#  ${\check{\rm C}}$etaev condition for nonlinear nonholonomic systems and
homogeneous constraints

F. Talamucci  DIMAI, Dipartimento di Matematica e Informatica “Ulisse Dini”,
Università degli Studi di Firenze, Italy
e-mail<EMAIL_ADDRESS>
###### Abstract

We first present a way to formulate the equations of motion for a nonholonomic
system with nonlinear constraints with respect to the velocities. The
formulation is based on the ${\check{\rm C}}$etaev condition which aims to
extend the practical method of virtual displacements from the holonomic case
to the nonlinear nonholonomic one. The condition may appear in a certain sense
artificial and motivated only to coherently generalize that concerning the
holonomic case. In the second part we show that for a specific category of
nonholonomic constraints (homogeneous functions with respect to the
generalized velocities) the ${\check{\rm C}}$etaev condition reveals the same
physical meaning that emerges in systems with holonomic constraints. In
particular the aspect of the mechanical energy associable to the system is
analysed.

2010 Mathematics Subject Classification: 37J60, 70F25, 70H03.

Keywords: Nonholonomic mechanical systems -Virtual displacements for nonlinear
kinematic constraints - ${\check{\rm C}}$etaev condition - Homogeneous
constraints - Mechanical energy of nonholonomic systems.

##  1 Introduction

The vast and growing interest in systems with kinematic constraints is
motivated by a series of significant applications, among which we quote the
motion of vehicles. From a modelling point of view it is a question of
extending the consolidated procedure inherent to holonomic systems (i. e.
those subject to geometric constraints which concern the spatial coordinates
only) by admitting the presence of restrictions on the velocities (kinematic
constraints) and in some cases even more complex limitations. The transition
from the linear case (in some way similar to the holonomic case), namely the
linear dependence of the constraints equations on the velocities, to that of
nonlinear kinematic constraints considerably complicates the treatment from a
formal point of view. If on the one hand the treatment of linear nonholonomic
systems dates back to over a century ago (for an exhaustive historical review,
see the fundamental text [12] and the more recent review [7]), on the other
the formal study of the nonlinear ones is recent and not definitive: even the
question of the actual implementation of such constraints, starting from the
Appell–Hamel example [1], is debated ([12], [19]).

In a very generic and not drastic way we can divide the mathematical
approaches to nonholonomic systems into three categories: a first method that
is certainly expanding and relies on high–profile publications ([2], [8],
[11], [10], [14], [17]) is based on differential geometry and extends the
basic theory of holonomic systems to more complex concepts (jet manifolds, jet
bundles).

A second method takes advantage of the possibility of deriving the equations
of motion from a variational principle ([15], [13]), likewise the Hamilton
principle in the holonomic case; as far as we know the situation is still open
and more than one publication warns about some critical aspects regarding this
procedure ([3], [6]).

The third approach we mention, which is the one we will develop in the paper,
takes into account the virtual displacements ([9] and again [13] for a
comprehensive analysis of the various types of displacements) associable to
the constrained system, as well as the directions tangential to the
configuration space establish a way to write a set of independent equations in
the range of holonomic systems. A nonholonomic system with linear kinematic
constraints can be treated essentially as a system with geometric constraints,
by considering a linear subspace of the admitted directions.

However, for nonlinear kinematic constraints a plain geometric description of
the virtual displacements does not exist: a frequently accepted hypothesis for
constituting the set of virtual displacements is the so-called ${\check{\rm
C}}$etaev condition ([4], [16]), which it is an accepted axiom leading to the
expected equations of motion but does not present a real justification from
the laws of mechanics, as far as we can understand ([9]).

The first part of the work (Section 1) is just right dedicated to the modeling
formulation the ${\check{\rm C}}$etaev condition, from which the equations for
nonholonomic nonlinear constrained systems are derived. A properly combination
of the equations lead to an information about the mechanical energy of the
system (as well as in the holonomic case occurs) and the energy equation is
presented in the same Section.

In the second part (Section 2) the intention is to highlight how the validity
of a property of the constraint equations (namely (24)) entails the removal of
discrepancies that emerge between the mathematical feature of the ${\check{\rm
C}}$etaev condition and the physical reading associated with it: actually, the
virtual displacements prescribed by the condition play in this case the same
role as the velocities consistent with the instantaneous configurations of the
system, as well as it occurs for holonomic systems. From the mathematical
point of view, it is shown that the above “conciliant” condition is satisfied
whenever the constraints are homogeneous equations with respect to the
generalized velocities; this is the central point of the work. Even if the few
examples described here do not cover the multiplicity of nonholonomic models,
it is true that a large part of them are implemented by conditions showing
homogeneity in the kinetic variable: this motivates the final consideration of
Section 3, where the question is reversed, that is whether the natural
condition (24) is an exclusive prerogative of those systems where the
nonholonomic constraints can be formulated via homogeneous equations.

###  1.1 The equations of motion

Let us consider a mechanical system whose configurations are determined by $n$
the lagrangian coordinates $q_{1}$, $\dots$, $q_{n}$; we recall the Lagrangian
equations of motion

$\dfrac{d}{dt}\dfrac{\partial T}{\partial{\dot{q}}_{j}}-\dfrac{\partial
T}{\partial q_{j}}={\cal F}^{(j)}+{\cal R}^{(j)}\qquad j=1,\dots,n$ (1)

where $T(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t)$ is the
kinetic energy, ${\cal F}^{(j)}$, $j=1,\dots,n$ the generalized forces acting
on the system and ${\cal R}^{(j)}$ the constraints forces due to the action of
the $k<n$ kinematic constraints

$\phi_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t)=0,\qquad\nu=1,\dots,k$
(2)

The constraints are independent with respect to the kinematic variables, in
the sense that the rank of the $k\times n$ matrix is full:

$rank\,\left(\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{j}}\right)_{\footnotesize{\begin{array}[]{l}\nu=1,\dots,k\\\
j=1,\dots,n\end{array}}}=k$ (3)

We can assume, since it is marginal in this discussion, that the active forces
admit a potential $U$:

${\mathcal{F}}^{(j)}=\dfrac{\partial}{\partial
q_{j}}U(q_{1},\dots,q_{n},t),\quad j=1,\dots,n\quad$ (4)

Hence the equations (1) take the form

$\dfrac{d}{dt}\dfrac{\partial{\cal
L}}{\partial{\dot{q}}_{j}}-\dfrac{\partial{\cal L}}{\partial q_{j}}={\cal
R}^{(j)}\qquad j=1,\dots,n.$ (5)

The system of $n+k$ equations (5), (2) contain the $2n$ unknown functions
$q_{j}$ and ${\cal R}^{(j)}$, $j=1,\dots,n$.

At this point, we outline two different procedures to deal with the system:

  * $(i)$

to formulate an hypothesis for the constraints forces based on the method of
lagrangian multipliers,

  * $(ii)$

to identify the set of virtual displacements similarly to the case of
holonomic constraints, that is to provide the admitted variations $\delta
q_{1}$, $\dots$, $\delta q_{n}$ of the coordinates consistent with the
restrictions (2) (virtual variations).

In the first case one sets

${\cal
R}^{(j)}=\sum\limits_{\nu=1}^{k}\lambda_{\nu}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{j}},\qquad
j=1,\dots,n$ (6)

where $\lambda_{\nu}$, $\nu=1,\dots,k$, are unknown coefficients; in this way
the unknown quantities equalize the number of equations. Concerning $(ii)$, a
generally accepted prospect is to define the virtual displacements as those
which verify

$\sum\limits_{i=1}^{n}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{i}}\delta
q_{i}=0,\qquad i=1,\dots,\nu$ (7)

known as ${\check{\rm C}}$etaev condition; the virtual work of the constraints
forces along the displacements verifying (7) is assumed to be null:

$\sum\limits_{i=1}^{n}{\cal R}^{(i)}\delta q_{i}=0.$

The two approaches have in common the fact that the virtual work of the forces
(6) is zero whenever (7) holds:

$\sum\limits_{i=1}^{n}{\cal R}^{(i)}\delta
q_{i}=\sum\limits_{\nu=1}^{k}\sum\limits_{i=1}^{n}\lambda_{\nu}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{i}}\delta
q_{i}=0$ (8)

Nevertheless, the starting point (7) yields an useful method which has the
advantage of not making the multipliers $\lambda_{\nu}$ appear and of
selecting a set of independent kinetic variables: actually, we see that
condition (3) makes possible the explicit writing of conditions (2) in the
following way (without losing generality, except for re-enumerating the
variables):

${\dot{q}}_{m+\nu}=\alpha_{1}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)\qquad\nu=1,\dots,k$
(9)

with $m=n-k$. The parameters ${\dot{q}}_{r}$, $r=1,\dots,m$, play the role of
independent kinetic variables, and correspondingly the virtual displacements
$\delta q_{1}$, $\dots$, $\delta q_{m}$ can be considered as independent; in
turn, (9) allows to express the dependent displacements as

$\delta
q_{m+\nu}=\sum\limits_{r=1}^{m}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\delta
q_{r},\qquad\nu=1,\dots,k$ (10)

To verify it, we use the role of ${\dot{q}}_{r}$, $r=1,\dots,m$ as independent
variables, differentiate (2) and write

$\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{r}}+\sum\limits_{\mu=1}^{k}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{m+\nu}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}=0\quad\Longrightarrow\quad\sum\limits_{r=1}^{m}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{r}}\delta
q_{r}+\sum\limits_{\mu=1}^{k}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{m+\mu}}\underbrace{\sum\limits_{r=1}^{m}\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{r}}\delta
q_{r}}_{\delta q_{m+\mu}}=0,\quad\nu=1,\dots,k.$

###### Remark 1

Having in mind a comparison with the holonomic case and referring to the case
of $N$ vectors ${\bf r}_{i}(q_{1},\dots,q_{n},t)$, $i=1,\dots,N$ locating the
points of the system, assumption (7) means that the customary condition

$\delta{\bf r}_{i}=\sum\limits_{j=1}^{n}\dfrac{\partial{\bf r}_{i}}{\partial
q_{j}}\delta q_{j},\quad i=1,\dots,N$

holding for a holonomic system is replaced in the case of kinematic
constraints (2) by

$\delta{\bf r}_{i}=\sum\limits_{j=1}^{n}\dfrac{\partial{\dot{\bf
r}}_{i}}{\partial{\dot{q}}_{j}}\delta q_{j},\quad i=1,\dots,N.$ (11)

###### Remark 2

For linear kinematic constraints

$\sum\limits_{i=1}^{n}\sigma_{\nu,i}(q_{1},\dots,q_{n},t){\dot{q}}_{i}+\zeta_{1}(q_{1},\dots,\dots,q_{n},t)=0,\quad\nu=1,\dots,k$

admitting the explicit expressions

${\dot{q}}_{m+\nu}=\sum\limits_{j=1}^{m}\alpha_{\nu,j}(q_{1},\dots,q_{n},t){\dot{q}}_{j}+\beta_{\nu}(q_{1},\dots,q_{n},t),\quad\nu=1,\dots,k$
(12)

the condition (10) reduces to

$\delta
q_{m+\nu}=\sum\limits_{r=1}^{m}\alpha_{\nu,r}(q_{1},\dots,q_{n},t)\delta
q_{r},\qquad\nu=1,\dots,k$

and corresponds to the usual assumption on virtual variations adopted in texts
dealing with nonholonomic theory for linear kinematic constraints ([12]).

The approach (10) requires to express coherently the Lagrangian function
${\cal L}$ by means of the only independent kinetic variables:

${\cal L}^{*}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)={\cal
L}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},\alpha_{1}(\cdot),\dots,\alpha_{k}(\cdot),t)$
(13)

where $\alpha_{\nu}(\cdot)$ stands for
$\alpha_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t)$,
$\nu=1,\dots,k$. In terms of ${\cal L}^{*}$ the equations of motion calculated
starting from (1) and considered along the virtual displacements (7), (10)
take the form (see [18] for details)

$\dfrac{d}{dt}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{r}}-\dfrac{\partial{\cal L}^{*}}{\partial
q_{r}}-\sum\limits_{\nu=1}^{k}\dfrac{\partial{\cal L}^{*}}{\partial
q_{m+\nu}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}_{r}}}-\sum\limits_{\nu=1}^{k}B_{r}^{\nu}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}=0$ (14)

for $r=1,\dots$, where

$B_{r}^{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)=\dfrac{d}{dt}\left(\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\right)-\dfrac{\partial\alpha_{\nu}}{\partial
q_{r}}-\sum\limits_{\mu=1}^{k}\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{r}}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}$ (15)

for $r=1,\dots,m$ and $\nu=1,\dots,k$. They represent an alternative formal
way (without multipliers) to the set (5), (6), (2). The linear case (12)
refers to the Voronec equations for nonholonomic systems with linear kinematic
constraints ([12]).

