# Energy balance for nonholonomic nonlinear systems

F. Talamucci  DIMAI, Dipartimento di Matematica e Informatica “Ulisse Dini”,
Università degli Studi di Firenze, Italy
e-mail<EMAIL_ADDRESS>
2010 Mathematics Subject Classification: 37J60, 70F25, 70H03.

Keywords: Lagrange’s equations of motion - Nonholonomic systems - Nonlinear
kinematic constraints - Equations of Voronec type - Generalized Theorem of
energy.

Abstract. We consider nonholonomic systems with nonlinear restrictions with
respect to the velocities. The mathematical problem is formulated by means of
the Voronec equations extended to the nonlinear case. The main point of the
paper is the balance of the mechanical energy induced by the equations of
motion; the conservation of the energy on the basis of the tipology of the
constraint equations is discussed. Several examples are performed.

##  1 Introduction

Nonholonomic systems are frequently encountered in mechanical and engineering
problems and attention to mathematical models concerning kinematic constraints
is increasingly paid.

The trait “nonholonomic” refers, in a general way, to restrictions imposed on
a system which are not expressible solely in terms of one or more equations
involving only the spatial coordinates which delineate the position of the
system.

In the wide range of restrictions with nonholonomic features, we focus the
interest on the kinematic constraints, that is restrictions which can be
formulated by equations containing the coordinates and components of the
velocities. As a matter of fact, this kind of constraints show a more suitable
relevance and evidence concerning applications and feasibility.

A remarkable difference exists between the case of linear kinematic
constraints and the case of nonlinear ones: even in literature, the starting
point of the theory of linear constraints is distant in time ([4], [7]) and in
parallel to the development of the lagrangian formalism, whereas the
advancements in nonlinear constrained systems are somewhat recent. The not
uniform progress is motivated both by modellistic reasons (actually, the
linear case can be convincingly considered as a natural extent of the
holonomic case, as we will see just below) and by practical reasons of
realization of physical models exhibiting nonlinear constraints (such a
question, arised in [1] and studied in [14], is still debated, see [2]).

In this paper we deal with systems submitted to nonlinear nonholonomic
constraints. The first part (Section 2) consists in recalling the equations of
motions proposed in a previous work ([11]) and based on an elementary
principle, formulated along the lines of the linear nonholonomic case. The
main requirements of this formulations are the explicit form of the constraint
equations and the use of some of the generalized velocities as independent
parameters. In this way, the formal path is simple and proceeds with extending
the classical theory of mechanics. We will also point out the correspondence
of the obtained equations of motion with models on the same topics present in
literature, where the equations derive from more refined and formally complex
methods. In literature, nonholonomic mechanics appear to be studied mainly via
geometric methods, as Lagrangian systems on fibered manifolds (see, among
others, [8]): the method proposed in the present paper is a handly approach
developing in analogy to the basic concepts of a holonomic system.

The second part (Section 3) is devoted to the main result, regarding the
expression for the rate of change of the mechanical energy of the system. The
selected typology of the equations of motion turns ut to be appropriate both
from the mathematical point of view and for the physical intepretation, since
the variables involved are the real velocities.

The energy balance is achieved via a somehow usual handling of the equations
of motion: in spite of the spontaneity of the method, the resulting formula
shows distintly each contribute in terms of energies and it is suitable in
order to investigate which categories of constraints entail the conservation
of generalized energy (Jacobi’s first integral). In Section 4 we identify some
cases of nonholonomic systems where the energy balance equation actually
infers the conservation of the mechanical energy.

Finally, various examples aimed at the implementation of the results on both
linear and nonlinear nonholonomic constrained systems have been added, by
drawing basically the most recurrent models in literature.

##  2 Mathematical model and notations

The starting point is a holonomic system ${\bf X}(q_{1},\dots,q_{n},t)$, where
${\bf X}$ locates the position of $N$ material points $(P_{i},M_{i})$,
$i=1,\dots,N$. In addition to the $3N-n$ holonomic conditions, the system
undergoes $k<n$ kinematic constraints

$\Phi_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t)=0,\quad\nu=1,\dots,k$
(1)

which can be linear or nonlinear and are assumed to be independent, in the
sense that the rank of the Jacobian matrix
$J_{({\dot{q}}_{1},\dots,{\dot{q}}_{n})}(\Phi_{1},\dots,\Phi_{k})$ is the
greatest value $k$.

Under that assumption, if for istance
$det\,J_{({\dot{q}}_{m+1},\dots,{\dot{q}}_{n})}(\Phi_{1},\dots,\Phi_{k})\not=0$,
$m=n-k$, conditions (1) can be made explicit:

${\dot{q}}_{m+\nu}=\alpha_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t),\quad\nu=1,\dots,k.$
(2)

with $m=n-k$. The case when all of (1) are linear is denoted by the functions
$\alpha_{\nu,j}$ such that

$\alpha_{\nu}=\sum\limits_{j=1}^{m}a_{\nu,j}(q_{1},\dots,q_{n},t){\dot{q}}_{j},\quad\nu=1,\dots,k.$
(3)

###  2.1 Equations of motion

Let ${\bf X}^{(\mathsf{M})}\in{\mathbb{R}}^{3N}$ list the $n$ triplets
$M_{i}P_{i}$, $i=1,\dots,N$, so that ${\cal Q}={\dot{\bf X}}^{(\mathsf{M})}$
is the $3N$–vector of linear momenta; furthermore, let us denote by
${\mathcal{F}}=({\mathcal{F}}_{1},\dots,{\mathcal{F}}_{N})$,
${\mathcal{R}}=({\mathcal{R}}_{1},\dots,{\mathcal{R}}_{N})$ the list in
${\mathbb{R}}^{3N}$ of the active forces ${\mathcal{F}}_{i}$ and of the
constraint forces ${\mathcal{R}}_{i}$ concerning each $P_{i}$, $i=1,\dots,N$.

We present initially the equations of motion in the newtonian form. The key
statement is

$\left({\dot{\mathcal{Q}}}-{\mathcal{F}}-{\mathcal{R}}\right)\cdot{\widehat{\dot{\bf
X}}}=0$ (4)

where

${\widehat{\dot{\bf
X}}}=\sum\limits_{r=1}^{m}{\dot{q}}_{r}\left(\dfrac{\partial{\bf X}}{\partial
q_{r}}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\dfrac{\partial{\bf
X}}{\partial q_{m+\nu}}\right)$ (5)

are all the possible displacements (see [2], [11]) consistent with the
kinematic restrictions (2), for arbitrary
$({\dot{q}}_{1},\dots,{\dot{q}}_{m})\in{\mathbb{R}}^{m}$.

At this point, the assumptions of ideal constraints demands ${\cal
R}\cdot{\hat{\dot{\bf X}}}=0$ for all possible (5), so that (4) is equivalent
to the equations

$\left({\dot{\mathcal{Q}}}-{\mathcal{F}}\right)\cdot{\bf X}_{r}=0,\quad
r=1,\dots,m,\quad{\bf X}_{r}=\dfrac{\partial{\bf X}}{\partial
q_{r}}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\dfrac{\partial{\bf
X}}{\partial q_{m+\nu}}$ (6)

which correspond explicitly to the $m$ differential equations (which have to
be joined with (2))

$\sum\limits_{r=1}^{m}\left(C_{i}^{r}{\mathop{q}\limits^{..}}_{r}+\sum\limits_{s=1}^{m}D_{i}^{r,s}{\dot{q}}_{r}{\dot{q}}_{s}+E_{i}^{r}{\dot{q}}_{r}\right)+G_{i}={\cal
F}^{(q_{i})}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}{\cal
F}^{(q_{m+\nu})},\qquad i=1,\dots,m$ (7)

where $\mathcal{F}^{(q_{j})}=\mathcal{F}\cdot\dfrac{\partial{\bf X}}{\partial
q_{j}}$ for each $j=1,\dots,n$ is the generalized force, $C_{i}^{r}$,
$D_{i}^{r,s}$, $E_{i}^{r}$ and $G_{i}$, $i,r,s=1,\dots,m$, depend on
$q_{1},\dots,q_{n}$, ${\dot{q}}_{1}$, $\dots$, ${\dot{q}}_{m}$, $t$ and are
defined by

$\displaystyle C_{i}^{r}$ $\displaystyle=$ $\displaystyle
g_{i,r}+\sum\limits_{\nu,\mu=1}^{k}\left(g_{i,m+\nu}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}+g_{m+\nu,r}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}+g_{m+\nu,m+\mu}\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{i}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\right),\quad
D_{i}^{r,s}=\xi_{r,s,i}+\sum\limits_{\nu=1}^{k}\xi_{r,s,m+\nu}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}},$
$\displaystyle E_{i}^{r}$ $\displaystyle=$
$\displaystyle\sum\limits_{\nu,\mu=1}^{k}\left(2\xi_{r,m+\nu,m+\mu}\alpha_{\nu}+g_{m+\nu,m+\mu}\dfrac{\partial\alpha_{\nu}}{\partial
q_{r}}+2\eta_{r,m+\mu}\right)\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{i}}+\sum\limits_{\nu=1}^{k}\left(2\xi_{r,m+\nu,i}\alpha_{\nu}+g_{m+\nu,i}\dfrac{\partial\alpha_{\nu}}{\partial
q_{r}}\right)+2\eta_{r,i},$ $\displaystyle G_{i}$ $\displaystyle=$
$\displaystyle\sum\limits_{\nu,\mu,p=1}^{k}\left(\xi_{m+\nu,m+\mu,m+p}\alpha_{\nu}\alpha_{\mu}+g_{m+\nu,m+p}\alpha_{\mu}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}+2\eta_{m+\nu,m+p}\alpha_{\nu}+g_{m+\nu,m+p}\dfrac{\partial\alpha_{\nu}}{\partial
t}+\zeta_{m+p}\right)\dfrac{\partial\alpha_{p}}{\partial{\dot{q}}_{i}}$
$\displaystyle+$
$\displaystyle\sum\limits_{\nu,\mu=1}^{k}\left(\xi_{m+\nu,m+\mu,i}\alpha_{\nu}\alpha_{\mu}+g_{m+\nu,i}\alpha_{\mu}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}\right)+\sum\limits_{\nu=1}^{k}\left(2\eta_{m+\nu,i}\alpha_{\nu}+g_{m+\nu,i}\dfrac{\partial\alpha_{\nu}}{\partial
t}\right)+\zeta_{i}.$

by setting, for any $i,j,k=1,\dots,n$:

$\begin{array}[]{ll}g_{i,j}(q_{1},\dots,q_{n},t)=\dfrac{\partial{\bf
X}^{(\mathsf{M})}}{\partial q_{i}}\cdot\dfrac{\partial{\bf X}}{\partial
q_{j}},&\xi_{i,j,k}(q_{1},\dots,q_{n},t)=\dfrac{\partial^{2}{\bf
X}^{(\mathsf{M})}}{\partial q_{i}\partial q_{j}}\cdot\dfrac{\partial{\bf
X}}{\partial q_{k}},\\\ \\\
\eta_{i,j}(q_{1},\dots,q_{n},t)=\dfrac{\partial^{2}{\bf
X}^{(\mathsf{M})}}{\partial q_{i}\partial t}\cdot\dfrac{\partial{\bf
X}}{\partial q_{j}},&\zeta_{i}(q_{1},\dots,q_{n},t)=\dfrac{\partial^{2}{\bf
X}^{(\mathsf{M})}}{\partial t^{2}}\cdot\dfrac{\partial{\bf X}}{\partial
q_{j}}.\end{array}$ (8)

The explicit form (6) is convenient in order to easily disclose the terms with
the second derivatives ${\mathop{q}\limits^{..}}_{r}$, $r=1,\dots,m$, which
appear only in the linear terms with coefficients $C_{i}^{r}$. In case of
scleronomic holonomous constraints ${\bf X}({\bf q})$, in (8) it is
$\eta_{i,j}=0$, $\zeta_{i}=0$ for any $i,j=1,\dots,n$.

###### Proposition 1

The $m\times m$ matrix $C$ with entries $C_{i}^{r}$, $r=1,\dots,m$, is
positive definite.

Proof: Indeed the terms originate from $C_{i}^{r}=\left(\dfrac{\partial{\bf
X}^{(M)}}{\partial
q_{r}}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\dfrac{\partial{\bf
X}^{(M)}}{\partial q_{m+\nu}}\right)\cdot\left(\dfrac{\partial{\bf
X}}{\partial
q_{i}}+\sum\limits_{j=1}^{k}\dfrac{\partial\alpha_{j}}{\partial{\dot{q}}_{i}}\dfrac{\partial{\bf
X}}{\partial q_{m+j}}\right)$. Setting ${\bf
X}_{r}^{(\sqrt{M})}=\dfrac{\partial{\bf X}^{(\sqrt{M})}}{\partial
q_{r}}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\dfrac{\partial{\bf
X}^{(\sqrt{M})}}{\partial q_{m+\nu}}$, where ${\bf X}^{(\sqrt{M})}$ stands for
$(\sqrt{m_{1}}P_{1},\dots,\sqrt{m_{n}}P_{n})$, one has $C_{\nu}^{i}={\bf
X}_{\nu}^{(\sqrt{M})}\cdot{\bf X}_{i}^{(\sqrt{M})}$, so that the matrix $C$ is
positive definite, since the vectors ${\bf X}_{j}^{(\sqrt{M})}$, $j=1,\dots,m$
are linearly independent, as it can be easily verified. $\quad\square$

Hence, the system of equations (7) is well posed, whatever the functions
$\alpha_{1}$, $\dots$, $\alpha_{\nu}$ are; the already explicit version (2)
does not require any further condition.

###### Remark 1

In support of the form (6) of the equations of motion (in comparison with the
more common equations in Lagrangian form, presented later on) is that they are
promptly attainable, whenever the coefficients (8) are simple: this is the
case, for istance, of cartesian coordinates: if no holonomic constraint is
present and $q_{1},\dots,q_{n}$ are the cartesian coordinates ${\bf X}$ of the
$N$ points, $n=3N$, formulae (2.1) are considerably simplified: actually, in
(8) the only non zero coefficients are $g_{i,i}$ containing (in triplets) the
masses $M_{1}$, $\dots$, $M_{N}$ and

$\begin{array}[]{l}C_{i}^{i}=g_{i,i}+\sum\limits_{\nu=1}^{k}g_{m+\nu,m+\nu}\left(\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}\right)^{2},\quad
C_{i}^{r}=C_{r}^{i}=\sum\limits_{\nu=1}^{k}g_{m+\nu,m+\nu}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}},\quad\textrm{for}\;\;i\not=r\\\
D_{i}^{r,s}=0,\quad
E_{i}^{r}=\sum\limits_{\nu}^{k}g_{m+\nu,m+\nu}\dfrac{\partial\alpha_{\nu}}{\partial
q_{r}}\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{i}},\quad
G_{i}=\sum\limits_{\nu=1}^{k}g_{m+\nu,m+\nu}\sum\limits_{\mu=1}^{k}\alpha_{\mu}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}.\end{array}$ (9)

###### Example 1

(nonholonomic pendulum, [3]). Two points $P_{1}$ and $P_{2}$ of mass $M_{1}$
and $M_{2}$ respectively are constrained on a $(x,y)$–plane, is such a way as
to verify the following nonholonomic condition:

${\dot{y}}_{2}(x_{1}{\dot{x}}_{1}+{\dot{y}}_{1}{\dot{y}}_{1})-{\dot{y}}_{1}(x_{2}{\dot{x}}_{2}+y_{2}{\dot{y}}_{2})=0$

where $(x_{i},y_{i})$ are the cartesian coordinates of $P_{i}$, $i=1,2$,
Graphically, this means that the lines orthogonal to the velocities
${\dot{P}}_{1}$, ${\dot{P}}_{2}$ intersect in a point of the $y$–axis. We set
$(q_{1},q_{2},q_{3},q_{4})=(x_{1},y_{1},y_{2},x_{2})$ and we put the kinematic
constraint in the form (2) by writing

${\dot{q}}_{4}=\dfrac{{\dot{q}}_{3}}{q_{4}}\left(q_{2}-q_{3}+\dfrac{q_{1}{\dot{q}}_{1}}{{\dot{q}}_{2}}\right)=\alpha_{1}(q_{1},q_{2},q_{3},q_{4},{\dot{q}}_{1},{\dot{q}}_{2},{\dot{q}}_{3}).$

Setting $\Phi=\frac{q_{1}{\dot{q}}_{3}}{q_{4}{\dot{q}}_{2}}$,
$\Psi=\frac{1}{q_{4}}\left(q_{2}-q_{3}+\frac{q_{1}{\dot{q}}_{1}}{{\dot{q}}_{2}}\right)$,
coefficients (9) are ($i$ denotes the row, $r$ the column)

$C_{i=1,2,3}^{r=1,2,3}=\left(\begin{array}[]{ccc}M_{1}+M_{2}\Phi^{2}&-M_{2}\frac{{\dot{q}}_{1}}{{\dot{q}}_{2}}\Phi^{2}&M_{2}\Phi\Psi\\\
&M_{1}+M_{2}\left(\frac{{\dot{q}}_{1}}{{\dot{q}}_{2}}\right)^{2}\Phi^{2}&-M_{2}\frac{{\dot{q}}_{1}}{{\dot{q}}_{2}}\Phi\Psi\\\
&&M_{2}+M_{2}\Psi^{2}\end{array}\right)$

$E_{i=1,2,3}^{r=1,2,3}=m_{2}\Phi\left(\begin{array}[]{ccc}\frac{{\dot{q}}_{1}}{q_{1}}\Phi&\frac{{\dot{q}}_{3}}{q_{4}}&-\frac{{\dot{q}}_{3}}{q_{4}}\\\
-\frac{{\dot{q}}_{1}}{q_{1}}\frac{{\dot{q}}_{1}}{{\dot{q}}_{2}}\Phi&-\frac{{\dot{q}}_{1}}{q_{1}}\Phi&\frac{{\dot{q}}_{1}}{q_{1}}\Phi\\\
\frac{{\dot{q}}_{1}}{q_{1}}\Psi&\frac{{\dot{q}}_{2}}{q_{1}}\Psi&-\frac{{\dot{q}}_{2}}{q_{1}}\Psi\end{array}\right)$
$G=-M_{2}\frac{{\dot{q}}_{3}^{2}}{q_{4}}\Psi^{2}\left(\Phi,\frac{{\dot{q}}_{1}}{{\dot{q}}_{2}}\Phi,\Psi\right).$

After specifying the dynamics (spontaneous motion, gravitational field, …) in
a way that ${\cal F}^{(q_{i})}$ can be written, the equations of motion (7)
are readily achieved.

