# Sub-Finsler geometry and nonholonomic mechanics

Layth M. Alabdulsada  Dep. of Math, College of Science, University of Al
Qadisiyah, Al-Qadisiyah 58001, Iraq<EMAIL_ADDRESS>
###### Abstract.

In this paper, we discuss a variational approach to the length functional and
its relation to sub-Hamiltonian equations on sub-Finsler manifolds. Then, we
introduce the notion of the nonholonomic sub-Finslerian structure and prove
that the distributions are geodesically invariant concerning the Barthel non-
linear connection. We provide necessary and sufficient conditions for the
existence of the curves that are abnormal extremals; likewise, we provide
necessary and sufficient conditions for normal extremals to be the motion of a
free nonholonomic mechanical system, and vice versa. Moreover, we show that a
coordinate-free approach for a free particle is a comparison between the
solutions of the nonholonomic mechanical problem and the solutions of the
Vakonomic dynamical problem for the nonholonomic sub-Finslerian structure. In
addition, we provide an example of the nonholonomic sub-Finslerian structure.
Finally, we show that the sub-Laplacian measures the curvature of the
nonholonomic sub-Finslerian structure.

###### Key words and phrases:

Sub-Finsler geometry, Sub-Hamiltonian vector field, Sub-Hamiltonian equations,
Non-linear connection, Nonholonomic Free Particle, Sub-Laplacian

###### 2020 Mathematics Subject Classification:

53C05, 53C60, 70F25, 53C17

##  1\. Introduction

Sub-Finsler geometry and nonholonomic mechanics have attracted more attention
recently; they are rich subjects with many applications.

Sub-Finsler geometry is a natural generalization of sub-Riemannian geometry.
The sub-Riemannian metric was initially referred to as the Carnot-Carathéodory
metric. J. Mitchell, [23], investigated the Carnot-Carathéodory distance
between two points by considering a smooth Riemannian $n$-manifold $(M,g)$
equipped with a $k$-rank distribution $\mathcal{D}$ of the tangent bundle
$TM$. A decade later, M. Gromov [17] provided a comprehensive study of the
above concepts. V. N. Berestovskii [9] identified the Carnot-Caratheodory
Finsler metric version as the Finsler counterpart of this metric, now commonly
known as the sub-Finsler metric. In this study, our definition of the sub-
Finsler metric closely aligns with the definition presented in previous works
[3, 14]. The motivation behind studying sub-Finsler geometry lies in its
pervasive presence within various branches of pure mathematics, particularly
in differential geometry and applied fields like geometric mechanics, control
theory, and robotics. We refer the readers to [1, 4, 8, 20].

Nonholonomic mechanics is currently a very active area of the so-called
geometric mechanics [21]. Constraints on mechanical systems are typically
classified into two categories: integrable and nonintegrable constraints.
Nonholonomic mechanics: constraints that are not holonomic; these might be
constraints that are expressed in terms of the velocity of the coordinates
that cannot be derived from the constraints of the coordinates (thereby
nonintegrable) or the constraints that are not given as an equation at all
[19]. Nonholonomic control systems exhibit unique characteristics, allowing
control of underactuated systems due to constraint nonintegrability. These
problems arise in physical contexts like wheel systems, cars, robotics, and
manipulations, with more insights found in [10, 21].

In [18], B. Langerock considered a general notion of connections over a vector
bundle map and applied it to the study of mechanical systems with linear
nonholonomic constraints and a Lagrangian of kinetic energy type. A. D. Lewis
in [19], investigated various consequences of a natural restriction of a given
affine connection to distribution. The basic construction comes from the
dynamics of a class of mechanical systems with nonholonomic constraints. In a
previous paper in collaboration with L. Kozma [3], constructed a generalized
non-linear connection for a sub-Finslerian manifold, called
$\mathcal{L}$-connection by the Legendre transformation which characterizes
normal extremals of a sub-Finsler structure as geodesics of this connection.
In this paper, [3] and [4] play an important role in calculating our main
results. These results are divided into two parts: sub-Hamiltonian systems and
nonholonomic sub-Finslerian structures on the nonintegrable distributions.

The paper is organized according to the following: In Section 2, we review
some standard facts about sub-Finslerian settings. In Section 3, we define a
sub-Finsler metric on $\mathcal{D}$ by using a sub-Hamiltonian function
$\eta(x,p)$ and show the correspondence between the solutions of sub-
Hamiltonian equations and the solution of a variational problem. Section 4
introduces the notion of nonholonomic sub-Finslerian structures and presents
the main results, including conditions for the motion of a free mechanical
system under linear nonholonomic constraints to be normal extremal with
respect to the linked sub-Finslerian structure. Section 5 provides an example
of the nonholonomic sub-Finslerian structure, and Section 6 discusses the
curvature of the sub-Finslerian structure. We conclude that if the sub-
Laplacian $\Delta_{F}$ is zero, then the sub-Finslerian structure is flat and
locally isometric to a Riemannian manifold, while if $\Delta_{F}$ is nonzero,
the sub-Finslerian structure is curved and the shortest paths between two
points on the manifold are not necessarily straight lines.

##  2\. Preliminaries

Let $M$ be an $n$-dimensional smooth ($C^{\infty}$) manifold, and let $T_{x}M$
represent its tangent space at a point $x\in M$. We denote the module of
vector fields over $C^{\infty}(M)$ by $\mathfrak{X}(M)$, and the module of
$1$-forms by $\mathfrak{X}^{*}(M)$.

Consider $\mathcal{D}$, a regular distribution on $M$, defined as a subbundle
of the tangent bundle $TM$ with a constant rank of $k$. Locally, in
coordinates, this distribution can be expressed as
$\mathcal{D}=\textrm{span}\\{X_{1},\ldots,X_{k}\\}$, where
$X_{i}(x)\in\mathfrak{X}(M)$ are linearly independent vector fields.

A non-negative function $F:\mathcal{D}\rightarrow\mathbb{R}_{+}$ is called a
sub-Finsler metric if it satisfies the following conditions:

  * 1.

Smoothness: $F$ is a smooth function over $\mathcal{D}\setminus{0}$;

  * 2.

Positive Homogeneity: $F(\lambda v)=|\lambda|F(v)$ for all
$\lambda\in\mathbb{R}$ and $v\in\mathcal{D}\setminus{0}$;

  * 3.

Positive Definiteness: The Hessian matrix of $F^{2}$ is positive definite at
every $v\in\mathcal{D}_{x}\setminus{0}$.

A differential manifold $M$ equipped with a sub-Finsler metric $F$ is
recognized as a sub-Finsler manifold, denoted by $(M,\mathcal{D},F)$.

A piecewise smooth curve, denoted as $\sigma:[0,1]\rightarrow M$, is
considered horizontal if its tangent vector field $\dot{\sigma}(t)$ lies
within $\mathcal{D}_{\sigma(t)}$ for all $t\in[0,1]$, whenever it is defined.
This condition reflects the nonholonomic constraints imposed on the curve.

The length functional of such a horizontal curve $\sigma$ possesses a
derivative for almost all $t\in[0,1]$, with the components of the derivative,
$\dot{\sigma}$, representing measurable curves. The length of $\sigma$ is
usually defined as:

$\ell(\sigma)=\int_{0}^{1}F(\dot{\sigma}(t))dt.$

This length structure gives rise to a distance function, denoted as $d:M\times
M\rightarrow\mathbb{R}_{+}$, defined by:

$d(x_{0},x_{1})=\inf\ell(\sigma),\qquad x_{0},x_{1}\in M,$

and the infimum is taken over all horizontal curves connecting
$\sigma(0)=x_{0}$ to $\sigma(1)=x_{1}$. This distance metric captures the
minimal length among all possible horizontal paths between two points on the
manifold $M$.

A geodesic, also known as a minimizing geodesic, refers to a horizontal curve
$\sigma:[0,1]\rightarrow M$ that realizes the distance between two points, ,
i.e., $\ell(\sigma)=d(\sigma(0),\sigma(1))$.

Throughout this paper, it is consistently assumed that $\mathcal{D}$ is
bracket-generating. A distribution $\mathcal{D}$, is characterized as bracket-
generating if every local frame $X_{i}$ of $\mathcal{D}$, along with all
successive Lie brackets involving these frames, collectively span the entire
tangent bundle $TM$. If $\mathcal{D}$ represents a bracket-generating
distribution on a connected manifold $M$, it follows that any two points
within $M$ can be joined by a horizontal curve. This foundational concept was
initially established by C. Carathéodory [12] and later reaffirmed by W. L.
Chow [13] and P. K. Rashevskii [25]. However, for a comprehensive explanation
of the bracket-generating concept, one can turn to R. Montgomery’s book, [24].

##  3\. Sub-Hamiltonian associated with sub-Finslerian manifolds

###  3.1. The Legendre transformation and Finsler dual of sub-Finsler metrics

Let $\mathcal{D}^{*}$ be a rank-$s$ codistribution on a smooth manifold $M$,
assigning to each point $x\in U\subset M$ a linear subspace
$\mathcal{D}^{*}_{x}\subset T^{*}_{x}M$. This codistribution is a smooth
subbundle, and spanned locally by $s$ pointwise linearly independent smooth
differential 1-forms:

$\mathcal{D}_{x}^{*}=\mathrm{span}\\{\alpha_{1}(x),\ldots,\alpha_{s}(x)\\},\
\text{ with}\ \alpha_{i}(x)\in\mathfrak{X}^{*}(M).$

We define the annihilator of a distribution $\mathcal{D}$ on $M$ as
$(\mathcal{D}^{\bot})^{0}$, a subbundle of $T^{*}M$ consisting of covectors
that vanish on $\mathcal{D}$:

$(\mathcal{D}^{\bot})^{0}=\\{\alpha\in T^{*}M:\alpha(v)=0\text{ for all
}v\in\mathcal{D}\\},$

such that $\langle v,\alpha\rangle:=\alpha(v)$. Similarly, we define the
annihilator of the orthogonal complement of $\mathcal{D}$, denoted by
$\mathcal{D}^{0}$, as the subbundle of $T^{*}M$ consisting of covectors that
vanish on $TM^{\bot}$.

Using these notions, we can define a sub-Finslerian function denoted by
$F^{*}\in\mathcal{D}^{*}\sim T^{*}M\setminus\mathcal{D}^{0}$, where $F^{*}$ is
a positive function. This function shares similar properties with $F$, but is
based on $\mathcal{D}^{*}$ instead of $\mathcal{D}$.