The unknown functions in (14) are $q_{1}$, $\dots$, $q_{n}$ but only the
derivatives ${\dot{q}}_{r}$, ${\mathop{q}\limits^{..}}_{r}$ $r=1,\dots,m$ are
present, owing to (9), once the variables
$({\dot{q}}_{k+1},\dots,{\dot{q}}_{n})$ that are present in $\frac{\partial
T}{\partial{\dot{q}}_{m+\nu}}$ of (14) have been expressed in terms of
$(q_{1},\dots,q_{n},$ ${\dot{q}}_{1},\dots,{\dot{q}}_{m},t)$. Evidently, the
case of merely holonomic constraints which corresponds to suppress all the
terms containing the functions $\alpha_{\nu}$ brings back to the ordinary
Euler–Lagrange equations of motion for ${\cal L}^{*}={\cal L}$.

###### Remark 3

The coefficients in (14) can be expressed in terms of the original functions
$\phi_{\nu}$ of (2) by making use of the relations

$\dfrac{\partial\phi_{\nu}}{\partial
q_{i}}+\sum\limits_{\mu=1}^{k}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{m+\mu}}\dfrac{\partial\alpha_{\mu}}{\partial
q_{i}}=0,\quad\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{r}}+\sum\limits_{\mu=1}^{k}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{m+\mu}}\dfrac{\partial\alpha_{\mu}}{\partial{\dot{q}}_{r}}=0$

holding for any $\nu=1,\dots,k$, $i=1,\dots,n$, $r=1,\dots,m$. This means that
the structure of the equations does not depend on the choice (9) for making
explicit (2).

###  1.2 An equation for the energy of the system

We recall the Hamiltonian function

${\cal
E}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t)=\sum\limits_{i=1}^{n}{\dot{q}}_{i}\dfrac{\partial{\cal
L}}{\partial{\dot{q}}_{i}}-{\cal L}$ (16)

and we ascribe to it the role of energy of the system. At the same time, we
are interested in the properties of the energy function formulated by only the
independent kinetic variables, that is

${\cal
E}^{*}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)=\sum\limits_{r=1}^{m}{\dot{q}}_{r}\dfrac{\partial{\cal
L}^{*}}{\partial{\dot{q}}_{r}}-{\cal L}^{*}.$ (17)

We can say that (16) is the energy that is naturally associated to (5) with
the Lagrangian ${\cal L}$, while (17) is spontaneously connected to (14),
where only the independent velocities are present. Although in a much more
complex formal context of geometric type, the two notions of energy can be
traced back to those present in [5].

Even if we express (16) and (17) by means of the same set of variables,
replacing in the first one the dependent velocities ${\dot{q}}_{m+\nu}$,
$\nu=1,\dots,k$ by virtue of (9), we do not find the same function: actually,
it can be seen without difficulty that the following relation occurs:

${\cal E}^{*}={\cal
E}+\sum\limits_{\nu=1}^{k}({\overline{\alpha}}_{\nu}-\alpha_{\nu})\dfrac{\partial{\cal
L}}{\partial{\dot{q}}_{m+\nu}}$ (18)

where we set

${\overline{\alpha}}_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)=\sum\limits_{r=1}^{m}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}{\dot{q}}_{r},\quad\nu=1,\dots,k$
(19)

At the same time, it is possible to achieve the two equations that ${\cal E}$
and ${\cal E}^{*}$ verify, coming from (5) and (14) by means of the standard
procedure of multiplying by the generalized velocities, summing up and
rearranging the expressions:

$\dfrac{d{\cal E}}{dt}=-\dfrac{\partial{\cal L}}{\partial
t}+\sum\limits_{\nu=1}^{k}\sum\limits_{i=1}^{n}\lambda_{\nu}\dfrac{\partial\phi_{\nu}}{\partial{\dot{q}}_{i}}{\dot{q}}_{i}.$
(20)

$\dfrac{d{\cal E}^{*}}{dt}=-\dfrac{\partial{\cal L}^{*}}{\partial
t}+\sum\limits_{\nu=1}^{k}({\overline{\alpha}}_{\nu}-\alpha_{\nu})\dfrac{\partial{\cal
L}^{*}}{\partial
q_{m+\nu}}+\sum\limits_{\nu=1}^{k}{\overline{B}}_{\nu}\dfrac{\partial
T}{\partial{\dot{q}}_{m+\nu}}$ (21)

where ${\overline{\alpha}}_{\nu}$ is defined in (19) and (see also (15))

${\overline{B}}_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)=\sum\limits_{r=1}^{m}B_{r}^{\nu}{\dot{q}}_{r}=\dfrac{d}{dt}({\overline{\alpha}}_{\nu}-\alpha_{\nu})-\sum\limits_{\mu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial
q_{m+\mu}}({\overline{\alpha}}_{\mu}-\alpha_{\mu})+\dfrac{\partial\alpha_{\nu}}{\partial
t},\qquad\nu=1,\dots,k.$ (22)

Evidently, if the constraints (2) are absent (holonomic case), both (20) and
(21) reduce to the standard energy balance for ${\cal L}^{*}={\cal L}$:

$\dfrac{d}{dt}\dfrac{\partial{\cal
L}}{\partial{\dot{q}}_{i}}-\dfrac{\partial{\cal L}}{\partial
q_{i}}=0\;\;\Rightarrow\;\;\dfrac{d}{dt}\left(\sum\limits_{i=1}^{n}{\dot{q}}_{i}\dfrac{\partial{\cal
L}}{\partial{\dot{q}}_{i}}-{\cal L}\right)=-\dfrac{\partial{\cal L}}{\partial
t}$ (23)

##  2 The case ${\overline{\alpha}}_{\nu}=\alpha_{\nu}$

As it is exptected from the governing equations introduced above, the special
situation when the functions in (9) verify the condition

${\overline{\alpha}}_{\nu}=\alpha_{\nu}\quad\textrm{for
any}\quad\nu=1,\dots,k$ (24)

where ${\overline{\alpha}}_{\nu}$ is defined in (19), deserves to be analyzed.

Many of the nonholonomic models studied in literature starting with the first
examples fall into the category (24): for istance, rigid bodies rolling
without sliding on a plane or on a surface, knife edges or sleighs sliding on
a horizontal plane, some joints simulating the basic functioning of a vehicle
are formulated by linear homogeneous constraints of the type

$\sum\limits_{i=1}^{n}\sigma_{\nu,i}(q_{1},\dots,q_{n},t){\dot{q}}_{i}=0\qquad\nu=1,\dots,k$
(25)

which verify (24).

Again, the kinematic constraints with homogeneous quadratic functions

$\sum\limits_{i,j=1}^{n}a_{i,j}^{(\nu)}(q_{1},\dots,q_{n},t){\dot{q}}_{i}{\dot{q}}_{j}=0,\qquad\nu=1,\dots,k$
(26)

which include the conditions of parallel velocities
(${\dot{P}}_{1}\wedge{\dot{P}}_{2}={\bf 0}$), orthogonal velocities
(${\dot{P}}_{1}\cdot{\dot{P}}_{2}=0$) or same lenght
($|{\dot{P}}_{1}|=|{\dot{P}}_{2}|$), fulfill the request (24).

###### Remark 4

Simple examples of constaints that do not verify (24) are affine nonholonomic
constraints of degree $p$, with $p$ positive integer:

$\sum\limits_{j=1}^{n}\sigma_{\nu,j}(q_{1},\dots,q_{n},t){\dot{q}}_{j}^{p}+\zeta_{\nu}(q_{1},\dots,q_{n},t)=0,\quad\nu=1,\dots,k$
(27)

The constraint on the magnitude of the velocity $|{\dot{P}}|=C(t)$, with
$C(t)$ given nonnegative function, is part of (27) with $p=2$; for $p=1$ we
have linear affine constraint. The explicit form (9) is of the type

$\alpha_{\nu}={\dot{q}}_{m+\nu}=(\pm
1)^{p+1}\left(\sum\limits_{j=1}^{m}\alpha_{\nu,j}(q_{1},\dots,q_{n},t){\dot{q}}_{j}^{p}+\beta_{\nu}(q_{1},\dots,q_{n},t)\right)^{1/p},\qquad\nu=1,\dots,k$
(28)

for suitable coefficients $\alpha_{\nu,j}$ and $\beta_{\nu}$. The function
(19) can be put in the form

${\overline{\alpha}}_{\nu}=\alpha_{\nu}\dfrac{\sum\limits_{i=1}^{m}\alpha_{\nu,i}{\dot{q}}_{i}^{p}}{\sum\limits_{j=1}^{m}\alpha_{\nu,j}{\dot{q}}_{j}^{p}+\beta_{\nu}}\qquad\nu=1,\dots,k$

so that (24) is valid if and only if $\beta_{\nu}=0$, which means that the
constraints are homogeneous.

Focusing now on the class of constraints (24), at least three essential
properties have to be pointed out, whenever (24) holds:

  * $(1)$

the virtual displacements (11), which may appear somehow artifical in their
definition, reveal the physical meaning of settling along the virtual
velocities, in the following sense: formally referring to the $N$–points
system ${\bf r}_{i}(q_{1},\dots,q_{n},t)$ as in Remark 1, the velocity
${\dot{\bf r}_{i}}=\sum\limits_{j=1}^{n}\dfrac{\partial{\bf r}_{i}}{\partial
q_{r}}{\dot{q}}_{j}+\dfrac{\partial{\bf r}_{i}}{\partial t}$ makes us consider
the term ${\widehat{\dot{\bf r}}}_{i}=\sum\limits_{j=1}^{n}\dfrac{\partial{\bf
r}_{i}}{\partial q_{r}}{\dot{q}}_{j}$ as the component consistent with the
blocked configuration (virtual velocity). In the presence of (9) one has

${\widehat{\dot{\bf r}}}_{i}=\sum\limits_{r=1}^{m}\dfrac{\partial{\bf
r}_{i}}{\partial
q_{r}}{\dot{q}}_{r}+\sum\limits_{\nu=1}^{k}\dfrac{\partial{\bf
r}_{i}}{\partial
q_{m+\nu}}\alpha_{\nu}=\left(\sum\limits_{r=1}^{m}\dfrac{\partial{\bf
r}_{i}}{\partial q_{r}}+\sum\limits_{\nu=1}^{k}\dfrac{\partial{\bf
r}_{i}}{\partial
q_{m+\nu}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\right){\dot{q}}_{r},\qquad
i=1,\dots,N$ (29)

where the last equality is due to (24). On the other hand, the calculation of
(11) leads to

$\delta{\bf r}_{i}=\sum\limits_{j=1}^{n}\dfrac{\partial{\dot{\bf
r}}_{i}}{\partial{\dot{q}}_{j}}\delta
q_{j}=\sum\limits_{r=1}^{m}\left(\dfrac{\partial{\bf r}_{j}}{\partial
q_{r}}+\sum\limits_{\nu=1}^{k}\dfrac{\partial{\bf r}_{i}}{\partial
q_{m+\nu}}\dfrac{\partial\alpha_{\nu}}{\partial{\dot{q}}_{r}}\right)\delta
q_{r},\quad i=1,\dots,N$

where (10) has been taken into account. Hence, we see that, a part from the
formal appearance through $\delta q_{r}$ or ${\dot{q}}_{r}$, the two vectors
display the same direction.

  * $(2)$

At the same time, the vanishing $\sum\limits_{i=1}^{n}{\cal R}^{(i)}\delta
q_{i}=0$ (see (8)) does actually represent the absence of virtual work
$\sum\limits_{i=1}^{n}{\cal R}^{(i)}{\dot{q}_{i}}=0$: a wat to check that goes
through the relation

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

which can be deduced from (5).

  * $(3)$

The relation (18) shows that (16) and (17) overlap, that is calculating the
energy by considering the Lagrangian function with all the velocities
${\dot{q}}_{i}$, $i=1,\dots,n$ or the restricted function (13) of the
independent velocities only ${\dot{q}}_{r}$, $r=1,\dots,m$, leads to the same
result.