A different and more ordinary way to present the equations of motion for
constrained system refers to the kinetic energy

$T(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t)=\frac{1}{2}{\mathcal{Q}}\cdot{\dot{\bf
X}},\qquad{\mathcal{Q}}={\dot{\bf X}}^{(M)}$ (10)

and to its restriction due to (2)

$T^{*}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)=T(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},\alpha_{1}(\cdot),\dots,\alpha_{k}(\cdot),t)$
(11)

where each $\alpha_{j}(\cdot)$, $j=1,\dots,k$ depends on
$q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m}$ and $t$. In that case,
the equations of motion assume the form

$\dfrac{d}{dt}\dfrac{\partial T^{*}}{\partial{\dot{q}}_{i}}-\dfrac{\partial
T^{*}}{\partial q_{i}}-\sum\limits_{\nu=1}^{k}\dfrac{\partial T^{*}}{\partial
q_{m+\nu}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}_{i}}}-\sum\limits_{\nu=1}^{k}B_{i}^{\nu}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}={\cal
F}^{(q_{i})}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}{\cal
F}^{(q_{m+\nu})},\qquad i=1,\dots,m$ (12)

joined with (2), where in $\dfrac{\partial T}{\partial{\dot{q}}_{m+\nu}}$ the
variables ${\dot{q}}_{m+1}$, $\dots$, ${\dot{q}}_{n}$ are expressed in terms
of $(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)$ by using (2) and
the coefficients
$B_{i}^{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)$ are
defined by

$B_{i}^{\nu}=\sum\limits_{r=1}^{m}\left(\dfrac{\partial^{2}\alpha_{\nu}}{\partial{\dot{q}}_{i}\partial
q_{r}}{\dot{q}}_{r}+\dfrac{\partial^{2}\alpha_{\nu}}{\partial{\dot{q}}_{i}\partial{\dot{q}}_{r}}{\mathop{q}\limits^{..}}_{r}\right)-\dfrac{\partial\alpha_{\nu}}{\partial
q_{i}}+\sum\limits_{\mu=1}^{k}\left(\dfrac{\partial^{2}\alpha_{\nu}}{\partial{\dot{q}}_{i}\partial
q_{m+\mu}}\alpha_{\mu}-\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{i}}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}\right)+\dfrac{\partial^{2}\alpha_{\nu}}{\partial{\dot{q}}_{i}\partial
t}.$ (13)

In order to achieve (12), the key point is the well known relation
${\dot{\mathcal{Q}}}\cdot\dfrac{\partial{\bf X}}{\partial
q_{i}}=\dfrac{d}{dt}\dfrac{\partial T}{\partial{\dot{q}}_{i}}-\dfrac{\partial
T}{\partial q_{i}}$. We remark that an alternative expression for
$B_{i}^{\nu}$ is

$B_{i}^{\nu}=\dfrac{d}{dt}\left(\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}\right)-\dfrac{\partial\alpha_{\nu}}{\partial
q_{i}}-\sum\limits_{\mu=1}^{k}\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{i}}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}.$ (14)

###  2.2 Some remarks

Equations (6) or (12) hold for a system settled by $n$ parameters $q_{1}$,
$\dots$, $q_{n}$ and undergoing the $m<n$ nonlinear nonholonomic constraints
(2); it is worth it to dwell upon some significant points and remarks.

$\bullet$

Equations (7) trace the Gibbs–Appell equations, since the left side of (7)
corresponds to the calculus $\dfrac{\partial
S}{\partial{\mathop{q}\limits^{..}}_{i}}$, where $S=\frac{1}{2}{\dot{\cal
Q}}\cdot\mathop{\bf X}\limits^{..}$ is the acceleration energy (Gibbs–Appell
function); actually $\dfrac{\partial
S}{\partial{\mathop{q}\limits^{..}}_{i}}={\dot{\cal
Q}}\cdot\dfrac{\partial\mathop{\bf
X}\limits^{..}}{\partial{\mathop{q}\limits^{..}}_{i}}={\dot{\cal
Q}}\cdot(\frac{\partial{\bf X}}{\partial
q_{i}}+\sum\limits_{j=1}^{k}\frac{\partial\alpha_{j}}{\partial{\dot{q}}_{i}}\frac{\partial{\bf
X}}{\partial q_{m+j}})$, which sends back to (5).

$\bullet$

On the other hand, equations (7) extend to the nonlinear case the Voronec
equations (appeared in [13] and discussed in the important monography [9]) for
the linear nonholonomic constraints (3); in the latter case the terms
$\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}$ in (7) are simply
$\alpha_{\nu,i}$ and the equations of motion (12) are, for each $i=1,\dots,m$,

$\dfrac{d}{dt}\dfrac{\partial T^{*}}{\partial{\dot{q}}_{i}}-\dfrac{\partial
T^{*}}{\partial q_{i}}-\sum\limits_{\nu=1}^{k}\alpha_{\nu,i}\dfrac{\partial
T^{*}}{\partial
q_{m+\nu}}-\sum\limits_{\nu=1}^{k}\sum\limits_{r=1}^{m}\beta_{ir}^{\nu}{\dot{q}}_{r}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}={\mathcal{F}}^{(q_{i})}+\sum\limits_{\nu=1}^{k}\alpha_{\nu,i}{\cal
F}^{(q_{m+\nu})}$ (15)

where the coefficients (13) reduce to

$\begin{array}[]{l}B_{i}^{\nu}=\sum\limits_{r=1}^{m}\beta_{ir}^{\nu}(q_{1},\dots,q_{n},t){\dot{q}}_{r}+\dfrac{\partial\alpha_{\nu,i}}{\partial
t},\\\ \beta_{ir}^{\nu}=\dfrac{\partial\alpha_{\nu,i}}{\partial
q_{r}}-\dfrac{\partial\alpha_{\nu,r}}{\partial
q_{i}}+\sum\limits_{\mu=1}^{k}\left(\dfrac{\partial\alpha_{\nu,i}}{\partial
q_{m+\mu}}\alpha_{\mu,r}-\dfrac{\partial\alpha_{\nu,r}}{\partial
q_{m+\mu}}\alpha_{\mu,i}\right).\end{array}$ (16)

Although the explicit dependence of $\alpha_{\nu,i}$ on $t$ is absent in [9],
the widening to the rheonomic case is trivial.

$\bullet$

Equations (12) correspond to the ones derived in [15], as the most general
form of equations of motion in Poincaré–Chetaev variables extended to
nonlinear nonholonomic systems; the Voronec’s equations (12) are the same as
the Voronec’s equations pointed out in [15] as the special case of Poincaré’s
kinematic parameters chosen as the real generalized velocities. Also the
geometric approach for nonholonomic machanical systems (Lagrangian systems on
fibered manifolds) performed in [10] leads to the same equations of motion as
(12).

$\bullet$

Concerning the dependence of (2) on the variables, a special case is

$\begin{array}[]{ll}\alpha_{\nu}=\alpha_{\nu}(q_{1},\dots,q_{m},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)&\textrm{for
each}\;\nu=1,\dots,k,\\\ \\\ \
T=T(q_{1},\dots,q_{m},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t),&\mathcal{F}_{i}=\mathcal{F}_{i}(q_{1},\dots,q_{m},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t)\end{array}$
(17)

that is the coordinates $q_{m+1},\dots,q_{n}$ corresponding to the dependent
velocities do not occur; we may refer to these systems as nonlinear
$\check{\rm C}$aplygin systems. In this case, system (12) reduces to (see also
(14))

$\begin{array}[]{l}\dfrac{d}{dt}\dfrac{\partial
T^{*}}{\partial{\dot{q}}_{i}}-\dfrac{\partial T^{*}}{\partial
q_{i}}-\underbrace{\sum\limits_{\nu=1}^{k}\left(\sum\limits_{r=1}^{m}\left(\dfrac{\partial^{2}\alpha_{\nu}}{\partial{\dot{q}}_{i}\partial
q_{r}}{\dot{q}}_{r}+\dfrac{\partial^{2}\alpha_{\nu}}{\partial{\dot{q}}_{i}\partial{\dot{q}}_{r}}{\mathop{q}\limits^{..}}_{r}\right)-\dfrac{\partial\alpha_{\nu}}{\partial
q_{i}}+\dfrac{\partial^{2}\alpha_{\nu}}{\partial{\dot{q}}_{i}\partial
t}\right)}_{\frac{d}{dt}\frac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}-\alpha_{\nu}}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}\\\ ={\cal
F}^{(q_{i})}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}{\cal
F}^{(q_{m+\nu})}.\end{array}$ (18)

The remarkable advantage of the differential system (18) is that it contains
only the unknown functions $q_{1}$, $\dots$, $q_{m}$ and it is disentangled
from the constraints equations (3).

Within assumption (17), the linear stationary case

$\begin{array}[]{ll}\alpha_{\nu,j}=\alpha_{\nu,j}(q_{1},\dots,q_{m})&\textrm{for
each}\;\nu=1,\dots,k\;\textrm{and}\;j=1,\dots,m\\\ \\\ \
T=T(q_{1},\dots,q_{m},{\dot{q}}_{1},\dots,{\dot{q}}_{n}),&\mathcal{F}_{i}=\mathcal{F}_{i}(q_{1},\dots,q_{m},{\dot{q}}_{1},\dots,{\dot{q}}_{n})\end{array}$
(19)

leads to the $\check{\rm C}$aplygin’s equations (see ([9]))

$\dfrac{d}{dt}\dfrac{\partial T^{*}}{\partial{\dot{q}}_{i}}-\dfrac{\partial
T^{*}}{\partial
q_{i}}-\sum\limits_{\nu=1}^{k}\sum\limits_{r=1}^{m}\left(\dfrac{\partial\alpha_{\nu,i}}{\partial
q_{r}}-\dfrac{\partial\alpha_{\nu,r}}{\partial
q_{i}}\right){\dot{q}}_{r}\dfrac{\partial T}{\partial{\dot{q}}_{m+\nu}}={\cal
F}^{(q_{i})}+\sum\limits_{j=1}^{k}\alpha_{j,i}{\cal F}^{(q_{m+j})}\quad
i=1,\dots,m.$ (20)

$\bullet$

Whenever the kinetic energy (10) is, consistently with the usual mechanical
systems,

$T=\frac{1}{2}\sum\limits_{i,j=1}^{n}g_{i,j}{\dot{q}}_{i}{\dot{q}}_{j}+\sum\limits_{i=1}^{n}b_{i}{\dot{q}}_{i}+c,\quad
b_{i}(q_{1},\dots,q_{n},t)=\dfrac{\partial{\bf X}^{(\mathsf{M})}}{\partial
q_{i}}\cdot\dfrac{\partial{\bf X}}{\partial
t},\;c(q_{1},\dots,q_{n},t)=\frac{1}{2}\dfrac{\partial{\bf
X}^{(\mathsf{M})}}{\partial t}\cdot\dfrac{\partial{\bf X}}{\partial t}$

($g_{i,j}$ are defined in (8)) so that (11) writes

$T^{*}=\frac{1}{2}\left(\sum\limits_{r,s=1}^{m}g_{r,s}{\dot{q}}_{r}{\dot{q}}_{s}+\sum\limits_{\nu,\mu=1}^{k}g_{m+\nu,m+\mu}\alpha_{\nu}\alpha_{\mu}\right)+\sum\limits_{r=1}^{m}\sum\limits_{\nu=1}^{k}g_{r,m+\nu}{\dot{q}}_{r}\alpha_{\nu}+\sum\limits_{r=1}^{m}b_{r}{\dot{q}}_{r}+\sum\limits_{\nu=1}^{k}b_{m+\nu}\alpha_{\nu}+c$
(21)

then rearranging the terms in (12) one can easily check (see [11])) that all
the terms of $-\sum\limits_{\nu=1}^{k}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}B_{i}^{\nu}$, for each $i=1,\dots,m$, cancel out
with part of the addends of $\dfrac{d}{dt}\left(\dfrac{\partial
T^{*}}{\partial{\dot{q}}_{i}}\right)$, of $-\dfrac{\partial T^{*}}{\partial
q_{i}}$ and of $-\sum\limits_{\nu=1}^{k}\dfrac{\partial T^{*}}{\partial
q_{m+\nu}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}_{i}}}$. The remaining
terms of (12) coincide precisely with the terms of (7).