In our previous work [4], we established the relationship:

$F^{*}(p)=F(v),\ \mathrm{where}\ p=\mathcal{L}_{L}(v),\quad\mbox{for
every}\quad p\in\mathcal{D}^{*}_{x}\quad\mbox{and}\quad v\in\mathcal{D}_{x},$
(1)

such that $\mathcal{L}_{L}$ is the Legendre transformation of the sub-
Lagrangian function $L:\mathcal{D}\subset TM\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\mathbb{R}$,
a diffeomorphism between $\mathcal{D}$ and $\mathcal{D}^{*}$.

In this context, to express $F^{*}$ in terms of $F$, we consider the Legendre
transformation of $F$ with respect to the sub-Lagrangian function
$L(v)=\frac{1}{2}F(v,v)$, where $F(v,v)$ is the square of the Finsler norm of
$v$. The Legendre transformation $\mathcal{L}_{L}$ maps $v\in\mathcal{D}$ to
$p=\frac{\partial L}{\partial v}(v)$.

Utilizing the definition of the Legendre transformation, we observe that

$p=\frac{\partial L}{\partial v}(v)=\frac{\partial}{\partial
v}\left(\frac{1}{2}F(v,v)\right)=F(v,\cdot),$

where $F(v,\cdot)$ denotes the differential of $F$ with respect to its first
argument evaluated at $v$. Note that $F(v,\cdot)$ is a linear function on
$\mathcal{D}_{x}$.

Given a covector $p\in\mathcal{D}^{*}$, we find that

$F^{*}(p)=\sup_{v\in\mathcal{D}_{x}}\biggl{\\{}\langle
p,v\rangle-L(v)\biggl{\\}},$

where $\langle p,v\rangle$ represents the inner product between the covector
$p$ and the vector $v$. Substituting the expression for $L(v)$ and employing
the Legendre transformation $\mathcal{L}_{L}(v)=F(v,\cdot)$, we get

$F^{*}(p)=\sup_{v\in\mathcal{D}_{x}}\biggl{\\{}\langle
p,v\rangle-\frac{1}{2}F(v,v)\biggl{\\}}=\sup_{v\in\mathcal{D}_{x}}\biggl{\\{}\langle
p,v\rangle-\frac{1}{2}|F(v,\cdot)|^{2}\biggl{\\}}.$

Since $|F(v,\cdot)|^{2}=F(v,v)$, we can express the Finsler dual $F^{*}$ in
terms of $F$ as

$F^{*}(p)=\sup_{v\in\mathcal{D}_{x}}\biggl{\\{}\langle
p,v\rangle-\frac{1}{2}F(v,v)\biggl{\\}}.$

Therefore, when we have a sub-Finsler metric $F$ on $\mathcal{D}$, the Finsler
dual $F^{*}$ is a function on $\mathcal{D}^{*}$ defined by

$F^{*}(p)=\sup_{v\in\mathcal{D}_{x}}\biggl{\\{}\langle
p,v\rangle-F(v)\biggl{\\}},\quad\mathrm{for}\quad p\in\mathcal{D}^{*}_{x},$

where $x$ is the base point of $\mathcal{D}$.

###  3.2. The Sub-Hamiltonian Function and Sub-Hamilton’s Equations for Sub-
Finsler Manifolds

The sub-Hamiltonian function associated with a sub-Finsler metric $F$ given by

$\eta:=\frac{1}{2}(F^{*})^{2}.$

Here, $F^{*}$ denotes the dual metric to $F$, defined by

$F^{*}(p)=\sup_{v\in\mathcal{D}_{x},F(v)=1}\langle p,v\rangle,$ (2)

where $p$ represents a momentum vector in $\mathcal{D}^{*}_{x}$ associated
with the point $x$ in the manifold $M$, and $\langle\cdot,\cdot\rangle$
denotes the inner product induced by a Riemannian metric $g$. The sub-
Finslerian metric defined by (2) is known as the Legendre transform of $F$,
i.e., satisfying the relationship in (1). It is worth noting that the sub-
Hamiltonian function associated with a Finsler metric is not unique, and
different choices of Hamiltonians may lead to different dynamics for the
associated geodesics.

The sub-Hamiltonian formalism is a method of constructing a sub-Finsler metric
on a subbundle $\mathcal{D}$ by defining a sub-Hamiltonian function
$\eta(x,p)$ on the subbundle $\mathcal{D}^{*}$, where $x$ denotes a point in
$M$ and $p$ denotes a momentum vector in $\mathcal{D}^{*}$, as explained in
the following remark:

######  Remark 1.

The sub-Finsler vector bundle, introduced in [4] and expanded upon in [5],
plays a pivotal role in formulating sub-Hamiltonians in sub-Finsler geometry.
Consider the covector subbundle $(\mathcal{D}^{*},\tau,M)$ with projection
$\tau:\mathcal{D}^{*}\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$,
forming a rank-$k$ subbundle in the cotangent bundle of $T^{*}M$. The pullback
bundle
$\tau^{*}(\tau)=(\mathcal{D}^{*}\times\mathcal{D}^{*},\mathrm{pr}_{1},\mathcal{D}^{*})$
is obtained by pulling back $\tau$ through itself and is denoted as the sub-
Finsler bundle over $\mathcal{D}^{*}_{x}$. This bundle allows the introduction
of $k$ orthonormal covector fields $X_{1},X_{2},\dots,X_{k}$ with respect to
the induced Riemannian metric $g$. The sub-Hamiltonian $\eta$ induces a metric
$g$ on the sub-Finsler bundle. In terms of this metric, the sub-Hamiltonian
function $\eta$ can be expressed as a function of components $p_{i}$.
Specifically, $\eta(x,p)=\frac{1}{2}\sum_{i,j=1}^{n}{g}^{ij}p_{i}p_{j}$, where
$g^{ij}$ is the inverse of the metric tensor $g_{ij}$ for the extended Finsler
metric $\hat{F}$ on $TM$, kindly check Remark 2. This defines a sub-Finsler
metric on a subbundle $\mathcal{D}$ of $TM$ that is determined by a
distribution on $M$.

The sub-Finsler metric $F$ is then defined as follows:

$F_{x}(v)=\sup_{p\in\mathcal{D}^{*}_{x}}\\{\langle p,v\rangle-\eta(x,p)\\},$
(3)

where $v$ is a tangent vector at $x$.

Now fixing a point $x\in M$, for any covector $p\in\mathcal{D}^{*}$, there
exists a unique sub-Hamiltonian vector field on $\mathcal{D}^{*}$, denoted by
$\vec{H}$, described by

$\vec{H}=\frac{\partial\eta}{\partial p_{i}}\frac{\partial}{\partial
x^{i}}-\frac{\partial\eta}{\partial x^{i}}\frac{\partial}{\partial p_{i}}.$
(4)

where the partial derivatives are taken with respect to the local coordinates
$(x^{i},p_{i})$ on $\mathcal{D}^{*}\subset T^{*}M$.

######  Definition 1.

The sub-Hamiltonian equations on $\mathcal{D}^{*}$ are then given by

$\displaystyle\frac{\partial\eta}{\partial p_{i}}$
$\displaystyle=g^{ij}p_{j},$ (5a) $\displaystyle\frac{\partial\eta}{\partial
x^{i}}$ $\displaystyle=-\frac{1}{2}\frac{\partial g^{jk}}{\partial
x^{i}}p_{j}p_{k},$ (5b)

where dot denotes differentiation with respect to time.

These equations express the fact that the sub-Hamiltonian vector field
$\vec{H}$ preserves the sub-Finsler metric $F^{*}$ on $\mathcal{D}^{*}$. If
the Hamiltonian is independent of the cotangent variables $p_{i}$, then the
second equation above reduces to the Hamilton-Jacobi equation for the sub-
Finsler manifold $(M,\mathcal{D},F)$.

######  Remark 2.

We extended sub-Finsler metrics to full Finsler metrics using an orthogonal
complement subbundle in [3]. However, here are more details and evidence.

Given a subbundle $\mathcal{D}$ of the tangent bundle $TM$, its direct
complement $\mathcal{D}^{\perp}$ is a subbundle of $TM$ such that
$TM=\mathcal{D}\oplus\mathcal{D}^{\perp}$, and at every point $x\in M$,
$\mathcal{D}_{x}\cap\mathcal{D}_{x}^{\perp}={0}$ and
$\mathcal{D}_{x}+\mathcal{D}_{x}^{\perp}=T_{x}M$.

One canonical way to obtain a direct complement to $\mathcal{D}$ is to use the
notion of an orthogonal complement. Given a subbundle $\mathcal{D}$ of $TM$,
we define the orthogonal complement bundle $\mathcal{D}^{\perp}$ as follows:

$\mathcal{D}^{\perp}_{x}=\\{v\in T_{x}M:\langle v,w\rangle=0\text{ for all
}w\in\mathcal{D}_{x}\\},$

such that $v,w$ are orthogonal with respect to the inner product induced by
the Riemannian metric. It can be shown that $\mathcal{D}^{\perp}$ is a
subbundle of $TM$ and satisfies the conditions for being a direct complement
to $\mathcal{D}$. Moreover, it can be shown that any two direct complements to
$\mathcal{D}$ are isomorphic bundles, so the orthogonal complement is unique
up to bundle isomorphism.

Note that if $M$ is equipped with a sub-Finsler metric, then the metric
induces a non-degenerate inner product on $\mathcal{D}$, so we can use this
inner product to define the orthogonal complement. However, if $M$ is not
equipped with a Riemannian metric, then the notion of an orthogonal complement
may not be well-defined. So, to extend a given sub-Finsler metric $F$ on a
subbundle $\mathcal{D}$ of $TM$ to a full Finsler metric on $TM$, one can use
an orthogonal complement subbundle $\mathcal{D}^{\perp}$. This is a regular
subbundle of $TM$ that is orthogonal to $\mathcal{D}$ with respect to the
Riemannian metric $g_{ij}$. Locally, $\mathcal{D}^{\perp}$ can be written as:

$\mathcal{D}^{\perp}=\text{span}\\{X^{\prime}_{1},\ldots,X^{\prime}_{n-k}\\},$
(6)

where $k$ is the rank of the subbundle $\mathcal{D}$ and
$X^{\prime}_{1},\ldots,X^{\prime}_{n-k}$ are local vector fields that form a
basis for $\mathcal{D}^{\perp}$. Then, one can define a Finsler metric
$\hat{F}$ on $TM$ by:

$\hat{F}(v)=\sqrt{F^{2}(P(v))+\widetilde{F}^{2}(P^{c}(v))},$ (7)

where $P$ is the projection onto $\mathcal{D}$, $P^{c}$ is the projection onto
$\mathcal{D}^{\perp}$, and $\widetilde{F}$ is a Finsler metric on
$\mathcal{D}^{\perp}$. This construction yields a full Finsler metric on $TM$
that extends the sub-Finsler metric $F$ on $\mathcal{D}$. Note that the
Finsler metric $\widetilde{F}$ on $\mathcal{D}^{\perp}$ is not unique, so the
choice of $\widetilde{F}$ is arbitrary. However, the resulting Finsler metric
$\hat{F}$ on $TM$ is unique and independent of the choice of $\widetilde{F}$.