  * $(4)$

The balance equation (21) reduces to (see also (22))

$\dfrac{d{\cal E}^{*}}{dt}=-\dfrac{\partial{\cal L}^{*}}{\partial
t}+\sum\limits_{\nu=1}^{k}\dfrac{\partial\alpha_{\nu}}{\partial
t}\dfrac{\partial T}{\partial{\dot{q}}_{m+\nu}}=-\dfrac{\partial{\cal
L}}{\partial t}$ (30)

and it reveals the first integral of motion ${\cal E}={\cal E}^{*}$, whenever
the Lagrangian function ${\cal L}$ does not depend explicitly on time $t$.

###### Remark 5

The second equality in (30) shows that the energy may be conserved even if the
constraints (2) depend explicitly on time $t$ (rheonomic constraints): a
simple example is the motion of a point $P$ of mass $M$ whose velocity has to
at any time the direction of $\overrightarrow{PQ}$, where the motion of $Q$ is
assigned by the functions $x_{Q}(t),y_{Q}(t),z_{Q}(t)$ (pursuing motion). The
constraints (9) are

$\left\\{\begin{array}[]{ll}{\dot{q}}_{2}=\dfrac{y_{Q}(t)-q_{2}}{x_{Q}(t)-q_{1}}{\dot{q}}_{1}=\alpha_{1,1}(q_{1},q_{2},t){\dot{q}}_{1},&\\\
{\dot{q}}_{3}=\dfrac{z_{Q}(t)-q_{3}}{x_{Q}(t)-q_{1}}{\dot{q}}_{1}=\alpha_{2,1}(q_{1},q_{3},t){\dot{q}}_{1}\end{array}\right.$

and they verify (24), as it can be easily checked. If there are not active
forces, one has ${\cal
L}=T=\frac{1}{2}M({\dot{q}}_{1}^{2}+{\dot{q}_{2}}^{2}+{\dot{q}}_{3}^{2})$ and

${\cal E}={\cal
E}^{*}=\dfrac{1}{2}M{\dot{q}}_{1}^{2}\left(1+\dfrac{(q_{2}-\eta(t))^{2}+(q_{3}-\zeta(t))^{2}}{(q_{1}-\xi(t))^{2}}\right).$

is conserved by virtue of (30). This means that $|{\dot{P}|}$ is constant
during the motion.

The unifying and physically expressive role of (24) motivates the following
mathematical investigation.

As it occurs in the holonomic case, it is reasonable to expect that the same
set of nonholonomic restriction may be formulated by more than one set of
constraint equations: a simple instance is the following

###### Example 1

Consider a system of two points $P_{1}$ and $P_{2}$ constrained in a way that
their velocities are perpendicular to the straight line joining them:
${\dot{P}}_{1}\cdot\overrightarrow{P_{1}P_{2}}=0$,
${\dot{P}}_{2}\cdot\overrightarrow{P_{1}P_{2}}=0$; referring to (2), the
condition is formulated as

$\left\\{\begin{array}[]{l}(x_{1}-x_{2}){\dot{x}}_{1}+(y_{1}-y_{2}){\dot{y}}_{1}=0\\\
\\\ (x_{1}-x_{2}){\dot{x}}_{2}+(y_{1}-y_{2}){\dot{y}}_{2}=0\end{array}\right.$

(we keep the cartesian coordinates for clarity). Calling $B$ the midpoint of
the segment $P_{1}P_{2}$, the same effect can be achieved by imposing that the
distance between the points is constant and the velocity of the midpoint is
orthogonal to the joining line, namely the conditions
$|\overrightarrow{P_{1}P_{2}}|=\ell>0$,
${\dot{B}}\cdot\overrightarrow{P_{1}P_{2}}={\bf 0}$. Actually, the
corresponding system in terms of cartesian coordinates

$\left\\{\begin{array}[]{l}(x_{1}-x_{2})({\dot{x}}_{1}-{\dot{x}}_{2})+(y_{1}-y_{2})({\dot{y}}_{1}-{\dot{y}}_{2})=0\\\
\\\
(x_{1}-x_{2})({\dot{x}}_{1}+{\dot{x}}_{2})+(y_{1}-y_{2})({\dot{y}}_{1}+{\dot{y}}_{2})=0\end{array}\right.$

is evidently equivalent to the one written just above (the first constraint
expresses the invariable distance in the differential form).

Further equivalent conditions can be provided, but attention must be paid to
the fact that the equivalence could be lost in correspondence of some
particual motions: concerning this example, the pair of conditions
${\dot{B}}\cdot\overrightarrow{P_{1}P_{2}}={\bf 0}$,
${\dot{P}}_{1}\wedge{\dot{P}}_{2}={\bf 0}$ (parallel velocities) is equivalent
to the previous ones whenever the velocity ${\dot{B}}$ is not null (the
critical motions are the rotations around $B$). Again, the combination of
$|\overrightarrow{P_{1}P_{2}}|=\ell>0$, ${\dot{P}}_{1}\wedge{\dot{P}}_{2}={\bf
0}$ produces the same effect if ${\dot{P}}_{1}\not={\dot{P}}_{2}$ (the
critical motions are of translations). The analysis of the Jacobian matrices
of the various cartesian systems confirms the statements of the Remark without
difficulty.

###### Remark 6

In order to find equivalent sets of kinematic conditions the constraint
equations are not necessarily linear with respect to the velocities as in the
previous case: an example can be formulated by combining two of the three
conditions $|\overrightarrow{P_{1}P_{2}}|=\ell>0$,
${\dot{B}}\wedge\overrightarrow{P_{1}P_{2}}={\bf 0}$,
$|{\dot{P}}_{1}|=|{\dot{P}}_{2}|$ (same magnitude of the velocities)
corresponding respectively to the linear and nonlinear equations

$\begin{array}[]{l}(x_{1}-x_{2})({\dot{x}}_{1}-{\dot{x}}_{2})+(y_{1}-y_{2})({\dot{y}}_{1}-{\dot{y}}_{2})=0,\quad({\dot{x}}_{1}+{\dot{x}}_{2})({\dot{x}}_{1}-{\dot{x}}_{2})+({\dot{y}}_{1}+{\dot{y}}_{2})({\dot{y}}_{1}-{\dot{y}}_{2})=0,\\\
(x_{1}-x_{2})({\dot{y}}_{1}+{\dot{y}}_{2})-(y_{1}-y_{2})({\dot{x}}_{1}+{\dot{x}}_{2})=0.\end{array}$
(31)

###  2.1 The mathematical aspect

From the mathematical point of view it is quite simple to understand which
functions verify the condition (24). The context that emerges is that of
homogeneous functions: we start by reminding that a function
$f(\xi_{1},\dots,\xi_{n})$ defined on a domain ${\cal
D}\subseteq{\mathbb{R}}^{n}$ is a positive homogeneous function of degree
$\sigma\in{\mathbb{R}}$ if

$f(\lambda\xi_{1},\dots,\lambda\xi_{\ell})=\lambda^{\sigma}f(\xi_{1},\dots,\xi_{\ell})\quad\forall\;(\xi_{1},\dots,\xi_{\ell})\in{\cal
D}\;\;\textrm{and}\;\;\forall\;\lambda>0.$ (32)

The functions (32), when differentiable, are distinguished by the following

###### Property 1

(Euler’s homogeneous function theorem) A function $f\in{\cal C}^{1}({\cal D})$
is a positive homogeneous function of degree $\sigma$ if and only if

$\sum\limits_{i=1}^{\ell}\xi_{i}\dfrac{\partial
f}{\partial\xi_{i}}(\xi_{1},\dots,\xi_{\ell})=\sigma
f(\xi_{1},\dots,\xi_{\ell})\quad\forall\;(\xi_{1},\dots,\xi_{\ell})\in{\cal
D}.$ (33)

If we differentiate (32) with respect to $\xi_{i}$, we run into the following
consequence:

###### Property 2

If a function $f(\xi_{1},\dots,\xi_{n})\in{\cal C}^{1}({\cal D})$, ${\cal
D}\subseteq{\mathbb{R}}^{\ell}$ is a positive homogeneous function of degree
$\sigma>0$, then each derivative $\dfrac{\partial f}{\partial x_{i}}$,
$i=1,\dots,n$, is a positive homogeneous function of degree $\sigma-1$.

A further property that we use immediately after and that is easy to verify is
the

###### Property 3

If $F_{1}$, $F_{2}$ are two homogeneous functions of degree $\sigma_{1}$ and
$\sigma_{2}$ respectively, then the product $F_{1}F_{2}$ is a homogeneous
function of degree $\sigma_{1}+\sigma_{2}$, the ratio $F_{1}/F_{2}$ (where
defined) is a homogeneous function of degree $\sigma_{1}-\sigma_{2}$.

The main result of our mathematical digression is the following

###### Proposition 1

Consider a system of $k$ equations

$\left\\{\begin{array}[]{l}F_{1}(\xi_{1},\dots,\xi_{\ell},\eta_{1},\dots,\eta_{k})=0\\\
\qquad\dots\quad\dots\dots\\\ \qquad\dots\quad\dots\dots\\\
F_{k}(\xi_{1},\dots,\xi_{\ell},\eta_{1},\dots,\eta_{k})=0\end{array}\right.$

where each $F_{\nu}\in{\cal C}^{1}({\cal D})$, ${\cal
D}\subseteq{\mathbb{R}}^{\ell+k}$ is a homogeneous function of degree
$\sigma_{\nu}$, $\nu=1,\dots,k$, that is

$F_{\nu}(\lambda\xi_{1},\dots,\lambda\xi_{\ell},\lambda y_{1},\dots,\lambda
y_{k})=\lambda^{\sigma_{\nu}}F(\xi_{1},\dots,\xi_{\ell},y_{1},\dots,y_{k})\quad\;\lambda>0$
(34)

in any point of $D$. Then, assuming that the set of zeros can be written by
means of the $k$ implicitly defined functions

$\left\\{\begin{array}[]{l}y_{1}=\psi_{1}(\xi_{1},\dots,\xi_{\ell})\\\
\qquad\dots\quad\dots\dots\\\ \qquad\dots\quad\dots\dots\\\
y_{k}=\psi_{k}(\xi_{1},\dots,\xi_{\ell})\end{array}\right.$

the functions $\psi_{1}$, $\dots$, $\psi_{k}$ turn out to be positive
homogeneous functions of degree $1$.

Proof. Since $F_{\nu}$ is a homogeneous function of degree $\sigma_{\nu}$,
(33) implies

$\dfrac{\partial F_{\nu}}{\partial\xi_{1}}\xi_{1}+\,\dots\,+\dfrac{\partial
F_{\nu}}{\partial\xi_{\ell}}\xi_{\ell}+\dfrac{\partial
F_{\nu}}{\partial\eta_{1}}\eta_{1}+\,\dots\,+\dfrac{\partial
F_{\nu}}{\partial\eta_{k}}\eta_{k}=\sigma_{\nu}F_{\nu}(\xi_{1},\dots,\xi_{\ell},y_{1},\dots,y_{k}),\quad\nu=1,\dots,k$

The calculation of $\eta_{1}$, $\dots$, $\eta_{k}$ from the previous
identities consists in solving a $k\times k$ linear system whose solutions are

$\eta_{1}=\dfrac{1}{D}\left|\begin{array}[]{llll}\sigma_{1}F_{1}-\sum\limits_{r=1}^{m}\dfrac{\partial
F_{1}}{\partial\xi_{r}}\xi_{r}&\dfrac{\partial
F_{1}}{\partial\eta_{2}}&\dots&\dfrac{\partial F_{1}}{\partial\eta_{k}}\\\
\quad\dots\dots\dots&\dots&\dots&\dots\\\
\sigma_{k}F_{k}-\sum\limits_{r=1}^{m}\dfrac{\partial
F_{k}}{\partial\xi_{r}}\xi_{r}&\dfrac{\partial
F_{k}}{\partial\eta_{2}}&\dots&\dfrac{\partial
F_{k}}{\partial\eta_{k}}\end{array}\right|\qquad\dots\qquad\eta_{k}=\dfrac{1}{D}\left|\begin{array}[]{llll}\dfrac{\partial
F_{1}}{\partial\eta_{1}}&\dots&\dfrac{\partial
F_{1}}{\partial\eta_{k-1}}&\sigma_{1}F_{1}-\sum\limits_{r=1}^{m}\dfrac{\partial
F_{1}}{\partial\xi_{r}}\xi_{r}\\\ \dots&\dots&\dots&\quad\dots\dots\dots\\\
\dfrac{\partial F_{k}}{\partial\eta_{1}}&\dots&\dfrac{\partial
F_{k}}{\partial\eta_{k-1}}&\sigma_{k}F_{k}-\sum\limits_{r=1}^{m}\dfrac{\partial
F_{k}}{\partial\xi_{r}}\xi_{r}\end{array}\right|$

where the vertical bars stand for the determinant of the contained matrix and
$D=\left|\begin{array}[]{lll}\dfrac{\partial
F_{1}}{\partial\eta_{1}}&\dots&\dfrac{\partial F_{1}}{\partial\eta_{k}}\\\
\dots&\dots&\dots\\\ \dfrac{\partial
F_{k}}{\partial\eta_{1}}&\dots&\dfrac{\partial
F_{k}}{\partial\eta_{k}}\end{array}\right|$.