In order to clarify the latter statement, we illustrate the following

###### Example 2

On a plane $(x,y)$ we consider two points $P_{1}\equiv(x_{1},y_{1})$ and
$P_{2}\equiv(x_{2},y_{2})$ whose velocities are perpendicular:

${\dot{x}}_{1}{\dot{x}}_{2}+{\dot{y}}_{1}{\dot{y}}_{2}=0.$ (22)

We set ${\bf X}(q_{1},q_{2},q_{3},q_{4})$, $q_{1}=x_{1}$, $q_{2}=y_{2}$,
$q_{3}=x_{2}$, $q_{4}=y_{2}$, so that (2) is

${\dot{q}}_{4}=\alpha_{1}({\dot{q}}_{1},{\dot{q}}_{2},{\dot{q}}_{3})=-\frac{{\dot{q}}_{1}{\dot{q}}_{3}}{{\dot{q}}_{2}}.$

The system runs into the case (9) with $g_{1,1}=g_{2,2}=M_{1}$,
$g_{3,3}=g_{4,4}=M_{2}$ and the coeffcients are

$\begin{array}[]{lll}C_{1}^{1}=M_{1}+M_{2}({\dot{q}}_{3}/{\dot{q}}_{2})^{2}&C_{2}^{2}=M_{1}+M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{2})^{2}&C_{3}^{3}=M_{1}+M_{2}({\dot{q}}_{1}/{\dot{q}}_{2})^{2}\\\
C_{1}^{2}=C_{2}^{1}=-M_{2}{\dot{q}}_{1}{\dot{q}}_{3}^{2}/{\dot{q}}_{2}^{3}&C_{1}^{3}=C_{3}^{1}=M_{2}{\dot{q}}_{1}{\dot{q}}_{3}^{2}/{\dot{q}}_{2}^{2}&C_{2}^{3}=C_{3}^{2}=-M_{2}{\dot{q}}_{1}^{2}{\dot{q}}_{3}/{\dot{q}}_{2}^{3}\\\
D_{i}^{r,s}=E_{i}^{r}=G_{i}=0&&\end{array}$

so that (7) are (let us leave the forces terms unspecified)

$\begin{array}[]{l}(M_{1}+M_{2}({\dot{q}}_{3}/{\dot{q}}_{2})^{2}){\mathop{q}\limits^{..}}_{1}-M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{3})({\dot{q}}_{3}{\mathop{q}\limits^{..}}_{2}-{\dot{q}}_{2}{\mathop{q}\limits^{..}}_{3})={\cal
F}^{(q_{1})}-{\cal F}^{(q_{4})}{\dot{q}}_{3}/{\dot{q}}_{2}\\\ \\\
(M_{1}+M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{2})^{2}){\mathop{q}\limits^{..}}_{2}-M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{3})({\dot{q}}_{3}{\mathop{q}\limits^{..}}_{1}-{\dot{q}}_{1}{\mathop{q}\limits^{..}}_{3})={\cal
F}^{(q_{1})}+{\cal F}^{(q_{4})}{\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{2}\\\
\\\
(M_{1}+M_{2}({\dot{q}}_{1}/{\dot{q}}_{2})^{2}){\mathop{q}\limits^{..}}_{3}+M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{3})({\dot{q}}_{2}{\mathop{q}\limits^{..}}_{1}-{\dot{q}}_{1}{\mathop{q}\limits^{..}}_{2})={\cal
F}^{(q_{1})}-{\cal F}^{(q_{4})}{\dot{q}}_{1}/{\dot{q}}_{2}\end{array}$ (23)

On the other hand, by writing (11) as
$T^{*}=\frac{1}{2}M_{1}({\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2})+\frac{1}{2}M_{2}{\dot{q}}_{3}^{2}(1+({\dot{q}}_{1}/{\dot{q}}_{2})^{2})$
and calculating (13) as

$B_{1}^{1}=({\dot{q}}_{3}{\mathop{q}\limits^{..}}_{2}-{\dot{q}}_{2}{\mathop{q}\limits^{..}}_{3})/{\dot{q}}_{2}^{2},\quad
B_{2}^{1}=({\dot{q}}_{1}{\mathop{q}\limits^{..}}_{3}+{\dot{q}}_{3}{\mathop{q}\limits^{..}}_{1})/{\dot{q}}_{2}^{2}-({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{3}){\mathop{q}\limits^{..}}_{2},\quad
B_{3}^{1}=(-{\dot{q}}_{2}{\mathop{q}\limits^{..}}_{1}+{\dot{q}}_{1}{\mathop{q}\limits^{..}}_{2})/{\dot{q}}_{2}^{2}$

the calculation of (12) leads to

$\left\\{\begin{array}[]{l}\dfrac{d}{dt}[{\dot{q}}_{1}(M_{1}+M_{2}({\dot{q}}_{3}/{\dot{q}}_{2})^{2})]+M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{3})({\dot{q}}_{3}{\mathop{q}\limits^{..}}_{2}-{\dot{q}}_{2}{\mathop{q}\limits^{..}}_{3})={\cal
F}^{(q_{1})}-{\cal F}^{(q_{4})}{\dot{q}}_{3}/{\dot{q}}_{2}\\\ \\\
\dfrac{d}{dt}[{\dot{q}}_{2}(M_{1}-M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{2})^{2})]+M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{3})({\dot{q}}_{3}{\mathop{q}\limits^{..}}_{1}+{\dot{q}}_{1}{\mathop{q}\limits^{..}}_{3}-({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}){\mathop{q}\limits^{..}}_{2})={\cal
F}^{(q_{2})}+{\cal F}^{(q_{4})}{\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{2}\\\
\\\
\dfrac{d}{dt}[M_{2}{\dot{q}}_{3}(1+({\dot{q}}_{1}/{\dot{q}}_{2})^{2})]-M_{2}({\dot{q}}_{1}{\dot{q}}_{3}/{\dot{q}}_{2}^{3})({\dot{q}}_{2}{\mathop{q}\limits^{..}}_{1}-{\dot{q}}_{1}{\mathop{q}\limits^{..}}_{2})={\cal
F}^{(q_{3})}-{\cal F}^{(q_{4})}{\dot{q}}_{1}/{\dot{q}}_{2}.\end{array}\right.$
(24)

According to the assertion claimed after (21), all the terms $-\dfrac{\partial
T}{\partial{\dot{q}}_{4}}B_{i}^{1}$, $i=1,2,3$, which are the terms beginning
with $+m_{2}$ in each of (24), cancel out with opposite terms of the explicit
calculation of $d/dt$. The rest of the terms are those appearing in (23).

###### Example 3

A different example, discussed in [12], draws attention to the fact that
equations (12) are still valid even though the function $T^{*}$ defined in
(21) degenerates w. r. t. the restricted variables ${\dot{q}}_{1}$, $\dots$,
${\dot{q}}_{m}$: let us consider one point of mass $M$ of cartesian
coordinates $(x,y,z)$ and whose velocity is constant in module:
${\dot{x}}^{2}+{\dot{y}}^{2}+{\dot{z}}^{2}=C^{2}\not=0$. In terms of
$(q_{1},q_{2},q_{3})=(x,y,z)$ we write (2) as
${\dot{q}}_{3}=\pm\sqrt{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}=\alpha_{1}({\dot{q}}_{1},{\dot{q}}_{2})$,
where the sign derives from the initial conditions (this example presents
$m=2$, $k=1$). Concerning (7), we use (9) with $g_{1,1}=g_{2,2}=g_{3,3}=m$,
$g_{i,j}=0$ for $i\not=j$, so that

$C_{i}^{i}=M\left(1+\frac{M{\dot{q}}_{i}^{2}}{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}\right),\quad
i=1,2,\qquad
C_{2}^{1}=C_{1}^{2}=\frac{M{\dot{q}}_{1}{\dot{q}}_{2}}{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}$

so that equations (12) are

$\left\\{\begin{array}[]{l}M\frac{C^{2}-{\dot{q}}_{2}^{2}}{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}_{2}^{2}}}{\mathop{q}\limits^{..}}_{1}+M\frac{{\dot{q}}_{1}{\dot{q}}_{2}}{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}_{2}^{2}}}{\mathop{q}\limits^{..}}_{2}={\cal
F}^{(q_{1})}\mp{\cal
F}^{(q_{3})}\frac{{\dot{q}}_{1}}{\sqrt{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}},\\\
M\frac{{\dot{q}}_{1}{\dot{q}}_{2}}{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}{\mathop{q}\limits^{..}}_{1}+M\frac{C^{2}-{\dot{q}}_{1}^{2}}{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}{\mathop{q}\limits^{..}}_{2}={\cal
F}^{(q_{1})}\mp{\cal
F}^{(q_{3})}\frac{{\dot{q}}_{2}}{\sqrt{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}},\end{array}\right.$
(25)

On the other hand, (21) is $T^{*}=\frac{1}{2}mC^{2}$, hence the only
contributions in the left hand side of (12) are

$-B^{1}_{i}\dfrac{\partial
T}{\partial{\dot{q}}_{3}}=-m\left(\dfrac{\partial^{2}\alpha_{1}}{\partial{\dot{q}}_{i}\partial{\dot{q}}_{1}}{\mathop{q}\limits^{..}}_{1}+\dfrac{\partial^{2}\alpha_{1}}{\partial{\dot{q}}_{i}\partial{\dot{q}}_{2}}{\mathop{q}\limits^{..}}_{2}\right){\dot{q}}_{3}({\dot{q}}_{1},{\dot{q}}_{2}),\quad
i=1,2$

Calculating the second derivatives, one easily finds exactly equations (25).