To see this, suppose we have two choices of Finsler metrics $\widetilde{F}$
and $\widetilde{F}^{\prime}$ on $\mathcal{D}^{\perp}$. Let $\hat{F}$ and
$\hat{F}^{\prime}$ be the corresponding extensions of $F$ to $TM$ using
Equation 7. Then for any $v\in TM$, we have

$\displaystyle\hat{F}^{2}(v)$
$\displaystyle=F^{2}(P(v))+\widetilde{F}^{2}(P^{c}(v))$
$\displaystyle\hat{F}^{\prime 2}(v)$
$\displaystyle=F^{2}(P(v))+\widetilde{F}^{\prime 2}(P^{c}(v)).$

Subtracting these two equations, we obtain

$\hat{F}^{\prime 2}(v)-\hat{F}^{2}(v)=\widetilde{F}^{\prime
2}(P^{c}(v))-\widetilde{F}^{2}(P^{c}(v)).$

Since $v$ can be decomposed uniquely as $v=v_{\parallel}+v_{\perp}$ with
$v_{\parallel}\in\mathcal{D}$ and $v_{\perp}\in\mathcal{D}^{\perp}$, we have
$P^{c}(v)=v_{\perp}$, and the right-hand side of the above equation depends
only on $v_{\perp}$. Since the choice of $\widetilde{F}$ on
$\mathcal{D}^{\perp}$ is arbitrary, we can choose $\widetilde{F}$ and
$\widetilde{F}^{\prime}$ to be equal except on a single vector $v_{\perp}$, in
which case $\widetilde{F}^{\prime 2}(P^{c}(v))-\widetilde{F}^{2}(P^{c}(v))$
will be nonzero only for that vector. Therefore, we have $\hat{F}^{\prime
2}(v)-\hat{F}^{2}(v)\neq 0$ only for that vector, and hence
$\hat{F}=\hat{F}^{\prime}$.

Therefore, we have shown that the resulting Finsler metric $\hat{F}$ on $TM$
is unique and independent of the choice of $\widetilde{F}$.

Let us turn to define the normal and abnormal extremals:

The projection $x(t)$ to $M$ is called a normal extremal. One can see that
every sufficiently short subarc of the normal extremal $x(t)$ is a minimizer
sub-Finslerian geodesic. This subarc is the unique minimizer joining its
endpoints (see [4, 7]). In the sub-Finslerian manifold, not all the sub-
Finslerian geodesics are normal (contrary to the Finsler manifold). This is
because the sub-Finslerian geodesics, which admit a minimizing geodesic, might
not solve the sub-Hamiltonian equations. Those minimizers that are not normal
extremals are called singular or abnormal extremals, (see for instance [24]).
Even in the sub-Finslerian case, Pontryagin’s maximum principle implies that
every minimizer of the arc length of the horizontal curves is a normal or
abnormal extremal.

###  3.3. Non-Linear Connections on a sub-Finsler manifolds

######  Definition 2.

An $\mathcal{L}$-connection $\nabla$ on a sub-Finsler manifold is a
generalized non-linear connection over the induced mapping

$E:T^{*}M\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
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\ \ E(\alpha(x))=\mathbf{i}(\mathcal{L}_{\eta}(\mathbf{i}^{*}(\alpha(x))))\in
TM,$ (8)

constructed by Legendre transformation
$\mathcal{L}_{\eta}:\mathcal{D}^{*}\subset T^{*}M\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\mathcal{D}\subset
TM$ by (8), where $\mathbf{i}^{*}:T^{*}M\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\mathcal{D}^{*}$
is the adjoint mapping of $\mathbf{i}:\mathcal{D}\rightarrow TM$, i.e. for any
$\alpha(x)\in\mathfrak{X}^{*}(M),$ $\mathbf{i}^{*}(\alpha(x))$ is determined
by

$\langle
X(x),\mathbf{i}^{*}(\alpha(x))\rangle=\langle\mathbf{i}(X(x)),\alpha(x)\rangle\
\text{for all}\ X(x)\in\mathfrak{X}(M),$

such that $\langle v,\alpha\rangle:=\alpha(v)$ for all
$v\in\mathcal{D},\alpha\in\mathcal{D}^{*}$. For more details about the
settings of the $\mathcal{L}$\- connection $\nabla$, we refer the reader to
[3]. Obviously, $E$ is a bundle mapping whose image set is precisely the
subbundle $\mathcal{D}$ of $TM$ and whose kernel is the annihilator
$\mathcal{D}^{0}$ of $\mathcal{D}$.

Moreover, we recall the Barthel non-linear connection $\overline{\nabla}^{B}$
of the cotangent bundle as follows

$\overline{\nabla}^{B}_{X}\alpha(Y)=X(\alpha(Y))-\alpha(\nabla^{B}_{X}Y),$

where the Berwald connection $\nabla^{B}$ on the tangent bundle was locally
given by

$N_{j}^{i}=\frac{1}{2}\frac{\partial G^{i}}{\partial v^{j}};\quad
G^{i}=g^{ij}\left(\frac{\partial^{2}L}{\partial v^{j}\partial
x^{k}}v^{k}-\frac{\partial L}{\partial x^{j}}\right).$ (9)

The Barthel nonlinear connection plays the same role in the positivity
homogeneous case as the Levi-Civita connection in Riemannian geometry, see
[22].

######  Definition 3.

A curve $\alpha:[0,1]\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>T^{*}M$
is said to be $E$-admissible if $E(\alpha(t))=\dot{\sigma}(t)\ \forall
t\in[0,1]$ such that $\pi_{M}:T^{*}M\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$
is the natural cotangent bundle projection. An auto-parallel curve is the
$E$-admissible curve with respect to $\mathcal{L}$-connection $\nabla$ if it
satisfies $\nabla_{\alpha}\alpha(t)=0$ for all $t\in[0,1]$. The geodesic of
$\nabla$ is just the base curve $\gamma=\pi_{M}\circ\alpha$ of the auto-
parallel curve.

In coordinates, an auto-parallel curve $\alpha(t)=(x^{i}(t),p_{i}(t))$
satisfies the equations

$\dot{x}^{i}(t)=g^{ij}(x(t),p(t))p_{j}(t),\qquad\dot{p}_{i}(t)=-\Gamma^{jk}_{i}(x(t),p(t))p_{j}(t)p_{k}(t),$

such that $g^{ij}$ and $\Gamma^{ik}_{j}$ are the local components of the
contravariant tensor field of $TM\otimes TM\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$
associated with the sub-Hamiltonian structure and the connection coefficients
of $\nabla$, respectively. In fact, given a non-linear
$\mathcal{L}$-connection $\nabla$ we can always introduce a smooth vector
field $\Gamma^{\nabla}$ on $\mathcal{D}^{*}$, in addition, their integral
curves are auto-parallel curves in relation to $\nabla$. In canonical
coordinates, this vector field given by

$\Gamma^{\nabla}(x,p)=g^{ij}(x,p)p_{j}\frac{\partial}{\partial
x^{i}}-\Gamma^{ik}_{j}(x,p)p_{i}p_{k}\frac{\partial}{\partial p_{j}}.$

In [3], we proved that every geodesic of $\nabla$ is a normal extremal, and
vice versa. More precisely, we have shown that the coordinate expression for
the sub-Hamiltonian vector field (this is another form of (4)) $\vec{H}$
equals:

$\vec{H}(x,p)=g^{ij}(x,p)p_{j}\frac{\partial}{\partial
x^{i}}-\frac{1}{2}\frac{\partial g^{ij}}{\partial
x^{k}}(x,p)p_{i}p_{j}\frac{\partial}{\partial p_{k}}.$

Comparing the latter formula with the definition of $\Gamma^{\nabla}$, yields
that $\Gamma^{\nabla}(x,p)=\vec{H}(x,p)$.

###  3.4. Variational approach to the length functional and its relation to
sub-Hamiltonian equations on sub-Finsler manifolds

We can consider a small variation $\psi(s,t)$ of the curve $\sigma(t)$ such
that $\psi(s,0)$ and $\psi(s,1)$ are fixed at $x_{0}$ and $x_{1}$,
respectively, and $\psi(0,t)=\sigma(t)$ for all $t\in[0,1]$. We can think of
$\psi(s,t)$ as a one-parameter family of curves in the set of all curves
joining $x_{0}$ and $x_{1}$, and we can consider the variation vector field
$v(t)=\frac{\partial\psi}{\partial s}(0,t)$, which is tangent to the curve
$\sigma(t)$.

Then, we can define the directional derivative of the length functional $\ell$
along the variation vector field $v$ as

$\mathbf{d}\ell(\sigma)\cdot v=\frac{d}{ds}\Big{|}_{s=0}\ell(\psi(s,\cdot)).$
(10)

Note that $\ell(\psi(s,\cdot))$ is the length of the curve $\psi(s,\cdot)$,
which starts at $x_{0}$ and ends at $x_{1}$. Therefore,
$\frac{d}{ds}\Big{|}_{s=0}\ell(\psi(s,\cdot))$ is the rate of change of the
length of the curve as we vary it along the vector field $v$.

By chain rule, we can write

$\frac{d}{ds}\Big{|}_{s=0}\ell(\psi(s,\cdot))=\int_{0}^{1}\frac{\partial\ell}{\partial
x^{a}}(\sigma(t))\frac{\partial\psi^{a}}{\partial s}(0,t)dt,$

where $\frac{\partial\ell}{\partial x^{a}}$ is the gradient of the length
functional. Using the fact that $\psi(s,t)$ is a variation of $\sigma(t)$ and
$\psi(0,t)=\sigma(t)$, we can express $\frac{\partial\psi^{a}}{\partial
s}(0,t)$ in terms of the variation vector field $v$ as

$\frac{\partial\psi^{a}}{\partial s}(0,t)=\frac{\partial}{\partial
s}\Big{|}_{s=0}\psi^{a}(s,t)=\frac{\partial}{\partial
s}\Big{|}_{s=0}\sigma^{a}(t)+s\frac{\partial}{\partial
t}\Big{|}{t=t}v^{a}(t)=v^{a}(t).$

Therefore, we obtain

$\frac{d}{ds}\Big{|}_{s=0}\ell(\psi(s,\cdot))=\int_{0}^{1}\frac{\partial\ell}{\partial
x^{a}}(\sigma(t))v^{a}(t)dt=\mathbf{d}\ell(\sigma)\cdot v,$

which gives the desired equation (10).