Owing to Property 2, each derivative $\dfrac{\partial
F_{\nu}}{\partial\eta_{\mu}}$, $\nu,\mu=1,\dots,k$ is a homogeneous function
of degree $\sigma_{\nu}-1$; the Leibniz formula for the determinant prescribes
the algebric sum of terms of the type $\prod\limits_{i,j=1}^{k}\dfrac{\partial
F_{\nu_{i}}}{\partial{\mu_{j}}}$, where the suffixes $\nu_{i}$ are all
different. Hence, by virtue of Property 3 and considering that the sum of
homogeneous functions with common degree is evidently a homogeneous function
of same degree, we have that the determinant $D$ is a homogeneous function of
degree
$(\sigma_{1}-1)+(\sigma_{2}-1)+\dots+(\sigma_{k}-1)=\sum\limits_{\nu=1}^{k}\sigma_{\nu}-k$.
The same argument applies to the matrices at the numerator of $\eta_{\nu}$,
whose determinants turn out to be homogeneous function of degree
$\sum\limits_{\nu=1}^{k}\sigma_{\nu}-(k-1)$ (indeed the entries of the
$\nu$–th column of the matrix pertinent to $\eta_{\nu}$ are homogeneous
function of degree $\sigma_{\nu}$). Invoking again Property 3 this time for
the ratio, we overall obtain that each $\eta_{\nu}$ is a homogeneous function
of degree
$\sum\limits_{\nu=1}^{k}\sigma_{\nu}-(k-1)-\left(\sum\limits_{\nu=1}^{k}\sigma_{\nu}-k\right)=1$.
$\quad\square$

###  2.2 The physical application

If we compare the property (33) with the request (24), we recognise in the
latter one the characteristic condition for the function $\alpha_{\nu}$ to be
a homogeneous function of degree $1$ with respect to the variables
${\dot{q}}_{r}$: thus, setting $\ell=m$ and ${\dot{q}}_{r}=\xi_{r}$,
$r=1,\dots,m$, (19), (32) and (33) entail the following

###### Proposition 2

The function $\alpha_{\nu}$, $\nu=1,\dots,k$, verifies
${\overline{\alpha}}_{\nu}=\alpha_{\nu}$ if and only if $\alpha_{\nu}$ is a
positive homogeneous function of degree $1$ with respect to the kinetic
variables ${\dot{q}}_{1}$, $\dots$, ${\dot{q}}_{m}$, namely if and only if

$\alpha_{\nu}(q_{1},\dots,q_{n},\lambda{\dot{q}}_{1},\dots,\lambda{\dot{q}}_{m},t)=\lambda\alpha_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{m},t)\quad\textrm{for
any}\;\;\lambda>0.$ (35)

A more general and noteworthy result is provided by Proposition 1, which
refers directly to the structure of the assigned constraint functions and not
to the implicitly defined ones: actually, with respect to (34) we consider the
kinetic variables ${\dot{q}}_{i}$ of (2) as ${\dot{q}}_{r}=\xi_{r}$,
$r=1,\dots,m=\ell$ and ${\dot{q}}_{m+\nu}=\eta_{\nu}$, $\nu=1,\dots,k$ and we
make use of Proposition $1$ for the following application:

###### Proposition 3

If $\phi_{\nu}(q_{1},\dots,q_{n},{\dot{q}}_{1},\dots,{\dot{q}}_{n},t)$ of (2)
are homogeneous functions (even with different degrees) of the generalized
velocities ${\dot{q}}_{1}$, $\dots$, ${\dot{q}}_{n}$, then however the
explicit forms (9) are chosen, they satisfy condition (24).

As we have already highlighted, the same system can be treated either with
linear kinematic constraints or with nonlinear kinematic constraints (or a
mixture of them): this fact does not affect the definition and conservation of
the energy of the system, if the latter falls into the category admitted by
the previous Proposition.

For instance, the system we considered in (31) is described by homogeneous
functions, whichever pair of constraints is choosen, except for any points
where equivalence is lost, as we remarked in Example 1. In any case, the
explicit functions (9) will verify condition (24).

Among many other examples we can present, we add the following nonholonomic
system, studied in [20].

###### Example 2

Consider on a vertical plane two material points $P_{1}$, $P_{2}$ of mass
$M_{1}$, $M_{2}$ respectively: let us call attention on the three kinematic
restrictions

  * $(i)$

the velocities are perpendicular: ${\dot{P}}_{1}\cdot{\dot{P}}_{2}=0$,

  * $(ii)$

the velocity of one of them is perpendicular to the straight line joining the
points: ${\dot{P}}_{1}\cdot\overrightarrow{P_{1}P_{2}}=0$,

  * $(iii)$

the velocity of the other point is parallel to the joining line:
${\dot{P}}_{2}\wedge\overrightarrow{P_{1}P_{2}}={\bf 0}$.

By formulating the conditions as

$\begin{array}[]{lll}{\dot{x}}_{1}{\dot{x}}_{2}+{\dot{y}}_{1}{\dot{y}}_{2}=0,&(x_{1}-x_{2}){\dot{x}}_{1}+(y_{1}-y_{2}){\dot{y}}_{1}=0,&(x_{1}-x_{2}){\dot{y}}_{2}-(y_{1}-y_{2}){\dot{x}}_{2}=0\end{array}$

respectively, it can be easily seen that they are not independent and two of
them imply the third with the exceptions noted alongside (the explication for
the excluded configurations is clear):

$\begin{array}[]{lll}(i),\,(ii)\;\Rightarrow\;(iii)\;\;\textrm{if}\;\;{\dot{P}}_{1}\not={\bf
0},&(i),\,(iii)\;\Rightarrow\;(ii)\;\;\textrm{if}\;\;{\dot{P}}_{2}\not={\bf
0},&(ii),\,(iii)\;\Rightarrow\;(i)\;\;\textrm{if}\;\;P_{1}\not\equiv
P_{2}.\end{array}$

Even admitting that the velocities can vanish but excluding the overlapping of
the points, we opt for the latter possibility and write (9) as

$\left\\{\begin{array}[]{l}{\dot{q}}_{3}=\alpha_{1}(q_{1},q_{2},q_{3},q_{4},{\dot{q}}_{1})=-\alpha{\dot{q}}_{1}\\\
\\\
{\dot{q}}_{4}=\alpha_{2}(q_{1},q_{2},q_{3},q_{4},{\dot{q}}_{2})=\dfrac{1}{\alpha}{\dot{q}}_{2}\end{array}\right.$

where we have set $(q_{1},q_{2},q_{3},q_{4})=(x_{1},x_{2},y_{1},y_{2})$ and
defined $\alpha(q_{1},q_{2},q_{3},q_{4})=\dfrac{q_{1}-q_{2}}{q_{3}-q_{4}}$.
The assumption (24) holds because the constraints are part of (25). Assuming
that an internal elastic force of constant $\kappa$ and the weight are acting,
the function (13) is

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

where the terms concerning $U$ are clear. Equation (30) entails the
conservation of the quantity (17), coinciding in this case with the energy of
the system ${\cal E}=T-U$.

##  3 Conclusion and next investigation

For nonholonomic systems verifying (24), the formulation of the motion through
the virtual displacements provided by the ${\check{\rm C}}$hetaev condition
(7) appears natural and not devoid of physical meaning. This represents a
convincing extension of the holonomic case equipped by a well-established and
dated theory. The absence of a justification of the condition (7) starting
from the laws of mechanics sometimes claimed in literature, is overcome when
the kinematic constraints are of the type (24).

As we have already highlighted, for these systems the notion of virtual
displacement assumes a physical sense and also the energy of the system has a
correct meaning, as for holonomic systems. Likewise to the latter ones, when
the Lagrangian function does not depend on time the energy is conserved, as
(30) states.

To understanding which functions verify (24) is simple, since it is a plain
implementation of Euler’s theorem on homogeneous functions. The further step
we performed was to indicate a large class of constraints for which the
explicit form (9) shows the functions $\alpha_{\nu}$ of the category (24):
this occurs for the constraints formulated by homogeneous functions (with not
necessarily the same degree) of the generalized velocities ${\dot{q}}_{i}$,
$i=1,\dots,n$.

The incoming investigation will concern a sort of inverted question: if one
has to do with explicit functions $\alpha_{\nu}$ verifiying (24), necessarily
the originating functions $\phi_{\nu}$ which define them are homogeneous
functions in the kinematic variables? This is not an irrelevant point, since
the category of nonholonomic constraints satisfying the physical properties
$(1)$–$(4)$ listed just after Remark 4 would become entirely and clearly
defined, as the systems subject to homogeneous constraints, i. e. the
${\check{\rm C}}$etaev condition would be completely legitimated for this
category of systems. In other words, the constrained systems for which the
condition acquires a physical meaning and extends in a natural way the virtual
displacements of the holonomic systems are presumed to be all and only those
which admit at least one set of conditions (2) formulated with equations
homogeneous in the velocities.

However, the mathematical aspect is anything but straightforward: as it
incidentally emerged in some passage of the script, the geometric and
kinematic restrictions can be implemented in more than a way, so that
different lists of functions (2) can formulate the same system. Then, from the
mathematical point of view, the issue consists in investigating the zero set
defined by (2) and wonder if for each explicit set $\alpha_{\nu}$,
$\nu=1,\dots,k$ a list of generating functions $\phi_{\nu}$ which are
homogeneous in the velocities can always be found.

## References

  * [1] Appell P. , Sur les liasons exprimées par des relations non linéaires entre les vitesses, C. R. Acad. Sci. Paris 152, 1197–319, 1200, 1911.
  * [2] Bloch, A. M. , Mardsen, J. E. and Zenkov, D. V. , Quasivelocities and Symmetries in Non-Holonomic Systems, Dynamical Systems 24, 187–222, 2009.
  * [3] Cronström, C. , On the Compatibility of Nonholonomic Systems and Related Variational Systems, Acta Physica Universitatis comenianae Vol. L–LI, 1 and 2, 25–36, 2010.
  * [4] ${\check{\rm C}}$etaev, N. G. , On the Gauss Principles, Papers on Analytical Mechanics 323, Science Academy, 1962.
  * [5] Fassò, F. , Sansonetto, N. , Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints, Regular and Chaotic Dynamics 20 n. 4, 449–462, 2015.
  * [6] Flannery M. R. , The enigma of nonholonomic constraints, American Journal of Physics 73, 265–272, 2005.
  * [7] de León M. , A historical review on nonholonomic mechanics, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas 106(1), 2012.
  * [8] de León, M. , de Diego, D. M. , On the geometry of non-holonomic Lagrangian systems, Mechanical systems with nonlinear constraints, J. Math. Phys. 37, 3389–3414, 1996.
  * [9] S. M. Li and J. Berakadar: A generalization of the Chetaev condition for nonlinear nonholonomic constraints: The velocity-determined virtual displacement approach, Reports on Mathematical Physics 63 Issue 2, 179–189, 2009.
  * [10] Marle, C–M, Various approaches to conservative and nonconservative nonholonomic systems Reports on Mathematical Physics 42, Issues 1–-2, 211–229, 1998.
  * [11] Massa, E. , Pagani, E. , Classical Dynamics of non–holonomic systems: a geometric approach, Ann. Inst. H. Poincaré Phys. Théor. 55, 511–544, 1991.
  * [12] Ne${\check{\rm i}}$mark Ju. I. , Fufaev N. A. , Dynamics of Nonholonomic Systems, Providence: American Mathematical Society, Translations of Mathematical Monographs 33, 1972.
  * [13] J. G. Papastravidis, J. G. , Time–integral Variational Principles for Nonlinear Nonholonomic Systems, J. Appl. Math. 64, 985–991, 1997.
  * [14] Cariñena, J. F. , Rañada, M. F. , Lagrangian systems with constraints: a geometric approach to the method of Lagrange multipliers, J. Phys. A: Math. Gen. 26, 1335-1351, 1993.
  * [15] V. V. Rumyantsev, V. V. , Forms of Hamilton’s principle for nonholonomic systems, Mechanics, Automatic and Robotics 2, n. 10, 1035–1048, 2000.
  * [16] V. V. Rumyantsev, V. V. , On the Chetaev principle (In Russian), Dokl. Akad. Nauk SSSR 210 n. 4, 787–790, 1973.
  * [17] Salehani, M. K. , A Jet Bundle Approach to the Variational Structure of Nonholonomic Mechanical Systems, Reports on Mathematical Physics 83 Issue 3, 373–385, 2019.
  * [18] Talamucci, F., Rheonomic Systems with Nonlinear Nonholonomic Constraints: The Voronec Equations, Regular and Chaotic Dynamics 25 n. 6, 662–673, 2020.
  * [19] Zeković, D. N. , Dynamics of mechanical systems with nonlinear nonholonomic constraints – I The history of solving the problem of a material realization of a nonlinear nonholonomic constraint, Z. Angew. Math. Mech. 91 n. 11, 883–898, 2011.
  * [20] Zeković, D. N. , Linear integrals of non–holonomic systems with non–linear constraints, Journal of Appl. Math. and Mech. 69, 832–836, 2005.


