Motion of Nonholonomous Rheonomous Systems in the Lagrangian Formalism ()
1. Introduction
Nonholonomous systems are beyond a doubt more and more considered, mainly in view of the important implementations they exhibit for mechanical models.
From the mathematical point of view, the draft of the equations for such systems commonly matches the introduction of the quasi-velocities and, starting from the Euler-Poincaré equations [1] , several sets of equations have been formulated.
The time-dependent case is probably more disregarded in literature: we direct here our attention especially to rheonomic systems, admitting the holonomic and nonholonomic constraints and the applied forces to depend explicitly on time.
The nonholonomous restrictions are assumed to be linear, so that the equations of motion can be written in the linear space of the admissible displacements of the system, eliminating the Lagrangian multipliers connected to the constraints.
If on the one hand the use of quasi-velocities formally complicates calculations, on the other hand the final form of the system allows computing the equations merely by means of a list of particular matrices, once the Lagrangian function has been written and the quasi-velocities have been chosen.
We pay attention to keep separated the various contributions to the mobility of the system; the customary stationary case can be easily recovered from the general equations we will write.
An energy balance-type equation, which will be proposed in terms of the quasi-velocities, affirms the conservation of the energy in the full stationary case and shows the contributions of the different terms in the rheonomic context.
We will conclude by presenting some applications of the developed system of equations.
Most of the formal notation used onward is explained just below. For a given a list of variables
,
the operator
will compute the gradient
of a scalar funcion
, and
calculates the
Jacobian matrix of a vector
:
,
,
.
Anywhere, vectors are in bold type and are meant as columns: row vectors will be written by means of the
transposition symbol
. Moreover,
is the null column vector
,
is the
null matrix,
the
null matrix and
the unit matrix of size
.
2. Modelling the System
The theoretical frame we point and expand is contained in [2] .
Let us consider a system of n point particles
,
,
restricted both by
geometrical constraints and by
kinematic constraints,
,
,
:
(1)
(2)
where
is the representative vector of the system and, for each fixed t,
,
is a matrix of size
,
a vector in
. The constraint equations are assumed to be independent:
(3)
We first make use of the
integer relations (1) in order to write the system configuration by means of the parametrisation
, where
,
are the local Lagrangian coordi-
nates. The velocity of the system
agrees with (1), but it must be consistent also with the
differential constraints (2) which are rewritten, in terms of the Lagrangian coordinates
and of the generalized velocities
, as
(4)
and
in case of fixed constraints. The dynamics of the system is summarized in
by
, where
represents the momentum of the system,
,
respectively all active forces and all constraint reactions (the i-th triplet concerning
). The virtual displacements of the system at each time t and at each position
are the vectors in
such that [2]
(5)
giving in each
, t the
dimensional linear space
,
where
are the rows of
. At the same time, the assumption of smooth constraints
make us write
(6)
where
,
are unknown multipliers.
The projection of the dynamics equation on the subspace generated by the
vectors
,
(the
columns of
), although such as space strictly includes
, if
, is anyhow noteworthy:
(7)
where we assumed
and we defined the Lagrangian function
(8)
with
symmetric and positive definite matrix of size
and
. The
Equation (7) written for the
unknown quantities
,
have to be considered together with the
Equations (4).
In order to improve (7), we see from (4) and (5) that
(virtual displacements) is the set of vectors
such that
,
.
Owing to (3) and recalling (4), it is
, hence the solution of the come last linear system, which ex-
plicitly writes
,
is
(9)
with
appropriate coefficients and
arbitrary factors in
,
. We conclude that
, or, equivalently, the
vectors
,
form a basis for
.
At this stage, calling
the matrix of size
and elements
and noticing that the columns of
give the basis for
, the projection of the dynamics equation on
gives, by virtue also of (6):
(10)
where the effect of the nonholonomic constraints (through
) on the ordinary Lagrangian equations for hol-
onomic systems is evident (in the absence of (2), say
, both (10) and (7) are
).