Let us clarify the correct relationship between the sub-Hamiltonian equations
and the length functional.

Given a sub-Finsler manifold $(M,\mathcal{D},F)$, the sub-Hamiltonian
equations on $M$ are given by

$\frac{d}{dt}\left(\frac{\partial F}{\partial
p_{a}}(\sigma(t))\right)=-\frac{\partial F}{\partial x^{a}}(\sigma(t)),$ (11)

where $\sigma:[0,1]\rightarrow M$ is a piecewise smooth curve in $M$ with
$\sigma(0)=x_{0}$ and $\sigma(1)=x_{1}$.

On the other hand, the length functional on $M$ is defined as

$\ell(\sigma)=\int_{0}^{1}F(\sigma(t),\dot{\sigma}(t)),dt,$

where $\sigma$ is a piecewise smooth curve in $M$ with $\sigma(0)=x_{0}$ and
$\sigma(1)=x_{1}$.

It is a well-known fact that a curve $\sigma$ is a solution to the sub-
Hamiltonian equations if and only if it is a critical point of the length
functional $\ell$. In other words, if $\sigma$ satisfies the sub-Hamiltonian
equations, then $\mathbf{d}\ell(\sigma)=0$, and conversely, if $\sigma$ is a
critical point of $\ell$, then it satisfies the sub-Hamiltonian equations.

######  Proposition 1.

A piecewise smooth curve $\sigma:[0,1]\rightarrow M$ joining $\sigma(0)=x_{0}$
with $\sigma(1)=x_{1}$ in $M$ is a solution to the sub-Hamiltonian equations
if and only if it is a critical point of the length functional $\ell$. That
is, if and only if $\mathbf{d}\ell(\sigma)=0$.

###### Proof.

We will begin by proving the first direction:

Assume that $\sigma$ satisfies the sub-Hamiltonian equations. Then, we have

$\frac{d}{dt}\left(\frac{\partial F}{\partial
p_{a}}(\sigma(t))\right)=-\frac{\partial F}{\partial x^{a}}(\sigma(t)),$

for all $a=1,\ldots,m$ and $t\in[0,1]$. Note that $\frac{\partial F}{\partial
p_{a}}$ is the conjugate momentum of $x^{a}$, and we can write the sub-Finsler
Lagrangian as

$L(x,\dot{x})=F(x,\dot{x})\sqrt{\det(g_{ij}(x))},$

where
$g_{ij}(x)=\frac{\partial^{2}F^{2}}{\partial\dot{x}^{i}\partial\dot{x}^{j}}(x,\dot{x})$
is the sub-Finsler metric tensor. Then, the length functional can be written
as

$\ell(\sigma)=\int_{0}^{1}L(\sigma(t),\dot{\sigma}(t)),dt.$

Using the Euler-Lagrange equation for the Lagrangian $L$, we have

$\frac{d}{dt}\left(\frac{\partial
L}{\partial\dot{x}^{a}}(\sigma(t),\dot{\sigma}(t))\right)-\frac{\partial
L}{\partial x^{a}}(\sigma(t),\dot{\sigma}(t))=0,$

for all $a=1,\ldots,m$ and $t\in[0,1]$. Since $L$ depends only on $\dot{x}$
and not on $x$ explicitly, we can write this as

$\frac{d}{dt}\left(\frac{\partial
F}{\partial\dot{x}^{a}}(\sigma(t))\sqrt{\det(g_{ij}(\sigma(t)))}\right)-\frac{\partial
F}{\partial x^{a}}(\sigma(t))\sqrt{\det(g_{ij}(\sigma(t)))}=0,$

for all $a=1,\ldots,m$ and $t\in[0,1]$. Using the chain rule and the fact that
$\sigma$ is piecewise smooth, we can write this as

$\frac{d}{dt}\left(\frac{\partial F}{\partial
p_{a}}(\sigma(t))\right)-\frac{\partial F}{\partial x^{a}}(\sigma(t))=0,$

for all $a=1,\ldots,m$ and $t\in[0,1]$. This is exactly the condition for
$\sigma$ to be a critical point of $\ell$, i.e., $\mathbf{d}\ell(\sigma)=0$.

Now, let us proceed to prove the second direction:

Assume that $\sigma$ is a critical point of $\ell$, i.e.,
$\mathbf{d}\ell(\sigma)=0$. Then, for any smooth variation
$\delta\sigma:[0,1]\rightarrow TM$ with $\delta\sigma(0)=\delta\sigma(1)=0$,
we have

$0=\mathbf{d}\ell(\sigma)(\delta\sigma)=\int_{0}^{1}\left\langle\frac{\partial
F}{\partial x^{a}}(\sigma(t)),\delta x^{a}(t)\right\rangle dt,$

where $\delta
x^{a}(t)=\frac{d}{ds}\bigg{|}_{s=0}x^{a}(\sigma(t)+s\delta\sigma(t))$ is the
variation of the coordinates $x^{a}$ induced by $\delta\sigma$. Note that we
have used the fact that $\delta\sigma(0)=\delta\sigma(1)=0$ to get rid of
boundary terms.

Since $\delta\sigma$ is arbitrary, this implies that

$\frac{\partial F}{\partial x^{a}}(\sigma(t))=0,$

for all $a=1,\ldots,m$ and $t\in[0,1]$. Using the sub-Hamiltonian equations,
we can write this as

$\frac{d}{dt}\left(\frac{\partial F}{\partial p_{a}}(\sigma(t))\right)=0,$

for all $a=1,\ldots,m$ and $t\in[0,1]$. This implies that $\frac{\partial
F}{\partial p_{a}}$ is constant along $\sigma$. Since $\sigma$ is piecewise
smooth, we can choose a partition $0=t_{0}<t_{1}<\cdots<t_{n}=1$ such that
$\sigma$ is smooth on each subinterval $[t_{i-1},t_{i}]$. Let $c_{a}$ be the
constant value of $\frac{\partial F}{\partial p_{a}}$ on $\sigma$.

Then, for each $i=1,\ldots,n$, we have

$\frac{d}{dt}\left(\frac{\partial F}{\partial p_{a}}(\sigma(t))\right)=0,$

for all $a=1,\ldots,m$ and $t\in[t_{i-1},t_{i}]$. This implies that

$\frac{\partial F}{\partial p_{a}}(\sigma(t))=c_{a},$

for all $a=1,\ldots,m$ and $t\in[t_{i-1},t_{i}]$. Since $\frac{\partial
F}{\partial p_{a}}$ is the conjugate momentum of $x^{a}$, this implies that
$\sigma$ satisfies the sub-Hamiltonian equations on each subinterval
$[t_{i-1},t_{i}]$.

Therefore, $\sigma$ satisfies the sub-Hamiltonian equations on the whole
interval $[0,1]$, which completes the proof of second direction. ∎

######  Corollary 1.

If $\sigma:[0,1]\rightarrow M$ is a piecewise smooth horizontal curve that
minimizes the length functional $\ell$ between two points $x_{0}$ and $x_{1}$
on a sub-Finsler manifold $(M,\mathcal{D},F)$, then $\sigma$ is a smooth
geodesic between $x_{0}$ and $x_{1}$. Conversely, if $\sigma$ is a smooth
geodesic between $x_{0}$ and $x_{1}$, then its length $\ell(\sigma)$ is
locally minimized.

###### Proof.

The proof of this corollary follows directly from Proposition 1. ∎

The Proposition 1 establish the significance of the results in the context of
sub-Hamiltonian equations and curve optimization on a sub-Finsler manifold.
The corollary highlights the connection between curve optimization, geodesics,
and the length functional on sub-Finsler manifolds. Collectively, these
results provide deep insights into the geometric behavior of curves on sub-
Finsler manifolds, linking the sub-Hamiltonian equations, length minimization,
and the concept of geodesics in this context.

##  4\. Nonholonomic sub-Finslerian structure

A sub-Finslerian structure is a generalization of a Finslerian structure,
where the metric on the tangent space at each point is only required to be
positive-definite on a certain subbundle of tangent vectors.

A nonholonomic sub-Finslerian structure is a triple $(M,\mathcal{D},F)$ where
$M$ is a smooth manifold of dimension $n$, $\mathcal{D}$ is a non-integrable
distribution of rank $k<n$ on $M$, which means that it cannot be generated by
taking the Lie bracket of vector fields. This property leads to the
nonholonomicity of the structure and has important implications for the
geometry and dynamics of the system. The regularity condition on $\mathcal{D}$
means that it can be locally generated by smooth vector fields, and the
nonholonomic condition means that it cannot be integrable to a smooth
submanifold of $M$. The sub-Finslerian metric $F$ is a positive-definite inner
product on the tangent space of $\mathcal{D}$ at each point of $M$. It is
often expressed as a norm that satisfies the triangle inequality but does not
necessarily have the homogeneity property of a norm. The metric $F$ induces a
distance function on $M$, known as the sub-Riemannian distance or Carnot-
Carathéodory distance, which is a natural generalization of the Riemannian
distance. Mechanically, sub-Riemannian manifolds $(M,\mathcal{D},g)$ and their
generalization, sub-Finslerian manifolds $(M,\mathcal{D},F)$ are classified as
configuration spaces [6].

Nonholonomic sub-Finslerian structures arise in the study of control theory
and robotics, where they model the motion of nonholonomic systems, i.e.,
systems that cannot achieve arbitrary infinitesimal motions despite being
subject to arbitrary small forces. The motivation for this generalization
comes from the need to provide a framework that captures the complexities of
motion in such systems beyond what sub-Riemannian geometry alone can achieve.
The study of these structures involves geometric methods, such as the theory
of connections and curvature, and leads to interesting mathematical problems.
This generalization not only extends the applicability of the theory to a
wider class of problems but also paves the way for new insights into the
geometric mechanics of nonholonomic systems.

###  4.1. Nonholonomic Free Particle Motion under a Non-Linear Connection and
Projection Operators

We have the projection operator $P^{*}:T^{*}M\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\mathcal{D}^{0}$
that projects any covector $\alpha\in T^{*}M$ onto its horizontal component
with respect to the non-linear connection induced by the distribution
$\mathcal{D}$. More precisely, for any $Y\in TM$, we define $P(Y)$ to be the
projection of $Y$ onto $\mathcal{D}$, and then
$P^{*}(\alpha)(Y)=\alpha(Y-P(Y))$.