# Differential current noise as an identifier of Andreev bound states that
induce nearly quantized conductance plateaus

Zhan Cao  Corresponding author.
<EMAIL_ADDRESS>Beijing Academy of Quantum Information Sciences, Beijing
100193, China  Gu Zhang  Beijing Academy of Quantum Information Sciences,
Beijing 100193, China  Hao Zhang  State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084,
China  Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Frontier Science Center for Quantum Information, Beijing 100084, China  Ying-
Xin Liang  Beijing Academy of Quantum Information Sciences, Beijing 100193,
China  Wan-Xiu He  Beijing Computational Science Research Center, Beijing
100193, China  Ke He  State Key Laboratory of Low Dimensional Quantum Physics,
Department of Physics, Tsinghua University, Beijing, 100084, China  Beijing
Academy of Quantum Information Sciences, Beijing 100193, China  Frontier
Science Center for Quantum Information, Beijing 100084, China  Hefei National
Laboratory, Hefei 230088, China  Dong E. Liu  Corresponding author.
<EMAIL_ADDRESS>State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Frontier Science Center for Quantum Information, Beijing 100084, China  Hefei
National Laboratory, Hefei 230088, China

###### Abstract

Quantized conductance plateaus, a celebrated hallmark of Majorana bound states
(MBSs) predicted a decade ago, have recently been observed with small
deviations in iron-based superconductors and hybrid nanowires. Here, we
demonstrate that nearly quantized conductance plateaus can also arise from
trivial Andreev bound states (ABSs). To avoid ABS interruptions, we propose
identifying ABS-induced quantized conductance plateaus by measuring the
associated differential current noise $P$ versus bias voltage $V$.
Specifically, for a quantized conductance plateau induced by one or multiple
low-energy ABSs, the associated $P(V)$ curve exhibits a double-peak around
zero bias, with the peak positions at $e|V|\approx 3k_{B}T$ (where $T$ is the
temperature) and peak values larger than $2e^{3}/h$. These features greatly
contrast those of an MBS or quasi-MBS, whose $P(V)$ curve displays a broad
zero-bias dip and is consistently below $2e^{3}/h$. This protocol can be
practically implemented in a variety of MBS candidate platforms using an
electrode or STM tip as a probe.

Majorana bound states (MBSs) [1, 2] have attracted great attention due to
their potential realization of topological quantum computation [3, 4, 5]. They
have been searched in many platforms [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23] by tunneling spectroscopy, inspired by the
predicted MBS-induced quantized zero-bias conductance peak (ZBCP) [24, 25,
26]. In turn, a discernible quantized ZBCP is believed as a significant
benchmark of MBSs for implementing topological qubits [27]. Thus far, most of
the ZBCPs observed in Majorana candidate platforms are unconvincing due to
possible interruption from false-positive signals generated by topologically
trivial zero-energy Andreev bound states (ABSs) [28, 29, 30]. Recently, this
was tested by introducing a dissipative environment to suppress interrupting
signals without sabotaging the robust tunneling feature of MBSs [31, 32, 33,
34, 35, 36].

The quantized conductance plateau [37], formed by a quantized ZBCP robust to
the tuning of an experimental parameter, was firmly believed to be an MBS
feature. In addition, quantized conductance plateaus can arise from robust
quasi-MBSs (QMBSs) [38] that have MBS-like local characteristics but lack
topological protection. Differentiating between MBSs and QMBSs, both
characterized by Majorana-type wavefunctions, is not pursued in this work.
Recently, nearly quantized conductance plateaus have been observed in the
tunneling spectroscopy of iron-based superconductors [39, 40] by sweeping the
STM tip-sample distance and of hybrid InAs-Al nanowire [41] by sweeping the
magnetic field or gate voltages. However, a nearly quantized conductance
plateau may also be caused by disorder-induced zero-energy ABSs, though
suggested by numerical simulations [42, 43, 44]. These results seriously
challenge the validity of the conductance plateau hallmark of an MBS or robust
QMBS (MBS/QMBS).

Table 1: Features of $P(V)$ curves for different bound states that induce nearly quantized conductance plateaus or ZBCPs. Cases |  A broad zero-bias dip |  A double-peak at $e|V|\approx 3k_{B}T$ |  $P_{max}>$ $2e^{3}/h$
---|---|---|---
An MBS/QMBS | ✓ | $\times$ | $\times$
One or multiple ABSs | $\times$ | ✓ | ✓
An MBS/QMBS and multiple ABSs | ✓ | $\times$ | ✓

In this Letter, we theoretically study MBS/QMBS and zero-energy ABSs by
combining their tunneling conductance and current noise. We demonstrate that a
zero-energy ABS can also induce nearly quantized conductance plateaus in a
sizable and continuous parameter space, which expands as temperature
increases. We also show the possibility of nearly quantized ZBCPs when the
probe is simultaneously coupled to multiple low-energy ABSs with or without an
MBS/QMBS. We propose that the origin of a quantized conductance plateau or
ZBCP can be identified by the associated differential current noise curve
$P(V)$ (where $V$ is the bias voltage), which exhibits distinguishable
features summarized in Table 1. These features are attributed to the
strikingly distinct energy dependence of the eigenchannel transmissions in
different cases. As both tunneling conductance and differential current noise
can be accessed by the same probe with high precision [45, 46, 47, 48, 49],
this protocol can be practically implemented in various MBS candidate
platforms.

_Model and formalism_.—We consider a local tunneling model schematically shown
in Fig. 1(a) and described by the effective Hamiltonian $H=H_{P}+H_{B}+H_{T}$
[26, 50, 51, 52]. Here
$H_{P}=\sum_{k\sigma}\varepsilon_{k}c_{k\sigma}^{{\dagger}}c_{k\sigma}$
describes a metallic probe (either an electrode or STM tip).
$H_{B}=i\varepsilon_{B}\gamma_{1}\gamma_{2}$ describes a generic
superconducting bound state as a composition of two Majorana components
$\gamma_{1,2}$ with a hybridization energy $\varepsilon_{B}$.
$H_{T}=\sum_{mk\sigma}(t_{m\sigma}c_{k\sigma}^{{\dagger}}-t_{m\sigma}^{\ast}c_{k\sigma})\gamma_{m}$
describes the tunneling between the probe and $\gamma_{1,2}$. The spin-
dependent hopping amplitudes $t_{m\sigma}$ ($m=1,2$) can be parametrized as
[53] $t_{1\uparrow}=t_{1}\sin(\theta_{1}/2)$,
$t_{1\downarrow}=-t_{1}\cos(\theta_{1}/2)$,
$t_{2\uparrow}=-it_{2}\sin(\theta_{2}/2)$, and
$t_{2\downarrow}=-it_{2}\cos(\theta_{2}/2)$, with $t_{1,2}$ being real numbers
and $\theta_{1,2}$ the spin canting angles of the wave functions of
$\gamma_{1,2}$ at the position nearest to the probe. In practice, $t_{1,2}$
and $\theta_{1,2}$ depend on experimentally tunable knobs, e.g., external
magnetic field and gate voltages, and device details. As shown below,
$\theta\equiv(\theta_{1}+\theta_{2})/2$ is the relevant quantity for
transport. For later convenience, we define the level broadening
$\Gamma_{1,2}=2\pi\rho t_{1,2}^{2}$, with $\rho$ being the probe density of
states, and assume $\Gamma_{2}\leq\Gamma_{1}$ without loss of generality.

Figure 1: (a) Left: schematic of a probe coupled to a superconducting bound
state with energy $\varepsilon_{B}$. Physically, this bound state can be
viewed as two hybridized Majoranas $\gamma_{1}$ and $\gamma_{2}$ that
independently couple to the probe with spin-dependent amplitudes
$t_{1,2\sigma}$. Right: typical profiles of 1D spatial wavefunctions of
$\gamma_{1,2}$ for an ABS, MBS, and QMBS, respectively. They imply that
$t_{1,2\sigma}\neq 0$ for an ABS, while $t_{1\sigma}\neq 0$ and
$t_{2\sigma}=0$ for an MBS/QMBS. $t_{1,2\sigma}$ can be parametrized by real
numbers $t_{1,2}$ and spin canting angles $\theta_{1,2}$ (see text), from
which we define $\theta\equiv(\theta_{1}+\theta_{2})/2$ and $r\equiv
t_{2}^{2}/(t_{1}^{2}+t_{2}^{2})$. $r=0$ ($r>0$) describes an MBS/QMBS (ABS).
[(b) and (c)] The zero-bias conductance $G(0)$ for zero-energy states
($\varepsilon_{B}=0$) as functions of $r$ and $\theta$ are plotted at (b) zero
temperature and (c) $k_{B}T=0.15~{}\Gamma$, respectively. The red solid curves
highlight the lines with $G(0)=1$ and areas within the white dashed lines have
$0.9\leq G(0)\leq 1.1$.

In Fig. 1(a), we illustrate the typical profiles of one-dimensional (1D)
spatial wavefunctions of $\gamma_{1,2}$ for an ABS, MBS, and QMBS,
respectively, that can potentially be probed. The two wave functions exhibit
extensive overlap for an ABS, significant separation for an MBS, and partial
separation for a QMBS [54, 55, 51, 56, 57]. Consequently, both $\Gamma_{1}$
and $\Gamma_{2}$ are finite for an ABS, while $\Gamma_{2}=0$ for an MBS/QMBS.
This implies that our local tunneling protocol cannot unequivocally
distinguish MBS from robust QMBS. Nevertheless, our focus is to address
whether nearly quantized conductance plateaus can be induced by trivial zero-
energy ABSs and, if yes, how to experimentally identify them. For simplicity,
we set $\varepsilon_{B}=0$ to focus on zero-energy bound states. Indeed, an
ABS with zero energy can most closely resemble an MBS/QMBS. We further define
$r=\Gamma_{2}/\Gamma$ (with $\Gamma=\Gamma_{1}+\Gamma_{2}$): $r=0$ and $r>0$
for an MBS/QMBS and ABS, respectively. Notably, earlier works study only
limiting cases of the effective Hamiltonian under consideration: $r=0$ in Ref.
26, $\theta=\pi/2$ in Refs. 50, 51, and
$\sqrt{r(1-r)}\leq\varepsilon_{B}/\Gamma$ in Ref. 52. By contrast, we visit
the entire $(r,\theta)$ space that is experimentally accessible. The provided
features of ABSs and MBS/QMBS are thus more complete and convincing.

With the scattering matrix theory [58], we derive the formulas of current $I$
and current noise $S$, refer to the Supplemental Material (SM) [59]. At zero
temperature,

$\displaystyle I(V)=\frac{2e}{h}\sum_{s=\pm}\int^{eV}_{0}d\varepsilon
T_{s}(\varepsilon),$ (1) $\displaystyle
S(V)=\textrm{sgn}(V)\frac{8e^{2}}{h}\sum_{s=\pm}\int^{eV}_{0}d\varepsilon
T_{s}(\varepsilon)\big{[}1-T_{s}(\varepsilon)\big{]},$ (2)

with

$\displaystyle
T_{\pm}(\varepsilon)=\Gamma^{2}\big{[}\varepsilon^{2}p_{+}+p_{-}(\varepsilon_{B}^{2}+\Gamma^{2}q_{-}/2)\pm\Pi\big{]}/\Lambda,$
(3)
$\displaystyle\Pi=|\varepsilon|\big{[}\delta_{1}(\delta_{2}\varepsilon^{2}+2p_{-}\varepsilon_{B}^{2})+\delta_{2}p_{-}(1/2-\delta_{1})\Gamma^{2}\big{]}^{1/2},$
(4)
$\displaystyle\Lambda=(\varepsilon^{2}-\varepsilon_{B}^{2})^{2}+(q_{+}\varepsilon^{2}+\varepsilon_{B}^{2}q_{-})\Gamma^{2}+\Gamma^{4}q_{-}^{2}/4.$
(5)

Here $V$ is the bias voltage, $\delta_{1}=1/2-2r(1-r)\cos^{2}\theta$,
$\delta_{2}=(1-2r)^{2}/2$, $p_{\pm}=(\delta_{1}\pm\delta_{2})/2$, and
$q_{\pm}=(1\pm 2\delta_{2})/2$. $T_{\pm}(\varepsilon)$ are the transmissions
of the two eigenchannels “$\pm$” of the scattering problem. Notably,
$T_{-}(\varepsilon)=0$ for a spin-polarized ABS (i.e., $\theta=0$) or an
MBS/QMBS (i.e., $r=0$).