##  3 Energy balance

Let us assume that the active forces depend only on ${\bf X}$ and $t$ and come
from a potential ${\cal U}$:

${\mathcal{F}}=\nabla_{\bf X}{\cal U}({\bf X},t)$ (26)

so that the restriction $U(q_{1},\dots,q_{n},t)={\cal U}({\bf
X}(q_{1},\dots,q_{n},t),t)$ to the configuration manifold provides the
generalized forces as follows:

${\mathcal{F}}^{(q_{i})}=\dfrac{\partial U}{\partial q_{i}},\quad
i=1,\dots,n\quad$ (27)

It is known that in case of holonomic systems ${\bf
X}(q_{1},\dots,q_{\ell},t)$ (that is removing (1)) the equations of motion
$\dfrac{d}{dt}\dfrac{\partial{\cal
L}}{\partial{\dot{q}}_{i}}-\dfrac{\partial{\cal L}}{\partial q_{i}}=0$,
$i=1,\dots,n$, where ${\cal L}=T+U$ (see (10)) entail the energy balance
$\dfrac{d}{dt}\left(\sum\limits_{i=1}^{n}{\dot{q}}_{i}\dfrac{\partial{\cal
L}}{\partial{\dot{q}}_{i}}-{\cal L}\right)=-\dfrac{\partial{\cal L}}{\partial
t}$, which supplies the conservation of the quantity in brackets, whenever
${\cal L}$ does not depend on $t$ explicitly.

Now, if the constraints (1) are present, recalling $T^{*}$ defined in (11) we
set

${\cal
L}^{*}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)=T^{*}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)+U(q_{1},\dots,q_{n},t)$
(28)

as the Lagrangian in terms of the independent velocities. The following
Proposition generalizes the just mentioned balance of holonomic systems.

###### Proposition 2

The equations of motion (12) entail

$\dfrac{d}{dt}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{i}}-{\cal
L}^{*}\right)-\sum\limits_{\nu=1}^{k}({\overline{\alpha}}_{\nu}-\alpha_{\nu})\dfrac{\partial{\cal
L}^{*}}{\partial
q_{m+\nu}}-\sum\limits_{\nu=1}^{k}{\overline{B}}_{\nu}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}=-\dfrac{\partial{\cal L}^{*}}{\partial t}$ (29)

where

$\displaystyle{\overline{\alpha}}_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}},$
(30)
$\displaystyle{\overline{B}}_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)$
$\displaystyle=$
$\displaystyle\sum\limits_{r=1}^{m}{\dot{q}}_{r}\left(\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial
q_{r}}-\dfrac{\partial\alpha_{\nu}}{\partial
q_{r}}\right)+\sum\limits_{\mu=1}^{k}\left(\alpha_{\mu}\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial
q_{m+\mu}}-{\overline{\alpha}}_{\mu}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}\right)$ $\displaystyle+$
$\displaystyle\sum\limits_{r=1}^{m}{\mathop{q}\limits^{..}}_{r}\left(\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial{\dot{q}}_{r}}-\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\right)+\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial
t}$ $\displaystyle=$
$\displaystyle\dfrac{d}{dt}({\overline{\alpha}}_{\nu}-\alpha_{\nu})-\sum\limits_{\mu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}({\overline{\alpha}}_{\mu}-\alpha_{\mu})+\dfrac{\partial\alpha_{\nu}}{\partial
t}.$

Proof. Basing on the formula

$\dfrac{d}{dt}F(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)=\sum\limits_{i=1}^{m}\dfrac{\partial
F}{\partial q_{i}}{\dot{q}}_{i}+\sum\limits_{\nu=1}^{k}\dfrac{\partial
F}{\partial q_{m+\nu}}\alpha_{\nu}+\sum\limits_{i=1}^{m}\dfrac{\partial
F}{\partial{\dot{q}}_{i}}{\mathop{q}\limits^{..}}_{i}+\dfrac{\partial
F}{\partial t}$ (32)

implemented with $F=T^{*}$ one finds

$\sum\limits_{i=1}^{m}{\dot{q}}_{i}\left(\dfrac{d}{dt}\dfrac{\partial
T^{*}}{\partial{\dot{q}}_{i}}-\dfrac{\partial T^{*}}{\partial
q_{i}}-\sum\limits_{\nu=1}^{k}\dfrac{\partial T^{*}}{\partial
q_{m+\nu}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}_{i}}}\right)=\dfrac{d}{dt}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial
T^{*}}{\partial{\dot{q}}_{i}}-T^{*}\right)+\dfrac{\partial T^{*}}{\partial
t}-\sum\limits_{\nu=1}^{k}\dfrac{\partial T^{*}}{\partial
q_{m+\nu}}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}-\alpha_{\nu}\right)$
$=\dfrac{d}{dt}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial
T^{*}}{\partial{\dot{q}}_{i}}-T^{*}\right)+\dfrac{\partial T^{*}}{\partial
t}-\sum\limits_{\nu=1}^{k}\dfrac{\partial T^{*}}{\partial
q_{m+\nu}}\left({\overline{\alpha}}_{\nu}-\alpha_{\nu}\right).$ (33)

Furthermore, recalling (14) it is

$\displaystyle\sum\limits_{i=1}^{m}B_{i}^{\nu}{\dot{q}}_{i}$ $\displaystyle=$
$\displaystyle\dfrac{d}{dt}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}\right)-\sum\limits_{i=1}^{m}\left({\dot{q}}_{i}\dfrac{\partial\alpha_{\nu}}{\partial
q_{i}}+{\mathop{q}\limits^{..}}_{i}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}\right)-\sum\limits_{\mu=1}^{k}\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{i}}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}$ $\displaystyle=$
$\displaystyle\dfrac{d{\overline{\alpha}}_{\nu}}{dt}-\sum\limits_{i=1}^{m}\left({\dot{q}}_{i}\dfrac{\partial\alpha_{\nu}}{\partial
q_{i}}+{\mathop{q}\limits^{..}}_{i}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}\right)-\sum\limits_{\mu=1}^{k}{\overline{\alpha}}_{\mu}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}$
$=\sum\limits_{r=1}^{m}{\dot{q}}_{r}\left(\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial
q_{r}}-\dfrac{\partial\alpha_{\nu}}{\partial
q_{r}}\right)+\sum\limits_{\mu=1}^{k}\left(\alpha_{\mu}\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial
q_{m+\mu}}-{\overline{\alpha}}_{\mu}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}\right)+\sum\limits_{r=1}^{m}{\mathop{q}\limits^{..}}_{r}\left(\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial{\dot{q}}_{r}}-\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\right)+\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial
t}.$ (34)

Concerning the forces, under assumption (27) and having in mind
$\dfrac{dU}{dt}=\sum\limits_{i=1}^{m}\dfrac{\partial U}{\partial
q_{i}}{\dot{q}}_{i}+\sum\limits_{\nu=1}^{k}\dfrac{\partial U}{\partial
q_{m+\nu}}\alpha_{\nu}+\dfrac{\partial U}{\partial t}$, we can write

$\sum\limits_{i=1}^{m}{\dot{q}}_{i}\left({\cal
F}^{(q_{i})}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{i}}{\cal
F}^{(q_{m+\nu})}\right)=\dfrac{dU}{dt}-\dfrac{\partial U}{\partial
t}+\sum\limits_{\nu=1}^{k}\dfrac{\partial U}{\partial
q_{m+\nu}}\left({\overline{\alpha}}_{\nu}-\alpha_{\nu}\right).$ (35)

By virtue of (33), (34) and (35), multiplying each of (12) by ${\dot{q}}_{i}$
and summing up with respect to $i$ one gets the statement (29). The second
equality for ${\overline{B}}_{\nu}$ in (2) is obtained by applying (32) with
$F={\overline{\alpha}}_{\nu}-\alpha_{\nu}$. $\quad\square$

###### Corollary 1

For a system verifying assumption (17) (nonlinear $\check{\rm C}$aplygin’s
systems) and assumption (27) for $i=1,\dots,m$, equation (29) takes the
simpler form

$\dfrac{d}{dt}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{i}}-{\cal
L}^{*}\right)-\sum\limits_{\nu=1}^{k}\left(\dfrac{d}{dt}({\overline{\alpha}}_{\nu}-\alpha_{\nu})+\dfrac{\partial\alpha_{\nu}}{\partial
t}\right)\dfrac{\partial T}{\partial{\dot{q}}_{m+\nu}}=-\dfrac{\partial{\cal
L}^{*}}{\partial t}$ (36)

Indeed, the terms containing $\dfrac{\partial{\cal L}^{*}}{\partial
q_{m+\nu}}$ in (30) cancel out for each $\nu=1,\dots,k$.

Whenever $T^{*}$ is the function (21), the energy of the system is

$\displaystyle\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{i}}-{\cal L}^{*}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{r,s}^{m}g_{r,s}{\dot{q}}_{r}{\dot{q}}_{s}+\sum\limits_{\nu,\mu=1}^{k}g_{m+\nu,m+\mu}({\overline{\alpha}}_{\nu}\alpha_{\mu}-\frac{1}{2}\alpha_{\nu}\alpha_{\mu})$
$\displaystyle+$
$\displaystyle\sum\limits_{r=1}^{m}\sum\limits_{\nu=1}^{k}g_{r,m+\nu}{\overline{\alpha}}_{\nu}{\dot{q}}_{r}+\sum\limits_{\nu=1}^{k}b_{m+\nu}({\overline{\alpha}}_{\nu}-\alpha_{\nu})-c-U.$

We refer to (3) as the energy of the system.