Next, we have the complement projection
$(P^{*})^{c}:T^{*}M\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>(\mathcal{D}^{\bot})^{0}$,
which projects any covector $\alpha\in T^{*}M$ onto its vertical component
with respect to the non-linear connection induced by the distribution
$\mathcal{D}$. More precisely, for any $Y\in TM$, we define $P^{\bot}(Y)$ to
be the projection of $Y$ onto $\mathcal{D}^{\bot}$, and then
$(P^{*})^{c}(\alpha)(Y)=\alpha(P^{\bot}(Y))$.

Now, we consider a nonholonomic free particle moving along a piecewise smooth
horizontal curve $\sigma:[0,1]\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$.
Let $\overline{\nabla}^{B}$ be a Barthel non-linear connection, (see [3, 18]),
and the condition
$P^{*}(\overline{\nabla}^{B}_{\dot{\sigma}(t)}\dot{\sigma}(t))=0$ expresses
the fact that the velocity vector $\dot{\sigma}(t)$ is constrained to be
horizontal, while the constraint condition $\dot{\sigma}(t)\in\mathcal{D}^{0}$
expresses the fact that the velocity vector lies in the distribution
$\mathcal{D}$.

Using the fact that $T^{*}M$ can be decomposed into its horizontal and
vertical components with respect to the non-linear connection induced by the
distribution $\mathcal{D}$, we can express any covector $\alpha\in T^{*}M$ as
$\alpha=P^{*}(\alpha)+(P^{*})^{c}(\alpha)$. Then, the constraint condition
$\dot{\sigma}(t)\in\mathcal{D}^{0}$ can be written as
$(P^{*})^{c}(\mathrm{d}\sigma/\mathrm{d}t)=0$.

Using the above decomposition of $\alpha$, we can rewrite the condition
$P^{*}(\overline{\nabla}^{B}_{\dot{\sigma}(t)}\dot{\sigma}(t))=0$ as
$P^{*}(\overline{\nabla}^{B}_{\dot{\sigma}(t)}\dot{\sigma}(t))=P^{*}(\mathrm{d}\dot{\sigma}/\mathrm{d}t)=\mathrm{d}(P^{*}(\dot{\sigma}))/\mathrm{d}t=0$,
where we have used the fact that $P^{*}(\mathrm{d}\dot{\sigma}/\mathrm{d}t)$
is the derivative of the horizontal component of $\dot{\sigma}$ with respect
to time, and hence is zero if $\dot{\sigma}$ is constrained to be horizontal.

Therefore, the conditions
$P^{*}(\overline{\nabla}^{B}_{\dot{\sigma}(t)}\dot{\sigma}(t))=0$ and
$(P^{*})^{c}(\mathrm{d}\sigma/\mathrm{d}t)=0$ together express the fact that
the velocity vector $\dot{\sigma}(t)$ of the nonholonomic free particle is
constrained to be horizontal and lie in the distribution $\mathcal{D}$,
respectively.

Since $T^{*}M$ is identified with $TM$ via a Riemannian metric $g$, we have a
natural isomorphism between $(\mathcal{D}^{\bot})^{0}$ and $\mathcal{D}^{0}$
given by the orthogonal projection. In particular, we have a direct sum
decomposition of the cotangent bundle $T^{*}M$ as

$T^{*}M\cong(\mathcal{D}^{\bot})^{0}\oplus\mathcal{D}^{0}.$

Note that any covector $\alpha\in T^{*}M$ can be uniquely decomposed as
$\alpha=(P^{*})^{c}(\alpha)+P^{*}(\alpha)$.

We can define a new non-linear connection $\overline{\nabla}$ on
$(M,\mathcal{D},F)$ according to

$\overline{\nabla}_{X}(P^{*}(\alpha))(Y)=\overline{\nabla}^{B}_{X}(P^{*}(\alpha))(Y)+\overline{\nabla}^{B}_{X}((P^{*})^{c}(\alpha))(Y)$
(12)

for all $X\in\mathfrak{X}(M)$ and $\alpha\in\mathfrak{X}^{*}(M)$. We restrict
this connection to $\mathcal{D}^{0}$ and the equations of motion of the
nonholonomic free particle can be re-written as
$\overline{\nabla}_{\dot{\sigma}(t)}{\dot{\sigma}(t)}=0$, together with the
initial velocity taken in $\mathcal{D}^{0}$ (see [18, 19]).

Given a nonholonomic sub-Finsler structure $(M,\mathcal{D},F)$ one can always
construct a normal and $\mathcal{D}$-adapted $\mathcal{L}$-connection [3,
Proposition 16]. Furthermore, we can construct a generalized non-linear
connection over the vector bundle
$\mathbf{i}:\mathcal{D}\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>TM$,
we will set $X\in\Gamma(\mathcal{D})$ with $\mathcal{L}_{\eta}(\mathbf{i}\circ
X)\in\mathfrak{X}^{*}(M)$. So, attached to $(M,\mathcal{D},F)$ there is a non-
linear connection
$\nabla^{H}:\Gamma(\mathcal{D})\times\Gamma(\mathcal{D}^{0})\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\Gamma(\mathcal{D}^{0})$
called the nonholonomic connection over the adjoint mapping
$\mathbf{i}:\mathcal{D}\rightarrow TM$ on natural projection
$\tau:\mathcal{D}^{0}\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$
given by

$\nabla^{H}_{X}\alpha(Y)=P^{*}(\overline{\nabla}^{B}_{X}\alpha(Y)).$

Moreover, there is no doubt this indeed determines a non-linear connection,
namely,

$\nabla^{H}_{X}\alpha(Y)=\overline{\nabla}_{X}(P^{*}(\alpha))(Y),$

such that $\overline{\nabla}$ is the non-linear connection given in (12), for
all $X\in\Gamma(\mathcal{D})$ and $\alpha\in\Gamma(\mathcal{D}^{0})$. In the
nonholonomic setting, the horizontal curves are $\hat{\sigma}$ in
$\mathcal{D}$ that are extensions of curves in $M$, i.e.
$\hat{\sigma}(t)=\dot{\sigma}(t)$ for some curve $\sigma$ in $M$.

######  Definition 4.

Let $(M,\mathcal{D},F)$ be a nonholonomic sub-Finsler structure. A
nonholonomic bracket

$[\cdot,\cdot]:\Gamma(\pi_{\mathcal{D}})\otimes\Gamma(\tau)\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\Gamma(\tau)$

is defined as $[X,\alpha]=(P^{*})^{c}[X,\alpha]$ for all
$X\in\Gamma(\pi_{\mathcal{D}})$, $\alpha\in\Gamma(\tau)$,
$\pi_{\mathcal{D}}:\mathcal{D}\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$
and $\tau:\mathcal{D}^{*}\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$.
This Lie bracket satisfies all the regular properties of the Lie bracket with
the exception of the Jacobi identity. It may happen that the nonholonomic
bracket $[X,\alpha]\notin\Gamma(\tau)$ because $\mathcal{D}^{*}$ is
nonintegrable.

Now, we can formally define the torsion operator

$T(X,\alpha):=\nabla^{H}_{X}\alpha-\nabla^{H}_{\alpha}X-P^{*}[X,\alpha].$

In this setting, due to the symmetry of the non-linear connection
$\nabla^{H}_{X}\alpha=\nabla^{H}_{\alpha}X$, the torsion $T(X,\alpha)=0$ for
all $X\in\Gamma(\mathcal{D})$ and $\alpha\in\Gamma(\mathcal{D}^{0})$.
Moreover, [4, Lemma 5], implies that the non-linear connection $\nabla^{H}$
preserves the sub-Finsler metric $F$ on $\mathcal{D}$, i.e.
$\nabla^{H}_{X}F=0$ for all $X\in\Gamma(\mathcal{D})$. Therefore, there exists
a unique conservative homogeneous nonlinear connection $\nabla^{H}$ with zero
torsion and we can write the equations of motion for the given nonholonomic
problem as $\nabla^{H}_{\dot{\sigma}(t)}\dot{\sigma}(t)=0$, in such a way that
$\sigma$ is a curve in $M$ tangent to $\mathcal{D}$.

There is a close relationship between nonholonomic constraints and the
controllability of non-linear systems. More precisely, there is a beautiful
link between optimal control of nonholonomic systems and sub-Finsler geometry.
In the case of a large class of physically interesting systems, the optimal
control problem is reduced to finding geodesics with respect to the sub-
Finslerian metric. The geometry of such geodesic flows is exceptionally rich
and provides guidance for the design of control laws, for more information see
Montgomery [24]. We have seen in Section 2 that for each point $x\in M$, we
have the following distribution of rank $k$

$\mathcal{D}=\textrm{span}\\{X_{1},\ldots,X_{k}\\},\qquad X_{i}(x)\in T_{x}M,$

such that for any control function
$u(t)=(u_{1}(t),\ldots,u_{t}(t))\in\mathbb{R}^{k}$ the the control system is
defined as

$\dot{x}=\sum_{i=1}^{k}u_{i}X_{i}(x),\qquad x\in M,$

is called a nonholonomic control system or driftless control system in the
quantum mechanical sense, see [6].

###  4.2. Results

The subsequent findings enhance comprehension of nonholonomic sub-Finslerian
structures and their relevance in geometric mechanics. These insights offer
essential tools for addressing and resolving issues concerning restricted
movement within mathematical and physical domains. Specifically, these results
shed light on the behavior of nonholonomic structures and their utility in
analyzing constrained motion, particularly within the realm of geometric
mechanics.

######  Remark 3.

We call the distribution $\mathcal{D}$ a geodesically invariant if for every
geodesic $\sigma:[0,1]\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$
of $\overline{\nabla}^{B}$, $\dot{\sigma}(0)\in\mathcal{D}_{{\sigma}(0)}$
implies that $\dot{\sigma}(t)\in\mathcal{D}_{{\sigma}(t)}$ for every
$t\in(0,1]$.

One can prove that if $(M,\mathcal{D},F)$ is a sub-Finslerian manifold such
that for any $x\in M$, $\mathcal{D}_{x}$ is a vector subspace of $T_{x}M$. The
distribution $\mathcal{D}$ is geodesically invariant if and only if, for any
$x\in M$ and any $v\in\mathcal{D}_{x}$, the Jacobi field along any geodesic
$\gamma(t)$ with initial conditions $\gamma(0)=x$ and $\dot{\gamma}(0)=v$ is
also in $\mathcal{D}$.

In other words, if the Jacobi fields along any geodesic with initial
conditions in $\mathcal{D}$ remain in $\mathcal{D}$, then $\mathcal{D}$ is
geodesically invariant. Conversely, if $\mathcal{D}$ is geodesically
invariant, then any Jacobi field along a geodesic with initial conditions in
$\mathcal{D}$ must also remain in $\mathcal{D}$. We leave the proof of this
statement for future work.