Theoretically, ZBCPs may also arise from multiple low-energy superconducting
bound states [60]. We examine this situation by extending $H_{T}$ and $H_{B}$
to include multiple bound states [59]. As the probe involves two spin-
subbands, a maximum of two eigenchannels are available for transport.
Therefore, Eqs. (1) and (2) remain applicable and $T_{\pm}(\varepsilon)$ could
be evaluated numerically. In this work, we mainly focus on the differential
conductance $G(V)\equiv dI(V)/dV$ and differential current noise
$P(V)\equiv|dS(V)/dV|$.

_Nearly quantized conductance plateau induced by a single zero-energy ABS_.—At
zero temperature, Eqs. (1)–(5) predict $G(0)=2e^{2}/h$ for $r=0$ and
$G(0)=4e^{2}\sin^{2}\theta/h$ for $r>0$, which are shown in Fig. 1(b) as
functions of $r$ and $\theta$. Along the red solid lines, $G(0)$ reaches the
quantized value, and in the region between the two white dashed lines
$0.9<G(0)<1.1$, showing a tolerance of $\pm 10\%$. This narrow parameter
region agrees with previous simulations that a few percentages of disorder-
induced zero-energy ABSs can lead to nearly quantized ZBCPs [43]. In Fig.
1(c), we show $G(0)$ at a finite temperature $k_{B}T=0.15~{}\Gamma$. The
nearly quantized ZBCP region is larger than Fig. 1(b), indicating that false-
positive ZBCPs are more likely to emerge for larger $k_{B}T/\Gamma$. The
values of $k_{B}T/\Gamma$ in Refs. [40] and [41], which reported nearly
quantized conductance plateaus, are estimated as $0.27$ and $0.03$,
respectively [59]. The parameter set $(r,\theta)$ depends on experimentally
tunable knobs and device details. Figures 1(b) and 1(c) suggest that nearly
quantized conductance plateaus induced by zero-energy ABSs could appear if the
tuning of one or more knobs occasionally results in $(r,\theta)$ trajectories
restricted between the white dashed lines. This highlights that a nearly
quantized conductance plateau is not an exclusive hallmark of MBS/QMBS.
Additionally, in the SM [59] we demonstrate that the Fano factor, calculated
by $S$ and $I$, approaching $2$ ($1$) in the zero (large) bias limit at low
temperatures is also not exclusive to MBS/QMBS, providing a possible
explanation for a very recent experiment [61].

Figure 2: [(a) and (b)] Zero-temperature differential conductance $G(V)$ and
differential current noise $P(V)$ for zero-energy ABSs ($r>0$) with
$\theta=0.25\pi$ and MBS/QMBS ($r=0$). For $r=0.4$, $0.2$, and $0$, the
[(c)–(e)] transmissions and [(f)–(h)] differential current noises associated
with the two eigen channels “$\pm$” [see Eqs. (1)–(3)] are shown.

_Distinguish MBS/QMBS and ABSs induced quantized ZBCPs with differential
current noise_.—As shown in Fig. 1(b), zero-energy ABSs with $\theta=0.25\pi$
can induce a quantized zero-bias conductance, i.e., $G(0)=2e^{2}/h$. In Fig.
2(a), we further show the associated quantized ZBCPs in $G(V)$ curves for
different $r$ values (from $0$ to $0.5$). The lineshapes of $G(V)$ for an
MBS/QMBS ($r=0$) and ABS ($r>0$) are quite similar, making it hard to identify
the origin of a quantized ZBCP. By contrast, the corresponding differential
current noise $P(V)$ in Fig. 2(b) is quite distinct for an MBS/QMBS (a zero-
bias dip) and ABS (a zero-bias peak). Furthermore, $P_{max}$, the maximum
value of a $P(V)$ curve, is bounded by $2e^{3}/h$ for an MBS/QMBS, whereas it
exceeds $2e^{3}/h$ for a zero-energy ABS. Such qualitative features in $P(V)$
can be easily discerned in experiments. For completeness, in the SM [59] we
also present the correlations between $G(V)$ and $P(V)$ for ABSs with
unquantized ZBCPs.

Now we analyze the features displayed in Figs. 2(a) and 2(b). As indicated by
Eqs. (1) and (2), $G(V)$ and $P(V)$ are proportional to
$\sum_{s=\pm}T_{s}(eV)$ and $\sum_{s=\pm}T_{s}(eV)[1-T_{s}(eV)]$,
respectively. The lineshapes of $T_{\pm}$, importantly, are different for an
MBS/QMBS and ABS; see Figs. 2(c)–2(e). Specifically, the channel “$-$”
decouples [i.e., $T_{-}(\varepsilon)=0$] for an MBS/QMBS. The MBS/QMBS induced
$P(V)$ then follows the shape of $T_{+}(1-T_{+})$: a broad dip at $T_{+}=1$
and two maximums of $2e^{3}/h$ at $T_{+}=0.5$ [Fig. 2(h)]. By contrast, for a
generic ABS, the quantized ZBCP requires instead $T_{+}(0)=T_{-}(0)=0.5$ due
to $T_{\pm}(0)=\sin^{2}\theta$. In this case, both channels are imperfect,
leading to a peak of $P(0)=4e^{3}/h$ [with its components $P_{\pm}(V)$ shown
in Figs. 2(f) and 2(g)]. For a small $r$ value, the width of the zero-bias
peak in $P(V)$ decreases as $r\Gamma$ [59].

Figure 3: [(a)–(d)] Temperature impact on the $G(V)$ and $P(V)$ curves of
$r=0$ and $0.4$ in Fig. 2. Refer to the SM [59] for the results of the other
$r$ values. The vertical dotted lines mark $e|V|=3k_{B}T$. [(e) and (f)]
Temperature dependence of the position $|V_{p}|$ of the double peak in $P(V)$
for ABSs ($r>0$), the width $W$ of the zero-bias dip in $P(V)$ for an MBS/QMBS
($r=0$), and the maximum $P_{max}$ of a $P(V)$ curve for ABS or MBS. The black
solid line in (e) is a linear fit of $e|V_{p}|=3k_{B}T$.

Figure 3 shows the temperature impact on the $G(V)$ and $P(V)$ curves
presented in Fig. 2. As temperature increases, the ZBCPs are suppressed below
$2e^{2}/h$ by thermal broadening [Figs. 3(a) and 3(b)]. Moreover, for an
MBS/QMBS, the zero-bias dip in $P(V)$ is always broader than a constant
$\sim\Gamma$ [Figs. 3(c) and 3(e)]. On the other hand, for zero-energy ABSs,
even if the zero-bias peak in $P(V)$ splits into a doublepeak, the interpeak
distance ($\sim 6k_{B}T$) scales to zero as the temperature decreases [Figs.
3(d) and 3(e)]. Additionally, as shown in Fig. 3(f), $P_{max}>2e^{3}/h$ is
maintained for zero-energy ABSs when $k_{B}T/\Gamma\ll r$, while $P_{max}$
decreases from $2e^{3}/h$ with increasing temperature for an MBS/QMBS.

The temperature dependence of the peak position (dip width) of $P(V)$ for
zero-energy ABSs (MBS/QMBS), as shown in Fig. 3(e), can be qualitatively
understood as follows. In general, the current noise consists of shot noise,
inherent to nonequilibrium charge transport even at $T=0$, and thermal noise,
associated with equilibrium thermal fluctuations proportional to
$f(\varepsilon-eV)[1-f(\varepsilon-eV)]$, where $f(\varepsilon-eV)$ is the
Fermi distribution and the width of its derivative is roughly $3k_{B}T$.
Consequently, at $T=0$, thermal noise is absent, and shot noise dominates. At
$T>0$ and in the low-bias region $e|V|<3k_{B}T$, thermal noise emerges and
prevails over shot noise. When the bias increases to the threshold
$e|V|\approx 3k_{B}T$, above which shot noise dominates the current noise, the
characteristic peak in $P(V)$ predicted for zero-energy ABSs at $T=0$
reappears. Due to the same reason, the dip width in $P(V)$ for an MBS/QMBS
also increases by approximately $3k_{B}T$.

Figure 4: Examples of $G(V)$ and $P(V)$ curves for a probe simultaneously
coupled to [(a) and (c)] two ABSs with energies of $0$ and
$0.1\widetilde{\Gamma}$, respectively, or to [(b) and (d)] an MBS/QMBS and a
zero-energy ABS. The bias voltage is rescaled by $\widetilde{\Gamma}$ such
that $G=e^{2}/h$ for $e|V|/\widetilde{\Gamma}=1$ and $T=0$. The vertical
dotted lines mark $e|V|=3k_{B}T$. Refer to the SM [59] for the calculation
parameters. [(e) and (f)] Zero-temperature $G$ and $P$ as functions of
$T_{\pm}$, calculated by Eqs. (1) and (2). The lower limits of $G$ and $P$
within the dashed circle are indicated.

Essentially, a ZBCP can potentially be induced by multiple low-energy bound
states. For this scenario, we explore the correlations between $G(V)$ and
$P(V)$ for a probe simultaneously coupled to a pair of low-energy ABSs or to a
combination of an MBS/QMBS and a zero-energy ABS. Exemplary numerical results
are shown in Fig. 4, which demonstrate that nearly quantized ZBCPs can occur
in both cases [Figs. 4(a) and 4(b)]. The associated $P(V)$ curves exhibit a
double peak at $e|V|\approx 3k_{B}T$ [Fig. 4(c)] and a broad zero-bias dip
[Fig. 4(d)], respectively. Both eigenchannels “$\pm$” contribute to $G$ and
$P$, as shown in the SM [59]. The profiles of the $G(V)$ and $P(V)$ curves for
two bound states in Figs. 4(a) and 4(c) [Figs. 4(b) and 4(d)] can be observed
when a probe is coupled to multiple ABSs (an MBS/QMBS and multiple ABSs).
Indeed, even with multiple bound states, there are at most two eigenchannels
with transmissions $T_{\pm}$, as only two spin subbands are available in the
probe. Notably, the low-temperature $P(V)$ curves in Figs. 4(c) and 4(d) all
exhibit $P_{max}>2e^{3}/h$, which is actually a universal feature when the
probe is coupled to one or more ABSs. Specifically, as indicated by Figs. 4(e)
and 4(f), which are calculated by Eqs. (1) and (2), as long as both channels
contribute to the transport with the condition that the maximum of a $G(V)$
curve exceeds $0.29\times 2e^{2}/h$, we invariably find $P_{max}>2e^{3}/h$.

_Simulations of hybrid nanowires_.—The analyses above are based on the
effective models of isolated bound states. To validate the reliability, we
perform numerical simulations to obtain the $G$ and $P$ of MBSs and near-zero-
energy ABSs in hybrid nanowires, considering several physical effects in
experiments [28, 29, 59]. In Fig. 5, we compare the maps of $G$ and $P$, as
functions of the bias voltage $V$ and the Zeeman energy $V_{Z}$, for a clean
and disordered single-subband nanowire, respectively. Clearly, a nearly
quantized conductance plateau persists over a sizable range of $V_{Z}$ for
both nanowires [Figs. 5(a) and 5(b)]. The local density of states shown in the
SM [59] reveals that the two plateaus are associated with an MBS and near-
zero-energy ABS, respectively. The $P$ maps of an MBS and ABS are
qualitatively distinct [Figs. 5(c) and 5(d)], which is evident from the line
cuts in Figs. 5(e) and 5(f): while both $G(V)$ curves display a nearly
quantized ZBCP, the $P(V)$ curve for an MBS exhibits a broad zero-bias dip and
$P_{max}\leq 2e^{3}/h$, whereas that for an ABS shows a zero-bias peak and
$P_{max}>2e^{3}/h$. These $G(V)$ and $P(V)$ correlations qualitatively agree
with those in Figs. 2(a) and 2(b) predicted by the effective model.