The
differential Equation (10) are for the
unknown quantities
and they have to be combined together with the
Equation (4). With respect to (7), they have the advantage of not exhibiting the multipliers
.
Remark 2.1 Either Equation (7) or (10) can be employed not necessarily for discrete systems of point particles: once the Lagrangian coordinates have been selected and the Lagrangian function has been written, they can be the same calculated.
The expedience of introducing quasi-velocities (or pseudovelocities) which have to be chosen in a suitable way in order to disentangle the mathematical problem, is by custom performed in nonholonomic systems.
Following the adopted standpoint, the definition of the quasi-velocities steps in establishing a specific (and convenient) connection between
and ![]()
(11)
where
are required to guarantee that the square matrix of size
is invertible. In
this way, each set of kinetic variables
is linked to a singular set of quasi-velocities
, and vice versa. More precisely, (11) and (4) give
(12)
where
is the same as (9) and
is a
matrix. The first system in (12) shows both the selection on the coordinates
of the tangent space
necessary to fulfill the restrictions on the system’s velocity (leading to the subspace
) and the kinematic conditions themselves.
In order to express (10) as a function of the variables
,
and to eliminate
, it suffices to extract from (12)
(13)
and to define
(14)
where
(15)
By using the formulae (see (11))
(16)
where
is the
matrix whose elements are, for each
, ![]()
![]()
we can write (10) in terms of the demanded variables (we use
, see (12)):
(17)
Remark 2.2 Multiplying both sides of (17) by
and performing the customary steps leading to the energy balance one finds
(18)
In the stationary circumstance
,
,
and
the Legendre transform
of
is conserved.
Our next step is writing (17) explicitly, sorting the terms in a suitable way: we start from the calculation
(19)
so that (17) takes the structure
(20)
Provided that
means the
-th column of any matrix
and defining for any
the operation
(21)
for a matrix
of size
, the terms in (20) are defined by the following expressions, where
means the
-th component of any vector
and
:
![]()
![]()
![]()
Equation (20) is sorted on the strength of the quasi-velocities
:
is quadratic with respect to
,
is linear with respect to the same variables and
does not contain
.
Since A is a positive-definite square matrix and
, even
is a positive-definite ![]()
symmetric matrix. Hence, system (20) + (13) can be written in the normal form
, where
is
a list of
functions, whose regularity allows us to apply the standard theorems on existence and uniqueness of solutions to first-order equations with given initial conditions.
Before commenting Equation (20), we remark that the
entries of the matrix
defined in (21) are, for each
:
(22)
We see now that a certain number of significant cases are encompassed by (20):
・ merely geometric constraints, corresponding to
,
, so that (4) are not present and all the terms containing
,
and the related quantities
must be dropped in (20). Furthermore:
○ selecting
(quasi-velocities are the generalized velocities) in (11) and (13) means
![]()
so that in (20) are written with as
![]()
thus the Lagrangian equations for geometric constraints (bearing in mind (22))
![]()
, are achieved.
○ establishing (11) as
(quasi-velocities are the generalized momenta) means
![]()
In this case (13) together with (20) are the Hamiltonian equations for
:
indeed the first one is
, whereas (20) reduces to
(23)
with
![]()
![]()
![]()
(actually from
one deduces
and
so that, also considering
, many terms are cancelled).
Since
for any
, it is
![]()
therefore (23) is
, as stated.
・ Stationary case, where the different contributions producing the dependence on
must be dropped. If one is dealing with a scleronomic system (covering many of common instances), the constraints (1), (2) reduce to
(24)
(25)
Conditions (24) entail
and
(if even the forces are independent of
time), on the other hand (25) implies
.
Equation (11), if one reasonably chooses
and
independent of
(otherwise, changes will be obvious), is
. Since
, system (20) + (13) drastically simplifies to
(26)
or, index by index, calling
the entries of the matrix
,
and having in mind (22)
(27)
where
is, for each index
, the square matrix of order ![]()
![]()
Equations (27) are identified with the Boltzmann-Hamel Equations (17) for the Lagrangian function
(see [3] [4] ). In this case the Legendre transform
is a first integral of
motion, see Remark 1.2.