###### Example 4

Let us exert (29) for the system of Example 1 (nonholonomic pendulum):
assuming that the forces give rise to the potential
$U(q_{1},q_{2},q_{3},q_{4})$, the function (28) is, recalling that the only
one kinematic constraint is
${\dot{q}}_{4}=\dfrac{{\dot{q}}_{3}}{q_{4}}\left(q_{2}-q_{3}+\dfrac{q_{1}{\dot{q}}_{1}}{{\dot{q}}_{2}}\right)=\alpha_{1}(q_{1},q_{2},q_{3},q_{4},{\dot{q}}_{1},{\dot{q}}_{2},{\dot{q}}_{3})$,

${\cal
L}^{*}=\dfrac{1}{2}M_{1}({\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2})+\dfrac{1}{2}M_{2}{\dot{q}}_{3}^{2}\left(1+\dfrac{1}{q_{4}^{2}}\left(q_{2}-q_{3}+\dfrac{q_{1}{\dot{q}}_{1}}{{\dot{q}}_{2}}\right)^{2}\right)+U(q_{1},q_{2},q_{3},q_{4}).$

Calculating (30) for $\alpha_{1}$ one finds

${\overline{\bar{\alpha}}_{1}}=\cancel{\frac{q_{1}{\dot{q}}_{3}}{q_{4}{\dot{q}}_{2}}{\dot{q}}_{1}}+\cancel{\frac{{\dot{q}}_{3}}{q_{4}}\left(-\dfrac{q_{1}{\dot{q}}_{1}}{{\dot{q}}_{2}^{2}}\right){\dot{q}}_{2}}+\dfrac{1}{q_{4}}\left(q_{2}-q_{3}+\dfrac{q_{1}{\dot{q}}_{1}}{{\dot{q}}_{2}}\right){\dot{q}}_{3}=\alpha_{1}$

so that even ${\overline{B}}_{1}=0$ (see (2)). Therefore, (29) provides the
constant of motion (3) (in this example $g_{1,1}=g_{2,2}=M_{1}$,
$g_{3,3}=g_{4,4}=M_{2}$, $g_{i,j}=0$ for $i\not=j$)

$I(q_{1},q_{2},q_{3},q_{4},{\dot{q}}_{1},{\dot{q}}_{2},{\dot{q}}_{3})=\dfrac{1}{2}M_{1}({\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2})+\dfrac{1}{2}M_{2}+\dfrac{1}{2}M_{2}{\dot{q}}_{3}^{2}\left(1+\dfrac{1}{q_{4}^{2}}\left(q_{2}-q_{3}+\dfrac{q_{1}{\dot{q}}_{1}}{{\dot{q}}_{2}}\right)^{2}\right)-U(q_{1},q_{2},q_{3},q_{4}).$

###### Example 5

$N$ material points $(P_{i},M_{i})$, $i=1,\dots,N$ lie on the $(x,y)$–plane
and are constrained to move with parallel velocities; denoting by
$(x_{i},y_{i})$, $i=1,2,3$, the cartesian coordinates of the points, the
nonholonomic restrictions are

${\dot{x}}_{1}{\dot{y}}_{2}-{\dot{x}}_{2}{\dot{y}}_{1}=0,\quad{\dot{x}}_{1}{\dot{y}}_{3}-{\dot{x}}_{3}{\dot{y}}_{1}=0\quad\dots\quad{\dot{x}}_{1}{\dot{y}}_{N}-{\dot{x}}_{N}{\dot{y}}_{N}=0$

which can be exhibited in the explicit form (2) by setting, for $n=2N$,
$(q_{1},q_{2},\dots,q_{n})=(\overbrace{x_{1},x_{2},\dots,x_{N},y_{1}}^{q_{1},\dots,q_{m}},\underbrace{y_{2},\dots,y_{N}}_{q_{m+1},\dots,q_{n}})$,
and by writing

${\dot{q}}_{m+1}=\frac{{\dot{q}}_{2}}{{\dot{q}}_{1}}{{\dot{q}}_{m}},\quad\dots\;\;{\dot{q}}_{n}=\frac{{\dot{q}}_{m-1}}{{\dot{q}}_{1}}{{\dot{q}}_{m}}$

that is, for $\nu=1,\dots,k$,
${\dot{q}}_{m+\nu}=\alpha_{\nu}=\frac{{\dot{q}}_{\nu+1}}{{\dot{q}}_{1}}{{\dot{q}}_{m}}$;
in this example $m=N+1$, $k=N-1$. The coefficients (30) and (2) are, for each
$\nu=1,\dots,m$,

${\overline{\alpha}}_{\nu}=\cancel{-\dfrac{{\dot{q}}_{\nu+1}}{{\dot{q}}_{1}^{2}}{\dot{q}}_{m}{\dot{q}}_{1}}+\cancel{\dfrac{{\dot{q}}_{\nu+1}}{{\dot{q}}_{2}}}+\dfrac{{\dot{q}}_{m}}{{\dot{q}}_{1}}{\dot{q}_{\nu+1}}=\alpha_{\nu},\qquad{\overline{B}}_{\nu}=0.$

Therefore ${\cal
L}^{*}=\dfrac{1}{2}\sum\limits_{i=1}^{m-1}M_{i}{\dot{q}}_{i}^{2}(1+{\dot{q}}_{m}^{2}/{\dot{q}}_{1}^{2})$
gives by means of (29) the constant of motion (3): having in mind
$g_{1,1}=g_{m,m}=M_{1}$, $g_{2,2}=g_{m+1,m+1}=M_{2}$, $\dots$,
$g_{m-1,m-1}=g_{n,n}=M_{N}$, elsewhere null, the first integral is
$\dfrac{1}{2}\sum\limits_{i=1}^{m-1}M_{i}{\dot{q}}_{i}^{2}(1+{\dot{q}}_{m}^{2}/{\dot{q}}_{1}^{2})-U(q_{1},\dots,q_{n})$,
where the last term takes into account the possibile interactions.

###### Example 6

We go back to Example 3 (one point with constant norm of the velocity), in
order to write (29) for (25). Assuming that the active forces are connected to
the potential $U=U(q_{1},q_{2},q_{3})$, we have ${\cal
L}^{*}=\frac{1}{2}MC^{2}+U$ and (29) writes

$-\dfrac{dU}{dt}-({\overline{\alpha}}_{1}-\alpha_{1})\dfrac{\partial
U}{\partial q_{3}}-M\alpha_{1}{\overline{B}}_{1}=0$

with $\alpha_{1}=\pm\sqrt{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}$,
${\overline{\alpha}}_{1}=\mp\frac{{\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2}}{\sqrt{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}}$,
${\overline{B}}_{1}=\mp\frac{C^{2}}{(C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2})^{3/2}}({\dot{q}}_{1}{\mathop{q}\limits^{..}}_{1}+{\dot{q}}_{2}{\mathop{q}\limits^{..}}_{2})$.
Hence we get

$-\dfrac{\partial U}{\partial q_{1}}{\dot{q}}_{1}-\dfrac{\partial U}{\partial
q_{2}}{\dot{q}}_{2}\pm\frac{{\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2}}{\sqrt{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}}\dfrac{\partial
U}{\partial
q_{3}}=\dfrac{M}{2}\dfrac{C^{2}}{C^{2}-{\dot{q}}_{1}^{2}-{\dot{q}}_{2}^{2}}\dfrac{d}{dt}({\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2}).$

###### Example 7

We can modify the previous example by requiring two material points to have
the same norm of the velocity: ${\dot{P}}_{1}^{2}={\dot{P}}_{2}^{2}$. Calling
$(x_{i},y_{i},z_{i})$ the cartesian coordinate of $P_{i}$, $i=1,2$, and
setting
$(q_{1},q_{2},q_{3},q_{4},q_{5},q_{6})=(x_{1},y_{1},z_{1},x_{2},y_{2},z_{2})$,
the form (2) of the nonholonomic restriction is

${\dot{q}}_{6}=\pm\sqrt{{\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2}+{\dot{q}}_{3}^{2}-{\dot{q}}_{4}^{2}-{\dot{q}}_{5}^{2}}=\alpha_{1}({\dot{q}}_{1},{\dot{q}}_{2},{\dot{q}}_{3},{\dot{q}}_{4},{\dot{q}}_{5}).$

Even in this case ${\overline{\alpha}}_{1}=\alpha_{1}$, ${\overline{B}}_{1}=0$
and (29) entails the conservation of the quantity (3)
$\frac{1}{2}(M_{1}+M_{2})({\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2}+{\dot{q}}_{3}^{2})-U(q_{1},q_{2},q_{3},q_{4},q_{5},q_{6})$,
whenever the forces allow the access to the potential $U$.

##  4 Special classes of nonholonomic constraints

As it emerges from the Examples, the energy balance (29) deserves a
distinctive treatment whenever the constraint functions (2) take a specific
form. In particular, the circumstance ${\overline{\alpha}}_{\nu}=\alpha_{\nu}$
play the crucial role for the conservation of the energy of the system. Let us
start from the following

###### Lemma 1

For a fixed $\nu$ from $1$ up to $k$, ${\overline{\alpha}}_{\nu}=\alpha_{\nu}$
if and only if $\alpha_{\nu}$ is a homogeneous function of degree $1$ w. r. t.
${\dot{q}}_{1}$, $\dots$, ${\dot{q}}_{m}$.

Proof. We simply turn to the Euler’s theorem:
${\overline{\alpha}}_{\nu}=\sum\limits_{r=1}^{m}{{\dot{q}}_{r}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}=\alpha_{\nu}$
if and only if $\alpha_{\nu}$ is a homogeneous function of degree $1$ with
respect to the variables ${\dot{q}}_{1}$, $\dots$, ${\dot{q}}_{m}$.
$\quad\square$

Let us now assume that each of the nonholonomic constraints verifies

${\overline{\alpha}}_{\nu}=\alpha_{\nu}\quad\textrm{for
any}\;\;\nu=1,\dots,k.$ (38)

Then, the energy balance simplifies according to the following statement.