Figure 5: [(a)–(d)] Zero-temperature simulations of $G$ and $P$ versus the
bias voltage $V$ and the Zeeman energy $V_{Z}$ for a [(a) and (c)] clean and
[(b) and (d)] disordered hybrid nanowire. [(e) and (f)] Line cuts of (a)–(d)
at the $V_{Z}$ marked by the colored bars. Refer to the SM [59] for simulation
parameters.

Further simulations are presented in the SM [59]. We simulate four additional
scenarios exhibiting nearly quantized conductance plateaus: (i) an MBS in a
dissipated single-subband nanowire, (ii) a near-zero-energy ABS in a
dissipated single-subband nanowire, (iii) a near-zero-energy ABS in the top
subband of a two-subband nanowire, and (iv) two low-energy ABSs in different
subbands of a two-subband nanowire. It is noteworthy that the results of all
these scenarios consistently align with our predictions. For a comprehensive
summary of our effective model calculations and numerical simulations, we
refer the reader to Tables S3 and S4 in the SM [59].

_Summary and Discussion_.—We have demonstrated that the recently observed
nearly quantized conductance plateaus [39, 40, 41] can potentially arise from
topologically trivial zero-energy or near-zero-energy ABSs. Moreover, we have
proposed measuring the associated differential current noise to identify this
scenario. While our protocol cannot distinguish between MBSs and robust QMBSs,
it facilitates achieving a crucial near-term objective: assessing whether a
given material system attains satisfactory quality to rule out trivial zero-
energy ABSs. This is currently a major concern in hybrid nanowires [42, 43,
62, 63, 64, 65, 66] and superconducting vortices [67]. With improved material
quality, the identification of MBSs could be accomplished through
complementary evidence from different combined measurement protocols using
sophisticated devices, such as the combination of local and nonlocal tunneling
conductance in a three-terminal device [68, 69, 70, 71, 72, 73], local
tunneling and Coulomb conductance in a two-terminal island device [74], or
tunneling conductance and differential current noise in a quantum dot based
three-terminal device [75].

_Acknowledgements_.—We appreciate the anonymous referees’ suggestion to
consider the scenario involving multiple bound states. This work was supported
by the National Natural Science Foundation of China (Grants No. 12004040 and
No. 11974198), the Innovation Program for Quantum Science and Technology
(Grant No. 2021ZD0302400), the National Natural Science Foundation of China
(Grants No. 92065206 and No. 12374158), the National Key Research and
Development Program of China (Grant No. 2017YFA0303303), and the Tsinghua
University Initiative Scientific Research Program.

