x_{i}\|^{q+2},$

which yields (9.10). The inequalities in (9.11) and (9.12) follow from Lemmas
9.2 and 9.4, respectively. ∎

The next estimates follow from (7.6) and (7.5) being useful for our subsequent
analysis:

$\displaystyle\|x_{k+1}-\widehat{x}_{k+1}\|\leq$
$\displaystyle\max\big{\\{}\|\widehat{x}_{k+1}-\widehat{x}_{k+1}\|,\|\widetilde{x}_{k+1}-\widehat{x}_{k+1}\|\big{\\}}$
(9.13) $\displaystyle=$
$\displaystyle\|\widetilde{x}_{k+1}-\widehat{x}_{k+1}\|\leq\beta_{k+1}\|\widehat{x}_{k+1}-\widehat{x}_{k}\|.$

######  Lemma 9.6.

Under Assumption 7.1, the sequence $\\{\widehat{x}_{k}\\}$ generated by
Algorithm 1 satisfies

$\|\widehat{x}_{k+1}-\widehat{x}_{k}\|\leq\frac{1}{1-\zeta}\left[\frac{2(q+1)(q+2)}{(q+2)\overline{\sigma}-2L_{2}}(f(x_{0})-f^{*})\right]^{\frac{1}{q+2}}=:\gamma.$

###### Proof.

For any index $i\geq 1$, we deduce from (9.13) and the condition
$\beta_{k+1}\leq\zeta$ for all $k\geq 0$ that

$\displaystyle\|\widehat{x}_{i+1}-\widehat{x}_{i}\|\leq$
$\displaystyle\|\widehat{x}_{i+1}-x_{i}\|+\|x_{i}-\widehat{x}_{i}\|\leq\|\widehat{x}_{i+1}-x_{i}\|+\beta_{i}\|\widehat{x}_{i}-\widehat{x}_{i-1}\|\leq\|\widehat{x}_{i+1}-x_{i}\|+\zeta\|\widehat{x}_{i}-\widehat{x}_{i-1}\|.$
(9.14)

Recursively applying (9.14), we deduce from $\widehat{x}_{0}=x_{0}$ that

$\displaystyle\|\widehat{x}_{k+1}-\widehat{x}_{k}\|\leq$
$\displaystyle\zeta^{k}\|\widehat{x}_{1}-\widehat{x}_{0}\|+\sum_{i=1}^{k}\zeta^{k-i}\|\widehat{x}_{i+1}-x_{i}\|\leq\sum_{i=0}^{k}\zeta^{k-i}\|\widehat{x}_{i+1}-x_{i}\|$

whenever $k\in\mathbb{N}$. Then it follows from (9.10) that

$\displaystyle\|\widehat{x}_{k+1}-\widehat{x}_{k}\|$
$\displaystyle\leq\max_{i\in\\{0,\cdots,k\\}}\|\widehat{x}_{i+1}-x_{i}\|\sum_{i=0}^{k}\zeta^{k-i}\leq\frac{1}{1-\zeta}\max_{i\in\\{0,\cdots,k\\}}\|\widehat{x}_{i+1}-x_{i}\|$
$\displaystyle\leq\frac{1}{1-\zeta}\left[\frac{2(q+1)(q+2)}{(q+2)\overline{\sigma}-2L_{2}}(f(x_{0})-f^{*})\right]^{\frac{1}{q+2}},$

which justified the claim of the lemma. ∎

Now we are ready to finalize the Proof of Lemma 7.6:
To verify assertion (i) therein, it suffices to show that $\\{f(x_{k})\\}$ is
a decreasing sequence having a lower bound. By Assumption 7.1, $f$ is bounded
from below and so is $\\{f(x_{k})\\}$. Employing (7.6) and (9.9) yields

$f(x_{k+1})\leq f(\widehat{x}_{k+1})\leq
f(x_{k})-\frac{(q+2)\overline{\sigma}-2L_{2}}{2(q+1)(q+2)}\|\widehat{x}_{k+1}-x_{k}\|^{q+2}$
(9.15)

and allows us to deduce that $\\{f(x_{k})\\}$ converges to some $v$.
To verify (ii), we begin with using (9.13),
$\beta_{k+1}\leq\|\widehat{x}_{k+1}-x_{k}\|$, and Lemma 9.6 to get

$\displaystyle\|x_{k+1}-x_{k}\|\leq\|x_{k+1}-\widehat{x}_{k+1}\|+\|\widehat{x}_{k+1}-x_{k}\|\leq\beta_{k+1}\|\widehat{x}_{k+1}-\widehat{x}_{k}\|+\|\widehat{x}_{k+1}-x_{k}\|$
$\displaystyle\leq\|\widehat{x}_{k+1}-x_{k}\|(\|\widehat{x}_{k+1}-\widehat{x}_{k}\|+1)\leq(1+\gamma)\|\widehat{x}_{k+1}-x_{k}\|,$

which tells us together with (9.10) that

$\lim_{k\to\infty}\|\widehat{x}_{k+1}-x_{k}\|=0.$ (9.16)