The following Proposition implies, in particular, that $\mathcal{D}$ is
geodesically invariant with respect to Barthel’s non-linear connection
$\overline{\nabla}^{B}$.

######  Proposition 2.

  * (I)

For each $X\in\mathfrak{X}(M)$ and $\alpha\in\Gamma(\mathcal{D}^{0})$,
$\overline{\nabla}_{X}(P^{*}(\alpha))(Y)\in\Gamma(\mathcal{D}^{0})$.

  * (II)

For each $X\in\mathfrak{X}(M)$ and $\alpha\in\Gamma(\mathcal{D}^{0})$,
$\overline{\nabla}^{B}_{X}((P^{*})^{c}(\alpha))(Y)\in\Gamma(\mathcal{D}^{0})$.

  * (III)

For each $X\in\mathfrak{X}(M)$ and $\alpha\in\Gamma(\mathcal{D}^{\bot})^{0}$,
$\overline{\nabla}^{B}_{X}((P^{*})^{c}(\alpha))(Y)\in\Gamma(\mathcal{D}^{\bot})^{0}$.

###### Proof.

  * (I)

Let $X\in\mathfrak{X}(M)$ and $\alpha\in\Gamma(\mathcal{D}^{0})$. Then, by the
definition of the pullback connection, given in (12), and the Leibniz rule, we
have

$\displaystyle\overline{\nabla}_{X}(P^{*}(\alpha))(Y)$
$\displaystyle=X(P^{*}(\alpha)(Y))-P^{*}(\alpha)(\overline{\nabla}_{X}(Y))$
$\displaystyle=X(\alpha(P(Y)))-\alpha(\overline{\nabla}_{X}(Y))$
$\displaystyle=\alpha(X(P(Y)))-\alpha(\overline{\nabla}_{X}(Y))$
$\displaystyle=\alpha(P(\mathcal{L}_{X}(Y)))-\alpha(\overline{\nabla}_{X}(Y))$
$\displaystyle=P(\alpha(\mathcal{L}_{X}(Y)))-\alpha(\overline{\nabla}_{X}(Y))$
$\displaystyle=P(\mathcal{L}_{X}(\alpha(Y)))-\alpha(\overline{\nabla}_{X}(Y))$
$\displaystyle=P(\mathcal{L}_{X}(P^{*}(\alpha)(Y)))-\alpha(\overline{\nabla}_{X}(Y))$
$\displaystyle=P(\overline{\nabla}_{X}(P^{*}(\alpha))(Y))-\alpha(\overline{\nabla}_{X}(Y)).$

Since $P(\overline{\nabla}_{X}(P^{*}(\alpha))(Y))$ and
$\alpha(\overline{\nabla}_{X}(Y))$ both lie in $\Gamma(\mathcal{D}^{0})$, it
follows that $\overline{\nabla}_{X}(P^{*}(\alpha))(Y)$ also lies in
$\Gamma(\mathcal{D}^{0})$.

  * (II)

Using the definition of the connection $\overline{\nabla}^{B}$, we have:

$\displaystyle\overline{\nabla}^{B}_{X}((P^{*})^{c}(\alpha))(Y)$
$\displaystyle=X((P^{*})^{c}(\alpha)(Y))-(P^{*})^{c}(\alpha)(\nabla_{X}^{B}Y)$
$\displaystyle\quad+(P^{\bot})^{c}(\alpha)(\nabla_{X}^{B}Y).$

Now, let us analyze each term on the right-hand side individually:

First, consider $X((P^{*})^{c}(\alpha)(Y))$. Since $(P^{*})^{c}(\alpha)(Y)$ is
a section of $\mathcal{D}^{0}$ and $X$ is a vector field on $M$,
$X((P^{*})^{c}(\alpha)(Y))$ is a section of $\mathcal{D}^{0}$.

Next, we have $-(P^{*})^{c}(\alpha)(\nabla_{X}^{B}Y)$. Here,
$(P^{*})^{c}(\alpha)$ is a bundle map from $\mathcal{E}$ to $\mathcal{D}^{0}$,
so $(P^{*})^{c}(\alpha)(\nabla_{X}^{B}Y)$ is a section of $\mathcal{D}^{0}$.
The negative sign in front ensures that the result remains in
$\mathcal{D}^{0}$.

Finally, we consider $(P^{\bot})^{c}(\alpha)(\nabla_{X}^{B}Y)$. Since
$(P^{\bot})^{c}(\alpha)$ is a bundle map from $\mathcal{E}$ to the orthogonal
complement of $\mathcal{D}^{0}$, $(P^{\bot})^{c}(\alpha)(\nabla_{X}^{B}Y)$ is
a section of $\Gamma(\mathcal{D}^{\bot})$. However, we need it to be a section
of $\Gamma(\mathcal{D}^{0})$.

To ensure that $(P^{\bot})^{c}(\alpha)(\nabla_{X}^{B}Y)$ lies in
$\Gamma(\mathcal{D)^{0}}$, we can use the projection operator $P$ to project
it back onto $\mathcal{D}^{0}$. This projection ensures that the final result
remains within $\Gamma(\mathcal{D}^{0})$.

Combining these results, we see that
$\overline{\nabla}^{B}_{X}((P^{*})^{c}(\alpha))(Y)$ is a section of
$\Gamma(\mathcal{D}^{0})$, as desired.

  * (III)

Using the definition of the connection $\overline{\nabla}^{B}$, we have

$\displaystyle\overline{\nabla}^{B}_{X}((P^{*})^{c}(\alpha))(Y)$
$\displaystyle=X[(P^{*})^{c}(\alpha)(Y)]-(P^{*})^{c}(\alpha)(\overline{\nabla}_{X}Y)+(P^{*})^{c}(\overline{\nabla}^{B}_{X}\alpha)(Y)$
$\displaystyle=X[(P^{*})^{c}(\alpha)(Y)]-(P^{*})^{c}(\alpha)(\overline{\nabla}_{X}Y)+(P^{*})^{c}((\overline{\nabla}_{X}\alpha)^{\top})(Y)$
$\displaystyle=X[(P^{*})^{c}(\alpha)(Y)]-(P^{*})^{c}(\alpha)(\nabla_{X}^{B}Y)+(P^{*})^{c}((\overline{\nabla}_{X}\alpha)^{\top})(Y)$

where in the last step we used the fact that

$(P^{*})^{c}(\alpha)(\nabla_{X}^{B}Y)=-(P^{*})^{c}(\alpha)(\overline{\nabla}_{X}Y),$

which follows from the definition of the codifferential operator and the fact
that $(P^{*})^{c}=-(P^{*})^{c}$.

Now we need to show that the three terms on the right-hand side of this
expression lie in $\Gamma(\mathcal{D}^{\bot})^{0}$. We will do this term by
term. First, note that
$(P^{*})^{c}(\alpha)(Y)\in\Gamma(\mathcal{D}^{\bot})^{0}$ since
$(P^{*})^{c}(\alpha)$ maps $\Gamma(\mathcal{D}^{\bot})$ to itself and
$Y\in\Gamma(\mathcal{D}^{\bot})^{0}$.

Next, we need to show that
$(P^{*})^{c}(\alpha)(\nabla_{X}^{B}Y)\in\Gamma(\mathcal{D}^{\bot})^{0}$. Note
that

$(P^{*})^{c}(\alpha)(\nabla_{X}^{B}Y)=-(P^{*})^{c}(\alpha)(\overline{\nabla}_{X}Y),$

so it suffices to show that
$(P^{*})^{c}(\alpha)(\overline{\nabla}_{X}Y)\in\Gamma(\mathcal{D}^{\bot})^{0}$.
To see this, note that
$\overline{\nabla}_{X}Y\in\Gamma(\mathcal{D}^{\bot})^{0}$ since $X$ and $Y$
are both sections of $\mathcal{D}^{\bot}$, and that $(P^{*})^{c}(\alpha)$ maps
$\Gamma(\mathcal{D}^{\bot})^{0}$ to itself.

Finally, we need to show that
$(P^{*})^{c}((\overline{\nabla}_{X}\alpha)^{\top})(Y)\in\Gamma(\mathcal{D}^{\bot})^{0}$.
To see this, note that $(\overline{\nabla}_{X}\alpha)^{\top}$ is a tensor of
type $(1,1)$ that maps vectors tangent to $M$ to vectors tangent to $M$, so
$(P^{*})^{c}((\overline{\nabla}_{X}\alpha)^{\top})(Y)$ is a section of
$\mathcal{D}^{\bot}$. Moreover,
$(P^{*})^{c}((\overline{\nabla}_{X}\alpha)^{\top})$ maps
$\Gamma(\mathcal{D}^{\bot})^{0}$ to itself since
$(\overline{\nabla}_{X}\alpha)^{\top}$ maps $\Gamma(TM)$ to itself and
$(P^{*})^{c}$ maps $\Gamma(\mathcal{D}^{\bot})$ to itself.

Therefore, we have shown that
$\overline{\nabla}_{X}\alpha\in\Gamma(\mathcal{D}^{\bot})^{0}$, which implies
that $\alpha$ is a harmonic one-form with respect to the induced metric on
$\partial M$.

To summarize, we showed that if $\alpha$ is a closed one-form on $M$ such that
$\alpha|_{\partial M}=0$, then $\alpha$ is a harmonic one-form with respect to
the induced metric on $\partial M$.

∎

In the following, we shall present the nonholonomic sub-Finslerian structure
results. To begin, we define coordinate independent conditions for the motion
of a free mechanical system subjected to linear nonholonomic constraints to be
normal extremal with respect to the connected sub-Finslerian manifold, and
vice versa. Then, we address the problem of characterizing the normal and
abnormal extremals that validate both nonholonomic and Vakonomic equations for
a free particle subjected to certain kinematic constraints.

Let $(M,\mathcal{D},F)$ be a nonholonomic sub-Finslerian structure and
$\sigma:[0,1]\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$
be a piecewise smooth horizontal curve tangent to $\mathcal{D}$, then $\sigma$
is said to be a normal extremal if there exists $E$-admissible curve $\alpha$
with base curve $\sigma$ that is auto-parallel with respect to a normal
$\mathcal{L}$-connection (Definition 3). While the curve $\sigma$ is said to
be an abnormal extremal if there exists $\gamma\in\Gamma(\mathcal{D}^{0})$
along $\sigma$ such that $\nabla_{\alpha}\gamma(t)=0$ for all $t\in[0,1]$,
with $\alpha$ a $E$-admissible curve with base curve $\sigma$.