## References

  * Read and Green [2000] N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61, 10267 (2000).
  * Kitaev [2001] A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001).
  * Kitaev [2003] A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (Amsterdam) 303, 2 (2003).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  * Das Sarma  _et al._ [2015] S. Das Sarma, M. Freedman, and C. Nayak, Majorana zero modes and topological quantum computation, npj Quantum Inf. 1, 15001 (2015).
  * Mourik  _et al._ [2012] V. Mourik, K. Zuo, S. M. Frolov, S. Plissard, E. P. Bakkers, and L. P. Kouwenhoven, Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices, Science 336, 1003 (2012).
  * Deng  _et al._ [2016] M. T. Deng, S. Vaitiekėnas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, and C. M. Marcus, Majorana bound state in a coupled quantum-dot hybrid-nanowire system, Science 354, 1557 (2016).
  * Fornieri  _et al._ [2019] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Portolés, A. C. Drachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang, R. Kallaher, _et al._ , Evidence of topological superconductivity in planar Josephson junctions, Nature 569, 89 (2019).
  * Ren  _et al._ [2019] H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. Molenkamp, _et al._ , Topological superconductivity in a phase-controlled Josephson junction, Nature 569, 93 (2019).
  * Vaitiekėnas  _et al._ [2020] S. Vaitiekėnas, G. Winkler, B. van Heck, T. Karzig, M.-T. Deng, K. Flensberg, L. Glazman, C. Nayak, P. Krogstrup, R. Lutchyn, _et al._ , Flux-induced topological superconductivity in full-shell nanowires, Science 367, eaav3392 (2020).
  * Vaitiekėnas  _et al._ [2021] S. Vaitiekėnas, Y. Liu, P. Krogstrup, and C. Marcus, Zero-bias peaks at zero magnetic field in ferromagnetic hybrid nanowires, Nat. Phys. 17, 43 (2021).
  * Song  _et al._ [2022] H. Song, Z. Zhang, D. Pan, D. Liu, Z. Wang, Z. Cao, L. Liu, L. Wen, D. Liao, R. Zhuo, D. E. Liu, R. Shang, J. Zhao, and H. Zhang, Large zero bias peaks and dips in a four-terminal thin InAs-Al nanowire device, Phys. Rev. Research 4, 033235 (2022).
  * Cao  _et al._ [2023] Z. Cao, S. Chen, G. Zhang, and D. E. Liu, Recent progress on Majorana in semiconductor-superconductor heterostructures–engineering and detection, Sci. China: Phys. Mech. Astron. 66, 267003 (2023).
  * Liu  _et al._ [2018] Q. Liu, C. Chen, T. Zhang, R. Peng, Y.-J. Yan, C.-H.-P. Wen, X. Lou, Y.-L. Huang, J.-P. Tian, X.-L. Dong, G.-W. Wang, W.-C. Bao, Q.-H. Wang, Z.-P. Yin, Z.-X. Zhao, and D.-L. Feng, Robust and Clean Majorana Zero Mode in the Vortex Core of High-Temperature Superconductor $\mathbf{(}{\mathrm{Li}}_{0.84}{\mathrm{Fe}}_{0.16}\mathbf{)}\mathrm{OHFeSe}$, Phys. Rev. X 8, 041056 (2018).
  * Wang  _et al._ [2018] D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du, J. Schneeloch, _et al._ , Evidence for Majorana bound states in an iron-based superconductor, Science 362, 333 (2018).
  * Machida  _et al._ [2019] T. Machida, Y. Sun, S. Pyon, S. Takeda, Y. Kohsaka, T. Hanaguri, T. Sasagawa, and T. Tamegai, Zero-energy vortex bound state in the superconducting topological surface state of Fe (Se, Te), Nat. Mater. 18, 811 (2019).
  * Liu  _et al._ [2020] W. Liu, L. Cao, S. Zhu, L. Kong, G. Wang, M. Papaj, P. Zhang, Y.-B. Liu, H. Chen, G. Li, _et al._ , A new Majorana platform in an Fe-As bilayer superconductor, Nat. Commun. 11, 5688 (2020).
  * Kong  _et al._ [2021] L. Kong, L. Cao, S. Zhu, M. Papaj, G. Dai, G. Li, P. Fan, W. Liu, F. Yang, X. Wang, _et al._ , Majorana zero modes in impurity-assisted vortex of LiFeAs superconductor, Nat. Commun. 12, 4146 (2021).
  * Nadj-Perge  _et al._ [2014] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor, Science 346, 602 (2014).
  * Jeon  _et al._ [2017] S. Jeon, Y. Xie, J. Li, Z. Wang, B. A. Bernevig, and A. Yazdani, Distinguishing a Majorana zero mode using spin-resolved measurements, Science 358, 772 (2017).
  * Xu  _et al._ [2015] J.-P. Xu, M.-X. Wang, Z. L. Liu, J.-F. Ge, X. Yang, C. Liu, Z. A. Xu, D. Guan, C. L. Gao, D. Qian, Y. Liu, Q.-H. Wang, F.-C. Zhang, Q.-K. Xue, and J.-F. Jia, Experimental Detection of a Majorana Mode in the core of a Magnetic Vortex inside a Topological Insulator-Superconductor ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}/{\mathrm{NbSe}}_{2}$ Heterostructure, Phys. Rev. Lett. 114, 017001 (2015).
  * Sun  _et al._ [2016] H.-H. Sun, K.-W. Zhang, L.-H. Hu, C. Li, G.-Y. Wang, H.-Y. Ma, Z.-A. Xu, C.-L. Gao, D.-D. Guan, Y.-Y. Li, C. Liu, D. Qian, Y. Zhou, L. Fu, S.-C. Li, F.-C. Zhang, and J.-F. Jia, Majorana Zero Mode Detected with Spin Selective Andreev Reflection in the Vortex of a Topological Superconductor, Phys. Rev. Lett. 116, 257003 (2016).
  * Flensberg  _et al._ [2021] K. Flensberg, F. von Oppen, and A. Stern, Engineered platforms for topological superconductivity and Majorana zero modes, Nat. Rev. Mater. 6, 944 (2021).
  * Sengupta  _et al._ [2001] K. Sengupta, I. Žutić, H.-J. Kwon, V. M. Yakovenko, and S. Das Sarma, Midgap edge states and pairing symmetry of quasi-one-dimensional organic superconductors, Phys. Rev. B 63, 144531 (2001).
  * Law  _et al._ [2009] K. T. Law, P. A. Lee, and T. K. Ng, Majorana fermion induced resonant Andreev reflection, Phys. Rev. Lett. 103, 237001 (2009).
  * Flensberg [2010] K. Flensberg, Tunneling characteristics of a chain of Majorana bound states, Phys. Rev. B 82, 180516(R) (2010).
  * Sau [2020] J. Sau, Counting on Majorana modes, Science 367, 145 (2020).
  * Liu  _et al._ [2017a] C.-X. Liu, J. D. Sau, T. D. Stanescu, and S. Das Sarma, Andreev bound states versus Majorana bound states in quantum dot-nanowire-superconductor hybrid structures: Trivial versus topological zero-bias conductance peaks, Phys. Rev. B 96, 075161 (2017a).
  * Pan and Das Sarma [2020] H. Pan and S. Das Sarma, Physical mechanisms for zero-bias conductance peaks in Majorana nanowires, Phys. Rev. Research 2, 013377 (2020).
  * Prada  _et al._ [2020] E. Prada, P. San-Jose, M. W. de Moor, A. Geresdi, E. J. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado, and L. P. Kouwenhoven, From Andreev to Majorana bound states in hybrid superconductor–semiconductor nanowires, Nat. Rev. Phys. 2, 575 (2020).
  * Liu [2013] D. E. Liu, Proposed Method for Tunneling Spectroscopy with Ohmic Dissipation Using Resistive Electrodes: A Possible Majorana Filter, Phys. Rev. Lett. 111, 207003 (2013).
  * Liu  _et al._ [2022] D. Liu, G. Zhang, Z. Cao, H. Zhang, and D. E. Liu, Universal Conductance Scaling of Andreev Reflections Using a Dissipative Probe, Phys. Rev. Lett. 128, 076802 (2022).
  * Zhang  _et al._ [2022a] S. Zhang, Z. Wang, D. Pan, H. Li, S. Lu, Z. Li, G. Zhang, D. Liu, Z. Cao, L. Liu, L. Wen, D. Liao, R. Zhuo, R. Shang, D. E. Liu, J. Zhao, and H. Zhang, Suppressing Andreev Bound State Zero Bias Peaks Using a Strongly Dissipative Lead, Phys. Rev. Lett. 128, 076803 (2022a).
  * Wang  _et al._ [2022a] Z. Wang, S. Zhang, D. Pan, G. Zhang, Z. Xia, Z. Li, D. Liu, Z. Cao, L. Liu, L. Wen, D. Liao, R. Zhuo, Y. Li, D. E. Liu, R. Shang, J. Zhao, and H. Zhang, Large Andreev bound state zero-bias peaks in a weakly dissipative environment, Phys. Rev. B 106, 205421 (2022a).
  * Zhang  _et al._ [2023] G. Zhang, C. Li, G. Li, C.-L. Song, X. Liu, F.-C. Zhang, and D. E. Liu, Theoretical proposal to obtain strong Majorana evidence from scanning tunneling spectroscopy of a vortex with a dissipative environment, Phys. Rev. B 107, 195413 (2023).
  * Zhang  _et al._ [2022b] S. Zhang, Z. Wang, D. Pan, Z. Wang, Z. Li, Z. Zhang, Y. Gao, Z. Cao, G. Zhang, L. Liu, _et al._ , In situ tuning of dynamical Coulomb blockade in hybrid nanowire devices, arXiv:2211.07170 (2022b).
  * Wimmer  _et al._ [2011] M. Wimmer, A. Akhmerov, J. Dahlhaus, and C. Beenakker, Quantum point contact as a probe of a topological superconductor, New J. Phys. 13, 053016 (2011).
  * Moore  _et al._ [2018a] C. Moore, C. Zeng, T. D. Stanescu, and S. Tewari, Quantized zero-bias conductance plateau in semiconductor-superconductor heterostructures without topological Majorana zero modes, Phys. Rev. B 98, 155314 (2018a).
  * Chen  _et al._ [2019] C. Chen, Q. Liu, T. Zhang, D. Li, P. Shen, X. Dong, Z.-X. Zhao, T. Zhang, and D. Feng, Quantized conductance of Majorana zero mode in the vortex of the topological superconductor (Li0.84Fe0.16)OHFeSe, Chin. Phys. Lett. 36, 057403 (2019).
  * Zhu  _et al._ [2020] S. Zhu, L. Kong, L. Cao, H. Chen, M. Papaj, S. Du, Y. Xing, W. Liu, D. Wang, C. Shen, _et al._ , Nearly quantized conductance plateau of vortex zero mode in an iron-based superconductor, Science 367, 189 (2020).
  * Wang  _et al._ [2022b] Z. Wang, H. Song, D. Pan, Z. Zhang, W. Miao, R. Li, Z. Cao, G. Zhang, L. Liu, L. Wen, R. Zhuo, D. E. Liu, K. He, R. Shang, J. Zhao, and H. Zhang, Plateau Regions for Zero-Bias Peaks within $5\%$ of the Quantized Conductance Value $2{e}^{2}/h$, Phys. Rev. Lett. 129, 167702 (2022b).
  * Das Sarma and Pan [2021] S. Das Sarma and H. Pan, Disorder-induced zero-bias peaks in Majorana nanowires, Phys. Rev. B 103, 195158 (2021).
  * Pan  _et al._ [2021] H. Pan, C.-X. Liu, M. Wimmer, and S. Das Sarma, Quantized and unquantized zero-bias tunneling conductance peaks in Majorana nanowires: Conductance below and above $2{e}^{2}/h$, Phys. Rev. B 103, 214502 (2021).
  * Zeng  _et al._ [2022] C. Zeng, G. Sharma, S. Tewari, and T. Stanescu, Partially separated Majorana modes in a disordered medium, Phys. Rev. B 105, 205122 (2022).
  * Ferrier  _et al._ [2016] M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. Delagrange, R. Weil, R. Deblock, R. Sakano, A. Oguri, and K. Kobayashi, Universality of non-equilibrium fluctuations in strongly correlated quantum liquids, Nat. Phys. 12, 230 (2016).
  * Bastiaans  _et al._ [2018] K. M. Bastiaans, D. Cho, T. Benschop, I. Battisti, Y. Huang, M. Golden, Q. Dong, Y. Jin, J. Zaanen, and M. Allan, Charge trapping and super-Poissonian noise centres in a cuprate superconductor, Nat. Phys. 14, 1183 (2018).
  * Bastiaans  _et al._ [2021] K. M. Bastiaans, D. Chatzopoulos, J.-F. Ge, D. Cho, W. O. Tromp, J. M. van Ruitenbeek, M. H. Fischer, P. J. de Visser, D. J. Thoen, E. F. Driessen, _et al._ , Direct evidence for Cooper pairing without a spectral gap in a disordered superconductor above $T_{c}$, Science 374, 608 (2021).
  * Thupakula  _et al._ [2022] U. Thupakula, V. Perrin, A. Palacio-Morales, L. Cario, M. Aprili, P. Simon, and F. Massee, Coherent and Incoherent Tunneling into Yu-Shiba-Rusinov States Revealed by Atomic Scale Shot-Noise Spectroscopy, Phys. Rev. Lett. 128, 247001 (2022).
  * Liang [2022] Y.-X. Liang, Low-noise large-bandwidth transimpedance amplifier for measuring scanning tunneling shot noise spectra in cryogenic STM and its applications, Ultramicroscopy 234, 113466 (2022).
  * Golub and Horovitz [2011] A. Golub and B. Horovitz, Shot noise in a Majorana fermion chain, Phys. Rev. B 83, 153415 (2011).
  * Vuik  _et al._ [2019] A. Vuik, B. Nijholt, A. Akhmerov, and M. Wimmer, Reproducing topological properties with quasi-Majorana states, SciPost Phys. 7, 061 (2019).
  * Danon  _et al._ [2017] J. Danon, E. B. Hansen, and K. Flensberg, Conductance spectroscopy on Majorana wires and the inverse proximity effect, Phys. Rev. B 96, 125420 (2017).
  * Prada  _et al._ [2017] E. Prada, R. Aguado, and P. San-Jose, Measuring Majorana nonlocality and spin structure with a quantum dot, Phys. Rev. B 96, 085418 (2017).
  * Moore  _et al._ [2018b] C. Moore, T. D. Stanescu, and S. Tewari, Two-terminal charge tunneling: Disentangling Majorana zero modes from partially separated Andreev bound states in semiconductor-superconductor heterostructures, Phys. Rev. B 97, 165302 (2018b).
  * Cao  _et al._ [2019] Z. Cao, H. Zhang, H.-F. Lü, W.-X. He, H.-Z. Lu, and X. C. Xie, Decays of Majorana or Andreev Oscillations Induced by Steplike Spin-Orbit Coupling, Phys. Rev. Lett. 122, 147701 (2019).
  * Stanescu and Tewari [2019] T. D. Stanescu and S. Tewari, Robust low-energy Andreev bound states in semiconductor-superconductor structures: Importance of partial separation of component Majorana bound states, Phys. Rev. B 100, 155429 (2019).
  * Cao  _et al._ [2022a] Z. Cao, G. Zhang, H. Zhang, W.-X. He, C. Zeng, K. He, and D. E. Liu, Probing electron-hole weights of an Andreev bound state by transient currents, Phys. Rev. B 106, 075416 (2022a).
  * Anantram and Datta [1996] M. P. Anantram and S. Datta, Current fluctuations in mesoscopic systems with Andreev scattering, Phys. Rev. B 53, 16390 (1996).
  * [59] See Supplemental Material, which includes Refs. [58, 76, 77, 61, 41, 40, 28, 29, 42, 50, 78, 79, 80, 81, 82], for details of formula derivation, calculation parameters, and numerical simulations of hybrid nanowires .
  * Liu  _et al._ [2012] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Zero-Bias Peaks in the Tunneling Conductance of Spin-Orbit-Coupled Superconducting Wires with and without Majorana End-States, Phys. Rev. Lett. 109, 267002 (2012).
  * Ge  _et al._ [2023] J.-F. Ge, K. M. Bastiaans, D. Chatzopoulos, D. Cho, W. O. Tromp, T. Benschop, J. Niu, G. Gu, and M. P. Allan, Single-electron charge transfer into putative Majorana and trivial modes in individual vortices, Nat. Commun. 14, 3341 (2023).
  * Ahn  _et al._ [2021] S. Ahn, H. Pan, B. Woods, T. D. Stanescu, and S. Das Sarma, Estimating disorder and its adverse effects in semiconductor Majorana nanowires, Phys. Rev. Mater. 5, 124602 (2021).
  * Hess  _et al._ [2021] R. Hess, H. F. Legg, D. Loss, and J. Klinovaja, Local and nonlocal quantum transport due to Andreev bound states in finite Rashba nanowires with superconducting and normal sections, Phys. Rev. B 104, 075405 (2021).
  * Cao  _et al._ [2022b] Z. Cao, D. E. Liu, W.-X. He, X. Liu, K. He, and H. Zhang, Numerical study of PbTe-Pb hybrid nanowires for engineering Majorana zero modes, Phys. Rev. B 105, 085424 (2022b).
  * Hess  _et al._ [2023] R. Hess, H. F. Legg, D. Loss, and J. Klinovaja, Trivial Andreev Band Mimicking Topological Bulk Gap Reopening in the Nonlocal Conductance of Long Rashba Nanowires, Phys. Rev. Lett. 130, 207001 (2023).
  * Das Sarma  _et al._ [2023] S. Das Sarma, J. D. Sau, and T. D. Stanescu, Spectral properties, topological patches, and effective phase diagrams of finite disordered Majorana nanowires, Phys. Rev. B 108, 085416 (2023).
  * de Mendonça  _et al._ [2023] B. S. de Mendonça, A. L. R. Manesco, N. Sandler, and L. G. G. V. Dias da Silva, Near zero energy Caroli–de Gennes–Matricon vortex states in the presence of impurities, Phys. Rev. B 107, 184509 (2023).
  * Ménard  _et al._ [2020] G. C. Ménard, G. L. R. Anselmetti, E. A. Martinez, D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm, K. Flensberg, C. M. Marcus, L. Casparis, and A. P. Higginbotham, Conductance-Matrix Symmetries of a Three-Terminal Hybrid Device, Phys. Rev. Lett. 124, 036802 (2020).
  * Puglia  _et al._ [2021] D. Puglia, E. A. Martinez, G. C. Ménard, A. Pöschl, S. Gronin, G. C. Gardner, R. Kallaher, M. J. Manfra, C. M. Marcus, A. P. Higginbotham, and L. Casparis, Closing of the induced gap in a hybrid superconductor-semiconductor nanowire, Phys. Rev. B 103, 235201 (2021).
  * Wang  _et al._ [2022c] J.-Y. Wang, N. van Loo, G. P. Mazur, V. Levajac, F. K. Malinowski, M. Lemang, F. Borsoi, G. Badawy, S. Gazibegovic, E. P. A. M. Bakkers, M. Quintero-Pérez, S. Heedt, and L. P. Kouwenhoven, Parametric exploration of zero-energy modes in three-terminal InSb-Al nanowire devices, Phys. Rev. B 106, 075306 (2022c).
  * Aghaee  _et al._ [2023] M. Aghaee, A. Akkala, Z. Alam, R. Ali, A. A. Ramirez, M. Andrzejczuk, A. E. Antipov, P. Aseev, M. Astafev, B. Bauer, _et al._ (Microsoft Quantum), InAs-Al hybrid devices passing the topological gap protocol, Phys. Rev. B 107, 245423 (2023).
  * Pöschl  _et al._ [2022] A. Pöschl, A. Danilenko, D. Sabonis, K. Kristjuhan, T. Lindemann, C. Thomas, M. J. Manfra, and C. M. Marcus, Nonlocal conductance spectroscopy of Andreev bound states in gate-defined InAs/Al nanowires, Phys. Rev. B 106, L241301 (2022).
  * Banerjee  _et al._ [2023] A. Banerjee, O. Lesser, M. A. Rahman, C. Thomas, T. Wang, M. J. Manfra, E. Berg, Y. Oreg, A. Stern, and C. M. Marcus, Local and Nonlocal Transport Spectroscopy in Planar Josephson Junctions, Phys. Rev. Lett. 130, 096202 (2023).
  * Valentini  _et al._ [2022] M. Valentini, M. Borovkov, E. Prada, S. Marti-Sanchez, M. Botifoll, A. Hofmann, J. Arbiol, R. Aguado, P. San-Jose, and G. Katsaros, Majorana-like Coulomb spectroscopy in the absence of zero bias peaks, Nature 612, 442 (2022).
  * Smirnov [2022] S. Smirnov, Revealing universal Majorana fractionalization using differential shot noise and conductance in nonequilibrium states controlled by tunneling phases, Phys. Rev. B 105, 205430 (2022).
  * Mahaux and Weidenmüller [1969] C. Mahaux and H. A. Weidenmüller,  _Shell-model approach to nuclear reactions_ (North-Holland, Amsterdam, 1969).
  * Perrin  _et al._ [2021] V. Perrin, M. Civelli, and P. Simon, Identifying Majorana bound states by tunneling shot-noise tomography, Phys. Rev. B 104, L121406 (2021).
  * Sela  _et al._ [2006] E. Sela, Y. Oreg, F. von Oppen, and J. Koch, Fractional Shot Noise in the Kondo Regime, Phys. Rev. Lett. 97, 086601 (2006).
  * Lutchyn  _et al._ [2010] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures, Phys. Rev. Lett. 105, 077001 (2010).
  * Oreg  _et al._ [2010] Y. Oreg, G. Refael, and F. von Oppen, Helical liquids and Majorana bound states in quantum wires, Phys. Rev. Lett. 105, 177002 (2010).
  * Liu  _et al._ [2017b] C.-X. Liu, J. D. Sau, and S. Das Sarma, Role of dissipation in realistic Majorana nanowires, Phys. Rev. B 95, 054502 (2017b).
  * Groth  _et al._ [2014] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, Kwant: a software package for quantum transport, New J. Phys. 16, 063065 (2014).