Combining further Assumption 7.1 and Lemma 9.6 with (7.4), (9.13), and (9.11),
we arrive at

$\displaystyle\|\nabla f(x_{k+1})\|\leq\|\nabla
f(\widehat{x}_{k+1})\|+\|\nabla f(\widehat{x}_{k+1})-\nabla f(x_{k+1})\|$
(9.17) $\displaystyle\leq\|\nabla
f(\widehat{x}_{k+1})\|+L_{1}\|x_{k+1}-\widehat{x}_{k+1}\|\leq\|\nabla
f(\widehat{x}_{k+1})\|+L_{1}\beta_{k+1}\|\widehat{x}_{k+1}-\widehat{x}_{k}\|$
$\displaystyle\leq\|\nabla
f(\widehat{x}_{k+1})\|(1+L_{1}\|\widehat{x}_{k+1}-\widehat{x}_{k}\|)\|\leq\frac{3(1+\gamma
L_{1})L_{2}}{q+1}\|\widehat{x}_{k+1}-x_{k}\|^{q+1},$

and therefore verify assertion (ii).
To verify the final assertion (iii), note that (9.15) yields
$\\{x_{k}\\}\subset\mathcal{L}(f(x_{0}))$. Recalling that
$\mathcal{L}(f(x_{k_{0}}))$ is bounded for some $k_{0}\in\mathbb{N}$, we
conclude that $\\{x_{k}\\}$ is bounded as well, which ensures that
$\Omega\neq\emptyset$. Pick any $\overline{x}\in\Omega$ and observe that
justifying (7.22) requires to bound the minimum eigenvalue of the Hessian.
Recalling that $|\lambda_{\min}(C)-\lambda_{\min}(D)|\leq\|C-D\|$ holds for
any real symmetric matrices $C$ and $D$, we have

$\displaystyle\lambda_{\min}(\nabla^{2}f(x_{k+1}))\geq\lambda_{\min}(\nabla^{2}f(\widehat{x}_{k+1}))-\|\nabla^{2}f(x_{k+1})-\nabla^{2}f(\widehat{x}_{k+1})\|.$

This together with Assumption 7.1, Lemma 9.6, (9.12), (9.13), and
$\beta_{k+1}\leq\|\widehat{x}_{k+1}-x_{k}\|$ tells us that

$\displaystyle\lambda_{\min}(\nabla^{2}f(x_{k+1}))\geq\lambda_{\min}(\nabla^{2}f(\widehat{x}_{k+1}))-\|\nabla^{2}f(x_{k+1})-\nabla^{2}f(\widehat{x}_{k+1})\|\geq\lambda_{\min}(\nabla^{2}f(\widehat{x}_{k+1}))-L_{2}\|x_{k+1}-\widehat{x}_{k+1}\|^{q}$
$\displaystyle\geq\lambda_{\min}(\nabla^{2}f(\widehat{x}_{k+1}))-L_{2}\beta_{k+1}^{q}\|\widehat{x}_{k+1}-\widehat{x}_{k}\|^{q}\geq\lambda_{\min}(\nabla^{2}f(\widehat{x}_{k+1}))-L_{2}\gamma^{q}\|\widehat{x}_{k+1}-x_{k}\|^{q}$
$\displaystyle\geq-\frac{(q+3)L_{2}}{q+1}\|\widehat{x}_{k+1}-x_{k}\|^{q}-L_{2}\gamma^{q}\|\widehat{x}_{k+1}-x_{k}\|^{q}=-\left(\frac{q+3}{q+1}+\gamma^{q}\right)L_{2}\|\widehat{x}_{k+1}-x_{k}\|^{q}.$

Unifying the latter with (9.16) and (9.17) leads us to the inequalities

$\|\nabla f(\overline{x})\|\leq\limsup_{k\to\infty}\|\nabla f(x_{k+1})\|\leq
0\quad\mbox{and}\quad\lambda_{\min}(\nabla^{2}f(\overline{x}))\geq\liminf_{k\to\infty}\lambda_{\min}(\nabla^{2}f(x_{k+1}))\geq
0,$

which show that $\nabla f(\overline{x})=0$ and
$\nabla^{2}f(\overline{x})\succeq 0$. Employing finally Lemma 7.6(i), we
obtain $f(\overline{x})=v$ and therefore complete the proof of the lemma.