######  Remark 4.

Cortés et al. [15], made a comparison between the solutions of the
nonholonomic mechanical problem and the solutions of the Vakonomic dynamical
problem for the general Lagrangian system. The Vakonomic dynamical problem,
associated with a free particle with linear nonholonomic constraints, consists
of finding normal extremals with respect to the sub-Finsleriann structure
$(M,\mathcal{D},F)$. It is an interesting comparison because the equations of
motion for the mechanical problem are derived by means of d’Alembert’s
principle, while the normal extremals are derived from a variation principle.
Our next results are an alternate approach to the Cortés results, that is a
coordinate-free approach, for the free particle case in the sub-Finslerian
settings.

######  Definition 5.

Let $(M,\mathcal{D},F)$ be a nonholonomic sub-Finslerian structure, one can
establish new tensorial operators according to the following:

$\displaystyle T^{B}$
$\displaystyle:\Gamma(\mathcal{D})\otimes\Gamma(\mathcal{D}^{*})\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\Gamma(\mathcal{D}^{0}),\quad(X,\alpha)\mapsto
P^{*}(\overline{\nabla}_{X}^{B}\alpha);$ $\displaystyle T$
$\displaystyle:\Gamma(\mathcal{D})\otimes\Gamma(\mathcal{D}^{0})\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\Gamma(\mathcal{D}^{\bot})^{0},\quad(X,\gamma)\mapsto(P^{*})^{c}(\delta_{X}\gamma);$

such that

$\delta:\Gamma(\mathcal{D})\times\Gamma(\mathcal{D}^{0})\>\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>\mathfrak{X}^{*}(M),\quad(X,\gamma)\mapsto\delta_{X}\gamma=i_{X}d\gamma.$

In addition, these tensorial operators have the following properties:

  * (I)

$T^{B}$ and $T$ are $\mathcal{F}(M)$-bilinear in their independent variables;

  * (II)

The behavior of $T^{B}$ and $T$ can be identified pointwise;

  * (III)

$T_{x}^{B}(X,\alpha)$ and $T_{x}(X,\gamma)$ have a clear and unequivocal
meaning for all $X\in\mathcal{D},\alpha\in\mathcal{D}^{*}$ and
$\gamma\in\mathcal{D}^{0}$.

In the following, w show the relation between the operator $T^{B}$ and the
curvature of the distribution $\mathcal{D}$ using the following condition:

Suppose $X\in\mathcal{D},\alpha\in\mathcal{D}^{*}$, then one has

$\langle
T(X,\gamma),\alpha\rangle=\langle\delta_{X}\gamma,\alpha\rangle=-\langle\gamma,[X,\alpha]\rangle,$

for any $\gamma\in\Gamma(\mathcal{D}^{0}).$ Therefore, $T$ is trivial if and
only if $\mathcal{D}$ is involutive.

######  Definition 6.

Let $\nabla^{T}$ denote the non-linear connection over
$i:\mathcal{D}\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>TM$
on $\mathcal{D}^{0}$ by the following formula

$\nabla^{T}_{X}\gamma=P^{*}(\delta_{X}\gamma),$

such that $X\in\Gamma(\mathcal{D})$ and $\gamma\in\Gamma(\mathcal{D}^{0}).$

######  Proposition 3.

Let $(M,\mathcal{D},F)$ be a nonholonomic sub-Finslerian structure, assume
that $\sigma:[0,1]\>\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
4.49588pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{\hbox{\kern 4.49588pt\raise
0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 14.99176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\>M$
is a horizontal curve on $\mathcal{D}$ and let $\nabla$ be a
$\mathcal{D}$-adapted $\mathcal{L}$-connection. Then, the following properties
are satisfied:

  * (I)

If $p_{0}\in\mathcal{D}^{*}_{\sigma(0)}$ is a given initial point, then
$p(t)=\tilde{p}(t)$ for each $t\in[0,1]$ if and only if
$T^{B}(\dot{\sigma}(t),\tilde{p}(t))=0$, such that $p(t)$ and $\tilde{p}(t)$
are parallel transported curves along $\sigma$ w.r.t. $\overline{\nabla}^{B}$
and $\nabla^{H}$, respectively.

  * (II)

If $\gamma_{0}\in\mathcal{D}^{0}_{\sigma(0)}$ is a given initial point, then
$\gamma(t)=\tilde{\gamma}(t)$ for each $t\in[0,1]$ if and only if
$T(\dot{\sigma}(t),\tilde{\gamma}(t))=0$, such that $\gamma(t)$ and
$\tilde{\gamma}(t)$ are parallel transported curves along $\sigma$ w.r.t.
$\nabla$ and $\nabla^{T}$, respectively.

###### Proof.

It is sufficient to prove that the first case and the second one follow
similar arguments.

As a consequence of the definition of the tensorial operator $T^{B}$, for any
section $S(t)$ of $\mathcal{D}^{*}$ along $\sigma$, the next expression is
true

$\nabla^{H}_{{\dot{\sigma}(t)}}S(t)=\overline{\nabla}^{B}_{{\dot{\sigma}(t)}}S(t)-T^{B}({\dot{\sigma}(t)},S(t)).$

Now, suppose that $S(t)=\tilde{p}(t)=p(t)$, then we get,

$T^{B}({\dot{\sigma}(t)},S(t))=0.$

Conversely, it is well known that, regarding any connection, the parallel
transported curves are uniquely determined by their initial conditions. ∎

It is clear that the second property of the above Proposition yields necessary
and sufficient conditions for the existence of the curves that have abnormal
extremals. In other words, $\sigma$ is an abnormal extremal if and only if
there exists a parallel transported section $\tilde{\gamma}$ of
$\mathcal{D}^{0}$ along $\sigma$ with respect to $\nabla^{T}$ such that
$T(\dot{\sigma}(t),\tilde{\gamma}(t))=0$. Now, by the next Proposition, one
can derive the necessary and sufficient condition for normal extremals to be a
motion of a free nonholonomic mechanical system and vice versa.

######  Lemma 1.

Let $(M,\mathcal{D},F)$ be nonholonomic sub-Finslerian structures, and
$\nabla$ be a normal non-linear $\mathcal{L}$-connection. Then for any
$\alpha\in\mathfrak{X}^{*}(M)$ we have that $\nabla_{\alpha}\alpha=0$ if and
only if

$\displaystyle\nabla^{H}_{E(\alpha)}\alpha(P)=$
$\displaystyle-T(E(\alpha),(P^{*})^{c}(\alpha));$
$\displaystyle\nabla^{T}_{E(\alpha)}P^{*}(\alpha)=$
$\displaystyle-T^{B}(E(\alpha),\alpha(P)).$

###### Proof.

We proved in [3], that $\nabla_{\alpha}\alpha=0$ if and only if
$\nabla_{\alpha}\alpha=\overline{\nabla}^{B}_{E(\alpha)}(P^{*})^{c}(\alpha)+\delta_{E(\alpha)}P^{*}(\alpha)=0.$
Moreover, $P^{*}(\alpha)=\alpha(P)$ and the Barthel non-linear connection
preserves the metric, i.e.
$\nabla^{B}\circ\mathcal{L}_{\eta}=\mathcal{L}_{\eta}\circ\overline{\nabla}^{B}$,
therefore

$\displaystyle\overline{\nabla}^{B}_{E(\alpha)}P^{*}(\alpha)=$
$\displaystyle\nabla^{H}_{E(\alpha)}P^{*}(\alpha)+T^{B}(E(\alpha),P^{*}(\alpha)),$
$\displaystyle\delta_{E(\alpha)}P^{*}(\alpha)=$
$\displaystyle\nabla^{T}_{E(\alpha)}P^{*}(\alpha)+T(E(\alpha),(P^{*})^{c}(\alpha)).$

According to the fact that $T^{*}M$ can be written as the direct sum of
$(\mathcal{D}^{\bot})^{0}$ and $\mathcal{D}^{0}$, so the equivalence is pretty
clear. ∎

######  Theorem 1.

If $\sigma:[0,1]\rightarrow M$ is a solution of a free nonholonomic system
given by nonholonomic sub-Finslerian structures, then it is also a solution of
the corresponding Vakonomic problem, and vice versa, if and only if there
exists $\gamma\in\Gamma(\mathcal{D}^{0})$ along $\sigma$ such that

$\nabla^{T}_{\dot{\sigma}}\gamma(t)=-T^{B}(\dot{\sigma}(t),\mathcal{L}_{L}(\dot{\sigma}(t))),$
(13)

further, for all $t$,
$\gamma(t)\in{(\mathcal{D}_{\sigma(t)}+[\dot{\sigma}(t),\mathcal{D}_{\sigma(t)}])}^{0}$.

###### Proof.

$\nabla_{\alpha}\alpha(t)=0$ is the condition for any $E$-admissible curve
$\alpha(t)=\mathcal{L}_{L}(\dot{\sigma}(t))+\gamma(t)$ to be parallel
transported with respect to a normal $\mathcal{L}$-connection. In other words,

$\nabla^{H}_{\dot{\sigma}}\mathcal{L}_{L}(\dot{\sigma}(t))=-T(\dot{\sigma}(t),\gamma(t))$

and

$\nabla^{T}_{\dot{\sigma}}\gamma(t)=-T^{B}(\dot{\sigma}(t),\mathcal{L}_{L}(\dot{\sigma}(t))).$

Therefore, $\nabla^{H}_{\dot{\sigma}}\mathcal{L}_{L}(\dot{\sigma}(t))=0$ if
and only if $T(\dot{\sigma}(t),\gamma(t))=0$, such that $\gamma(t)$ is a
solution of (13). Since Remark 3 and Proposition 2 guaranteed that
$\mathcal{D}$ is geodesically invariant, therefore, given any $\gamma(t)$ in
${(\mathcal{D}_{\sigma(t)}+[\dot{\sigma}(t),\mathcal{D}_{\sigma(t)}])}^{0}$,
then (13) ensure that there is always a solution for all $t\in[0,1]$ not only
for $\gamma(0)$ in
${(\mathcal{D}_{\sigma(0)}+[\dot{\sigma}(0),\mathcal{D}_{\sigma(0)}])}^{0}$. ∎

##  5\. Examples from Robotics

Typically, nonholonomic systems occur when velocity restrictions are applied,
such as the constraint that bodies move on a surface without slipping.
Bicycles, cars, unicycles, and anything with rolling wheels are all examples
of nonholonomic sub-Finslerian structures.