###### Proposition 3

If (38): holds, the energy balance (29) takes the simpler form

$\dfrac{d}{dt}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{i}}-{\cal L}^{*}\right)=-\dfrac{\partial{\cal
L}^{*}}{\partial t}+\sum\limits_{\nu=1}^{k}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}\dfrac{\partial\alpha_{\nu}}{\partial t}.$ (39)

Proof. Owing to (38) the terms with $\dfrac{\partial{\cal L}^{*}}{\partial
q_{m+\nu}}$ in (29) are null. Moreover, the definition (30) shows
${\overline{B}}_{\nu}=\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial
t}=\dfrac{\partial\alpha_{\nu}}{\partial t}$, hence
$\sum\limits_{\nu=1}^{k}{\overline{B}}_{\nu}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}=\sum\limits_{\nu=1}^{k}\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial
t}\dfrac{\partial T}{\partial{\dot{q}}_{m+\nu}}$ and (39) is proved.
$\quad\square$

###### Remark 2

The presence of the rheonomic contributions on the right side of equality (40)
is easily explainable: $\dfrac{\partial{\cal L}^{*}}{\partial t}$ can be not
null either because of the non–stationarity of the holonomic constraints
(hence $\dfrac{\partial T^{*}}{\partial t}\not=0$) or because of the presence
of $t$ in the forces (then $\dfrac{\partial U}{\partial t}\not=0$). On the
other hand, the possible non–stationarity of the nonholonomic constraints (3)
gives rise to the terms containing $\dfrac{\partial\alpha_{\nu}}{\partial t}$.

###  4.1 Linear nonholonomic constraints

A significant circumstance of validity of assumption is the case of linear
nonholonomic constraints of the form (3). Indeed, the linear functions
$\sum\limits_{j=1}^{m}\alpha_{\nu,j}(q_{1},\dots,q_{n},t){\dot{q}}_{j}$,
$\nu=1,\dots,k$, are homogeneous functions of degree $1$ w. r. t.
${\dot{q}}_{1}$, $\dots$, ${\dot{q}}_{m}$ and Lemma 1 is applicable. By virtue
of (38) the coefficients (2) are
${\overline{B}}_{\nu}=\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial\alpha_{\nu,i}}{\partial
t}$, hence the energy balance (39) takes the form

$\dfrac{d}{dt}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{i}}-{\cal L}^{*}\right)=-\dfrac{\partial{\cal
L}^{*}}{\partial
t}+\sum\limits_{\nu=1}^{k}\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}\dfrac{\partial\alpha_{\nu,i}}{\partial t}.$ (40)

Concerning the linear stationary case, the balance (40) assumes the form
pertinent to holonomic systems:

###### Corollary 2

For a system such that $\dfrac{\partial{\cal L}^{*}}{\partial t}=0$ and
submitted to stationary linear kinematic constraints (3) with
$\alpha_{\nu,j}=\alpha_{\nu,j}(q_{1},\dots,q_{n})$ for each $\nu=1,\dots,k$
and $j=1,\dots,m$, the quantity
$I(q_{1},\dots,q_{m},{\dot{q}}_{1},\dots,{\dot{q}}_{m})=\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{i}}-{\cal L}^{*}$ is conserved. Assuming for
$T^{*}$ the form (21) with $g_{i,j}$, $b_{i}$ and $c$ not depending on $t$
explicitly, $i,j=1,\dots,n$, the constant of motion is

$I=\sum\limits_{r,s}^{m}\left(\frac{1}{2}g_{r,s}+\frac{1}{2}\sum\limits_{\nu,\mu=1}^{k}g_{m+\nu,m+\mu}\alpha_{\nu,r}\alpha_{\mu,s}+\sum\limits_{\nu=1}^{k}g_{r,m+\nu}\alpha_{\nu,s}\right){\dot{q}}_{r}{\dot{q}}_{s}-U-c.$
(41)

The conserved quantity, examined in [9] as well as in other textbooks, is the
generalized energy integral, or Jacobi integral, of the Lagrangian ${\cal
L}^{*}$.

###### Example 8

A very simple model for linear constraints is a point in a pair of points
$P_{1}$ and $P_{2}$ constrained on vertical plane, keeped at constant distance
$2\ell$ one from the other and moving in a way such that the middle point’s
velocity is along the direction $P_{1}-P_{2}$ ([6]). By employing the angle
$q_{1}$ that $P_{2}-P_{1}$ forms with the $x$–axis and the coordinates
$(q_{1},q_{2})$ of the middle point, the holonomic setup is
$x_{1}=q_{1}+\ell\cos q_{3}$, $y_{1}=q_{2}+\ell\sin q_{3}$,
$x_{2}=q_{1}-\ell\sin q_{3}$, $y_{2}=q_{2}-\ell\sin q_{3}$. The linear
nonholonomic constraint is ${\dot{q}}_{2}\sin q_{1}-{\dot{q}}_{3}\cos q_{1}=0$
which can be made explicit by writing ${\dot{q}}_{3}={\dot{q}}_{2}\tan q_{1}$.
The function (21) is
$T^{*}=\frac{1}{2}(M_{1}+M_{2})\left(\ell^{2}{\dot{q}}_{1}^{2}+(1+\tan^{2}q_{1}){\dot{q}}_{2}^{2}\right)$
and, assuming that the $y$–axis points at the upward vertical direction, (28)
is ${\cal L}^{*}=T^{*}-(M_{1}+M_{2}g)q_{3}$. The constant of motion (41) reads
$I=\frac{1}{2}(M_{1}+M_{2})\left(\ell^{2}{\dot{q}}_{1}^{2}+(1+\tan^{2}q_{1}){\dot{q}}_{2}^{2}\right)+(M_{1}+M_{2}g)q_{3}$.

We incidentally remark that the model is different form the one corresponding
to the requests $\overline{P_{1}P_{2}}=\ell$,
$|{\dot{P}}_{1}|=|{\dot{P}}_{2}|$ (equidistant points and equal intensity of
velocities, see [15]): actually, in that case the restrictions yield to the
nonlinear condition ${\dot{q}}_{1}({\dot{q}}_{2}\sin q_{1}-{\dot{q}}_{3}\cos
q_{1})=0$.

###### Example 9

We implement (41) for a frequently proposed system ([16], [2]), consisting in
two material points $(P_{1},M_{1})$ and $(P_{2},M_{2})$ on a plane whose
velocities are both orthogonal to the line joining $P_{1}$ with $P_{2}$. The
linear kinematic constraints for $P_{1}\equiv(x_{1},y_{1})$ and
$P_{2}\equiv(x_{2},y_{2})$ are

${\dot{x}}_{1}(x_{2}-x_{1})+{\dot{y}}_{1}(y_{2}-y_{1})=0,\quad{\dot{x}}_{2}(x_{2}-x_{1})+{\dot{y}}_{2}(y_{2}-y_{1})=0.$

Setting $(q_{1},q_{2},q_{3},q_{4})=(x_{1},x_{2},y_{1},y_{2})$ the explicit
kinematic restrictions are
${\dot{q}}_{3}=\frac{q_{2}-q_{1}}{q_{3}-q_{4}}{\dot{q}}_{1}$,
${\dot{q}}_{4}=\frac{q_{2}-q_{1}}{q_{3}-q_{4}}{\dot{q}}_{2}$ and one has, with
respect to (8), (3) (in this case it is $m=k=2$):

$g_{1,1}=g_{3,3}=m_{1},\quad g_{2,2}=g_{4,4}=m_{2},\quad
g_{i,j}=0\;\;\textrm{for}\;i\not=j,\quad\alpha_{1,1}=\alpha_{2,2}=\frac{q_{2}-q_{1}}{q_{3}-q_{4}},\;\;\alpha_{2,1}=\alpha_{2,2}=0$

so that (41) gives the conserved quantity

$I(q_{1},q_{2},q_{3},q_{4},{\dot{q}}_{1},{\dot{q}}_{2})=\frac{1}{2}(m_{1}{\dot{q}}_{1}^{2}+m_{2}{\dot{q}}_{2}^{2})\left(1+\left(\frac{q_{2}-q_{1}}{q_{3}-q_{4}}\right)^{2}\right)-U(q_{1},q_{2},q_{3},q_{4})$

where $U$ is calculated (according to the dynamics of the model) by means of
(27).

We judge wortwhile to remark that, imaging a transfer from Example 5 to
Example 10 (both regard parallel velocities) made by specifiying a request on
the velocity vectors, entails a very different structure of the nonholonomic
constraints (nonlinear towards linear); such an argument is properly
highlighted in (LABEL:benenti).

###### Example 10

A further model, recurrent in literature, consists in considering two material
points $P_{1}$ and $P_{2}$ on a plane whose velocities are perpendicular;
furthermore, the velocity of $P_{1}$ is orthogonal to the straight line
joining $P_{1}$ with $P_{2}$. The model is different from the one discussed in
Example 2, where only the nonlinear nonholonomic constraint (22) is
considered: in the present case the linear kinematic condition ${\cal C}_{1}:$
${\dot{x}}_{1}(x_{2}-x_{1})+{\dot{y}}_{1}(y_{2}-y_{1})=0$ is added, according
to the imposed restriction. The same restriction, holding (22), can be
formulated also stating that the velocity of $P_{2}$ is parallel to the line
joining the two points, that is ${\cal C}_{2}:$
${\dot{x}}_{2}(y_{2}-y_{1})-{\dot{y}}_{2}(x_{2}-x_{1})=0$.