We will discuss the simplest wheeled mobile robot, which is a single upright
rolling wheel, or unicycle, which is known as a kinematic penny rolling on a
plane. Assume this wheel is of radius 1 and does not allow sideways sliding.
Its configuration $M$ consists of the heading angle $\phi$, the wheel’s point
or the contact position $(x_{1},x_{2})$, and the rolling angle $\psi$ (see
Figure 1). Consequently, the space concerned has dimensions four, i.e.,
$M=\mathbb{R}^{2}\times S^{1}\times S^{1}$. There are two control functions
deriving the wheel [14, 21]:

Figure 1. A kinematic penny rolling on a plane

  * (I)

$u_{1}$ [rolling speed], the forward-backward rolling angular,

  * (II)

$u_{2}$ [turning speed], the speed of turning the heading direction $\phi$.

With these controls, the rate of change of the coordinates can be expressed as
follows:

$\displaystyle\dot{M}$ $\displaystyle=\begin{bmatrix}\dot{\phi}\\\
\dot{x}_{1}\\\ \dot{x}_{2}\\\ \dot{\psi}\end{bmatrix}=\begin{bmatrix}0&1\\\
\cos{\phi}&0\\\ \sin{\phi}&0\\\ 1&0\\\ \end{bmatrix}\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}=X(M)u.$ (14)

As we generally do not worry about the wheel’s rolling angle, we could drop
the fourth row from the above equation to get a simpler control system

$\displaystyle\dot{M}$ $\displaystyle=\begin{bmatrix}\dot{\phi}\\\
\dot{x}_{1}\\\ \dot{x}_{2}\end{bmatrix}=\begin{bmatrix}0&1\\\ \cos{\phi}&0\\\
\sin{\phi}&0\end{bmatrix}\begin{bmatrix}u_{1}\\\ u_{2}\end{bmatrix}=X(M)u,$
(15)

which can be written as the following equation:

$X(M)u=X_{1}(M)u_{1}+X_{2}(M)u_{2},$

such that $u_{1},u_{2}$ are called the controls and $X_{1}(w),X_{2}(w)$ are
called vector fields. Moreover, each vector field assigns a velocity to every
point $w$ in the configuration space, so these vector fields are sometimes
called velocity vector fields. Hence the velocity vector fields of any
solution curve should lie in $\mathcal{D}$ spanned by the following vector
fields:

$\displaystyle X_{1}(M)$ $\displaystyle=\cos{\phi}\frac{\partial}{\partial
x_{1}}+\sin{\phi}\frac{\partial}{\partial
x_{2}}+\frac{\partial}{\partial\psi}$ $\displaystyle X_{2}(M)$
$\displaystyle=\frac{\partial}{\partial\phi}.$

In a natural way, a sub-Riemannian metric on $\mathcal{D}$ is gained by
asserting the vector fields $X_{1}(M),X_{2}(M)$ to be orthonormal vectors,

$\langle
u_{1}X_{1}(M)+u_{2}X_{2}(M),u_{1}X_{1}(M)+u_{2}X_{2}(M)\rangle=u_{1}^{2}+u_{2}^{2}.$

The integral of this quadratic form measures the work completed in rolling the
heading angle $\phi$ at the rate $\dot{\phi}$ and propelling the wheel ahead
at the rate of $\dot{\psi}$. The sub-Riemannian structure will be adjusted as
specified by the notion that curvature is costly: namely, it takes more
attempts to steer the wheel in a tight circle with little forward or backward
movement than to steer it in a wide arc. Therefore, the curvature of the
projection $\sigma$ given by $\kappa=\frac{\dot{\phi}}{\dot{\psi}}$ this
brings us to assume sub-Finsler metrics of the body

$F=f(\kappa)\sqrt{d{\psi}^{2}+d{\phi}^{2}},$

such that $f$ grows larger but remains constrained as $\mid\kappa\mid$
increases. After we check the sub-Finslerian property, one finds the
nonholonomic of the rolling wheel, often known as a unicycle, by the equation
$\dot{M}=X(M)u$ which is the kinematic model of the unicycle.

##  6\. The sub-Laplacian associated with nonholonomic sub-Finslerian
structures

The sub-Laplacian is a differential operator that arises naturally in the
study of nonholonomic sub-Finslerian structures. These are geometric
structures that generalize Riemannian manifolds, allowing for non-integrable
distributions of tangent spaces.

On a sub-Finslerian manifold $M$, there is a distinguished distribution of
tangent spaces $\mathcal{D}$, which corresponds to the directions that are
accessible by moving along curves with bounded sub-Finsler length. The sub-
Finsler metric $F$ on $M$ measures the sub-Finsler length of curves with
respect to this distribution.

The sub-Laplacian is defined as a second-order differential operator that acts
on functions on $M$ and is defined in terms of the metric $F$ and the
distribution $\mathcal{D}$. It is given by

$\Delta_{F}=\mathrm{div}_{\mathcal{D}}(\mathrm{grad}_{F}),$

where $\mathrm{grad}_{F}$ is the gradient vector field associated with $F$
which is the unique vector field satisfying
$\mathrm{d}F(\mathrm{grad}_{F},X)=X(F)$ for all vector fields $X$ on $M$, and
$\mathrm{div}_{\mathcal{D}}$ is the divergence operator with respect to the
distribution $\mathcal{D}$, which is defined as the trace of the tangential
part of the connection on $\mathcal{D}$.

Our goal in this section is to show that the sub-Laplacian measures the
curvature of the sub-Finslerian structure. It captures the interplay between
the sub-Finsler metric $F$ and the distribution $\mathcal{D}$, and plays a
crucial role in many geometric and analytic problems on nonholonomic sub-
Finslerian manifolds.

For example, the heat kernel associated with the sub-Laplacian provides a way
to study the long-term behavior of solutions to the heat equation on sub-
Finslerian manifolds. The Hodge theory on sub-Finslerian manifolds is also
intimately related to the sub-Laplacian, and involves the study of
differential forms that are harmonic with respect to the sub-Laplacian.

######  Remark 5.

To see that the sub-Laplacian measures the curvature of the sub-Finslerian
structure, let us first recall some basic facts about Riemannian manifolds,
see [16]. On a Riemannian manifold $(M,g)$, the Laplace-Beltrami operator is
defined as

$\Delta_{g}=\mathrm{div}(\mathrm{grad}_{g}),$

where $\mathrm{grad}_{g}$ is the gradient vector field associated with the
Riemannian metric $g$, and $\mathrm{div}$ is the divergence operator. It is a
well-known fact that the Laplace-Beltrami operator measures the curvature of
the Riemannian structure in the sense that it is zero if and only if the
Riemannian manifold is flat.

The sub-Finslerian case is more complicated due to the presence of the
distribution $\mathcal{D}$ that is not integrable in general. However, the
sub-Laplacian $\Delta_{F}$ can still be understood as a curvature operator. To
see this, we need to introduce the notion of a horizontal vector field.

A vector field $X$ on $M$ is called horizontal if it is tangent to the
distribution $\mathcal{D}$. Equivalently, $X$ is horizontal if it is locally
of the form $X=\sum_{i=1}^{k}h_{i}X_{i}$, where $h_{i}$ are smooth functions
and $X_{1},\ldots,X_{k}$ are smooth vector fields that form a basis for
$\mathcal{D}$.

Given a horizontal vector field $X$, we can define its sub-Finsler length
$|X|_{F}$ as the infimum of the lengths of horizontal curves that are tangent
to $X$ at each point. Equivalently, $|X|_{F}$ is the supremum of the scalar
products $g(X,Y)$ over all horizontal vector fields $Y$ with $|Y|_{F}\leq 1$.

With these definitions in place, we can now show that the sub-Laplacian
measures the curvature of the sub-Finslerian structure. More precisely, we
have the following result:

######  Theorem 2.

The sub-Laplacian $\Delta_{F}$ is zero if and only if the sub-Finslerian
manifold $(M,F,\mathcal{D})$ is locally isometric to a Riemannian manifold.

###### Proof.

First, suppose that $(M,F,\mathcal{D})$ is locally isometric to a Riemannian
manifold $(M,g)$. Then we can choose a local frame of orthonormal horizontal
vector fields $X_{1},\ldots,X_{k}$ with respect to the Riemannian metric $g$.
In this frame, we have

$\mathrm{grad}_{F}h=\sum_{i=1}^{k}g(\mathrm{grad}_{h},X_{i})X_{i}$

for any function $h$ on $M$, and hence

$\Delta_{F}h=\sum_{i=1}^{k}\mathrm{div}_{\mathcal{D}}(g(\mathrm{grad}_{h},X_{i})X_{i}).$

Using the fact that the $X_{i}$ form a basis for $\mathcal{D}$, we can rewrite
this as

$\Delta_{F}h=\mathrm{div}(\mathrm{grad}_{h})=\Delta_{g}h,$

where $\Delta_{g}$ is the Laplace-Beltrami operator associated with the
Riemannian metric $g$. Since $\Delta_{g}$ is zero if and only if $(M,g)$ is
flat, it follows that $\Delta_{F}$ is zero if and only if $(M,F,\mathcal{D})$
is locally isometric to a Riemannian manifold, which implies that the sub-
Finslerian structure is also flat.

Conversely, suppose that $\Delta_{F}$ is zero. Let $X_{1},\ldots,X_{k}$ be a
local frame of horizontal vector fields such that $F(X_{i})=1$ for all $i$,
and let $\omega_{ij}=g(X_{i},X_{j})$ be the Riemannian metric induced by $F$
on $\mathcal{D}$. Using the definition of the sub-Laplacian and the fact that
$\Delta_{F}$ is zero, we have

$0=\Delta_{F}F=\mathrm{div}_{\mathcal{D}}(\mathrm{grad}_{F}F)=\sum_{i=1}^{k}\sum_{i=1}^{k}\frac{\partial^{2}F}{\partial
x_{i}\partial x_{j}}\omega_{ij},$

where $x_{1},\ldots,x_{k}$ are local coordinates on $M$ that are adapted to
$\mathcal{D}$ (i.e., $X_{1},\ldots,X_{k}$ form a basis for the tangent space
at each point). This implies that the Hessian of $F$ with respect to the
Riemannian metric $\omega$ is zero, so $F$ is locally affine with respect to
$\omega$. In other words, $(M,F,\mathcal{D})$ is locally isometric to a
Riemannian manifold. ∎

######  Remark 6.

In the above Theorem 2, we have shown that the sub-Laplacian $\Delta_{F}$
measures the curvature of the sub-Finslerian structure. If $\Delta_{F}$ is
zero, then the sub-Finslerian manifold is locally isometric to a Riemannian
manifold, and hence the sub-Finslerian structure is flat. If $\Delta_{F}$ is
nonzero, then the sub-Finslerian manifold is not locally isometric to a
Riemannian manifold, and the sub-Finslerian structure is curved. This means
that the shortest paths between two points on the manifold are not necessarily
straight lines, and the geometry of the manifold is more complex than that of
a Riemannian manifold.

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