We can easily show that either (22) coupled with ${\cal C}_{1}$ or (22)
coupled with ${\cal C}_{2}$ are equivalent to the pair of linear constraints
${\cal C}_{1}$, ${\cal C}_{2}$. Indeed, it suffices to write the three
conditions as
$\frac{{\dot{x}}_{1}}{{\dot{y}}_{1}}=-\frac{{\dot{y}}_{2}}{{\dot{x}}_{2}}$,
$\frac{{\dot{x}}_{1}}{{\dot{y}}_{1}}=-\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$,
$\frac{{\dot{y}}_{2}}{{\dot{x}}_{2}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ to
conclude. Thus, examining the linear nonholonomic system ${\cal C}_{1}$,
${\cal C}_{2}$ we set $(q_{1},q_{2},q_{3},q_{4})=(x_{1},x_{2},y_{1},y_{2})$ in
order to write (3) as
${\dot{q}}_{3}=-\frac{q_{2}-q_{1}}{q_{4}-q_{3}}{\dot{q}}_{1}$,
${\dot{q}}_{4}=\frac{q_{4}-q_{3}}{q_{2}-q_{1}}{\dot{q}}_{2}$ and, in a very
similar way to the previous example, one finds the conservation of the
quantity (41)

$I(q_{1},q_{2},q_{3},q_{4},{\dot{q}}_{1},{\dot{q}}_{2})=\frac{1}{2}M_{1}{\dot{q}}_{1}^{2}\left(1+\left(\frac{q_{2}-q_{1}}{q_{4}-q_{3}}\right)^{2}\right)+\frac{1}{2}M_{2}{\dot{q}}_{2}^{2}\left(1+\left(\frac{q_{4}-q_{3}}{q_{2}-q_{1}}\right)^{2}\right)-U(q_{1},q_{2},q_{3},q_{4}).$

###  4.2 Nonlinear constraints: homogeneous quadratic functions

A frequently encountered subcategory of nonholonomic constraints (1)
encompasses restrictions of the form

$\sum\limits_{i,j=1}^{n}\gamma_{i,j}^{\nu}(q_{1},\dots,q_{n},t){\dot{q}}_{i}{\dot{q}}_{j}=0,\quad\nu=1,\dots,k.$
(42)

Assuming that the explicit form which singles out ${\dot{q}}_{m+1}$, $\dots$,
${\dot{q}}_{n}$ can be achieved, (2) takes the form

${\dot{q}}_{m+\nu}=\dfrac{\sum\limits_{r,s=1}^{m}\gamma_{r,s}^{\nu}(q_{1},\dots,q_{n},t){\dot{q}}_{r}{\dot{q}}_{s}}{\sum\limits_{i=1}^{m}\beta_{i}^{\nu}(q_{1},\dots,q_{n},t){\dot{q}}_{i}}=\alpha_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t),\qquad\nu=1,\dots,k$
(43)

for appropriate functions $\beta_{i}^{\nu}$. The functions $\alpha_{\nu}$ of
(43) are homogeneous of degree 1 w. r. t. ${\dot{q}}_{1}$, $\dots$,
${\dot{q}}_{m}$, hence the energy balance which pertains to such systems is
(39), where the functions $\gamma_{r,s}^{\nu}$ and $\beta_{i}^{\nu}$ will
appear; in the stationary case the energy is conserved. In a natural way,
constraints (42) appear when the restrictions concern parallelism or
orthogonality of the velocities, or the assignment of equal intensity of the
velocities. A typical and simple instance takes into consideration two points
$P_{1}$, $P_{2}$, for which the three restrictions
${\dot{P}}_{1}\wedge{\dot{P}}_{2}=0$, ${\dot{P}}_{1}\cdot{\dot{P}}_{2}=0$,
$|{\dot{P}}_{1}|^{2}=|{\dot{P}}_{2}|^{2}$ respectively read, in cartesian
coordinates,

$\begin{array}[]{ll}{\dot{x}}_{1}{\dot{y}}_{2}-{\dot{y}}_{1}{\dot{x}}_{2},\;\;{\dot{x}}_{1}{\dot{z}}_{2}-{\dot{z}}_{1}{\dot{x}}_{2}=0&\textrm{parallelism}\\\
{\dot{x}}_{1}{\dot{x}}_{2}+{\dot{y}}_{1}{\dot{y}}_{2}+{\dot{z}}_{1}{\dot{z}}_{2}=0&\textrm{orthogonality}\\\
{\dot{x}}_{1}^{2}+{\dot{y}}_{1}^{2}+{\dot{z}}_{1}^{2}-{\dot{x}}_{2}^{2}-{\dot{y}}_{2}^{2}-{\dot{z}}_{2}^{2}=0&\textrm{same
norm of velocity}\end{array}$

belonging to class (42).

The already performed Examples 1,4 (nonholonomic pendulum), 2 (perpendicular
velocities), 5 (parallel velocities), 7 (equal velocity in norm) are part of
the type (42) of restrictions. It is worthwhile to stress, as some of the
Examples revealed, that adding further resctrictions or specifying mechanisms
regarding the nonholonomic constraints (42) may modify totally the typology of
the restrictions (see Example 9): in this sense, the equivalence in realizing
phisically nonholonomic restrictions by means of either linear or nonlinear
equations claimed in (LABEL:zek) needs to be debated.

###  4.3 Linear affine constraints

Finally, a special situation concerns the nonholonomic systems with affine
constraints, which can be assumed of the form (not encompassed by (3))

$\alpha_{\nu}=\sum\limits_{j=1}^{m}a_{\nu,j}(q_{1},\dots,q_{n},t){\dot{q}}_{j}+c_{\nu}(t),\quad\nu=1,\dots,k$
(44)

with $c_{\nu}$ non zero function. In that case
${\bar{\alpha}}_{\nu}=\alpha_{\nu}-c_{\nu}$, so that
${\overline{B}}_{\nu}=\dfrac{\partial{\overline{\alpha}}_{\nu}}{\partial t}$
and (29) is

$\dfrac{d}{dt}\left(\sum\limits_{i=1}^{m}{\dot{q}}_{i}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{i}}-{\cal L}^{*}\right)+c_{\nu}\dfrac{\partial{\cal
L}^{*}}{\partial
q_{m+\nu}}-\sum\limits_{\nu=1}^{k}\left(\sum\limits_{j=1}^{m}\dfrac{\partial\alpha_{\nu,j}}{\partial
t}{\dot{q}}_{j}+{\dot{c}}_{\nu}\right)\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}=-\dfrac{\partial{\cal L}^{*}}{\partial t}.$ (45)

The stationary case
$\alpha_{\nu,j}(_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m})$,
$c_{\nu}$ constant provides the conservation of the quantity in round brackets
(energy), whenever ${\cal L}^{*}$ does not depend explicitly on $t$ and the
forces verify special properties, as it is described in the following

###### Example 11

For a particle in ${\mathbb{R}}^{3}$ with mass $M$ and submitted to the affine
constraint $ax{\dot{y}}+b{\dot{x}}y+c-{\dot{z}}=0$, $c\not=0$, we set
$(q_{1},q_{2},q_{3})=(x,y,z)$ so that (44) writes
${\dot{q}}_{3}=aq_{1}{\dot{q}}_{2}+bq_{2}{\dot{q}}_{1}+c$ (in this case $m=2$
and $k=1$); moreover

${\cal
L}^{*}=\frac{1}{2}M[(1+a^{2}q_{1}^{2}){\dot{q}}_{1}^{2}+(1+b^{2}q_{2}^{2}){\dot{q}}_{2}^{2}+2abq_{1}q_{2}{\dot{q}}_{1}{\dot{q}}_{2}+2c(aq_{1}{\dot{q}}_{2}+bq_{2}{\dot{q}}_{1})]-V(q_{1},q_{2},q_{3}).$

Whenever $V=V(q_{1},q_{2})$, (29) supplies the conservation of the quantity

$\frac{1}{2}M({\dot{q}}_{1}^{2}+{\dot{q}}_{2}^{2})-\frac{1}{2}M(aq_{1}{\dot{q}}_{2}+bq_{2}{\dot{q}}_{1}+c)(aq_{1}{\dot{q}}_{2}+bq_{2}{\dot{q}}_{1}-c).$

A necessary and sufficient condition (in terms of geometrical properties of
the constraint manifold) in order that (29) provides the energy integral is
discussed and proved in [5].

##  5 Conclusions

The equations of motions for nonholonomic nonlinear systems are presented in
the double version (7) and (12), each of them showing advantageous points. The
context of nonlinear restrictions leads us to make use of the generalized
velocities as kinematic variables (instead of quasi–coordinates), thus
favouring the extension of the Voronec’s method for linear kinematic
constraints to the nonlinear case.

The calculation of the power of the forces, by way of the equations of motion,
generates the formula (29), showing in an unified and consistent way the rate
of change in time of the energy espressed by the independent velocities (the
function in round brackets in (29)) in terms of the contributions due to the
constraints forces (by means of the coefficients ${\overline{\alpha}}_{\nu}$
and ${\overline{B}}_{\nu}$) and of the possible explicit dependence of the
restriction or of the forces on time (term on the right hand side).

The arrangement of the energy balance equation is suitable in order to
identify the type of nonholonomic constraints exhibiting
${\overline{\alpha}}_{\nu}=\alpha_{\nu}$, which is the key point for the
purposes of inferring the first integral of energy. At the same time, in some
special modellistic circumstances the setup (29) shows directly the
appropriate simplifcations, as in the case (36) or in the linear case. Several
examples of simple but meaningful systems have been performed.

The condition (38) fits for kinematic constraints whose explicit formulation
(2) is a homogeneous function of degree $1$: the models frequently adopted in
literature and accessible for implementations concerning special restrictions
on the velocities (parallelism, equal norms, orthogonality) fall in this
category. Theoretically, it is easy to extend the category to restrictions (1)
which are homogeneous functions of arbitrary degree w. r. t. the generalized
velocities ${\dot{q}}_{1}$, $\dots$, ${\dot{q}}_{n}$; however, from the
experimental point of view this may produce a not concrete realisation.

Beyond the mere aspect of the balance of energy, the subject nonlinear
constraints presents interesting questions somehow unexplored in literature
and sometimes misleading. These aspects, just mentioned in the paper, concern
the equivalence of linear kinematics models with nonlinear restrictions, the
correctness of mergering part of the constraints giving rise to new conditions
(tipically, two linear constraints are joined to form a quadratic condition);
the Hamel–Appell example of a system with nonlinear nonholonomic constraint
obtained by linear kinematic condition is a point of reference in this sense.

A sistematic procedure for readily comparing the equations of motion whenever
different sets of independent velocities are selected is also not secondary in
order to take into the right consideration the local use of (2).

The just mentioned open points are the next purpose of the research on
nonlinear kinematic constraints.

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