・ Reduced Lagrangian function for geometric constraints: in case of ν cyclic variables
,
,
(4) can play the role of the
relations derived from the first integral of motion
,
,
that is
,
. Assuming that
,
, it is possible
to acquire, according to (13),
,
, where
,
and bj depend
only on
. At this point, setting
,
,
we have, with respect to (11) and (12),
and
(Kronecker’s delta),
. Equation (20), which writes simply
, are the equations of motion for the reduced Lagrangian
,
with
,
; on the other hand,
for
, are the so called reconstruction equations.
3. Some Applications
We adopt now Equation (20) in order to formulate a couple of remarkable mechanical systems, each of them in a double form, as scleronomous and rheonomous model.
3.1. Pendulum on a Skate
Consider a system of four points
,
and
equidistant and lying on a horizontal plane,
equidistant from
and
,
oscillating around
, equidistant from
and
and coplanar to the latter points and
(see Figure 1).
The system represents a simple model for the motion of a bicycle, as exhibited in [5] : the mass in
is added on order to sketch the rigid structure of the bicycle (just as
and
represent the front and the back wheels), as well as the pendulum
simulates the movement of a driver.
Let
be a fixed point on the horizontal plane containing
and
,
the ascending vertical versor,
the midpoint of the segment
and
perpendicular to the same segment: the geometrical constraints (1) are written by means of the constant assigned values
,
,
as
(28)
Since the constraints are independent and
, we have
,
. Setting a fixed reference system
and the angle
between
and
, the angle
between
and
, the angle
between
and
, one defines the orthonormal versors
, ,
so that
,
,
,
and choose the five parameters
as Lagrangian coordinates, where
.
Opting for considering the segment
as a rigid bar of mass M (instead of a discrete point system, although not significant), the Lagrangian function (8) is written with
,
,
and
![]()
Figure 1. A simple model for the motion of a bicycle.
![]()
where
is the total mass and
(29)
The only one kinetic constraint concerns with the velocity of the back “wheel”
, to be aligned with the segment:
(30)
or
, that is (4) for
,
,
.
Hence
and the four quasi-velocities (11) are selected by setting
and.
Furthermore, (12) gives
![]()
so that
![]()
By computing the first line in (26) one finds the four equations of motion
![]()
joined with the conservation of the quantity
.
3.2. Assignment of the Front Motion
We modify the previous model by forcing the velocity of the front “wheel” to be a known function of time (a simpler version was considered in [6] for the motion of a bike):
. With respect to (28), time
enters explicitly the geometrical constraints and the fourth one has to be removed. Hence, in this example we have
,
,
and we choose
. The midpoint
is located by
and the Lagrangian function (8) is written with
, ,
whereas
is the same function.
The constraint (30) is now
, that is (4) for
,![]()
. Choosing
,
we have simply
![]()
Equation (20) are written with
![]()
and correspond to
![]()
The energy balance (18) writes
and the function in the right side of the latter equality is
![]()
with
.
3.3. Rolling Disk with Pendulum
A different version of the model 3.1 lies in replacing the bar with a disk and obtaining the unicycle with rider model presented in [7] (see Figure 1 again, replacing the bar with the disk). The system we consider here is a disk of diameter
and mass
, in addition to the same points
(with mass
) and
(with mass
). We directly choose the coordinates (see Remark 2.1)
where the new parameter
is the angle of rotation of the disk around the axis perpendicular to the disk and passing through the centre. The Lagrangian function is written with
and
![]()
where
and (see (29))
![]()
The kinematic constraint of rolling without sliding entails the zero velocity of the contact point
:
(31)
which is (4) with
,
and
.
This time
and the choice
![]()
leads to
![]()
where
. Moreover
![]()
with
![]()
and the corresponding equations of motion (20) are
![]()
where
![]()
3.4. Assigned Rotational Velocity of the Disk
We finally consider the same system with the differential constraint (31), but
assigned (we may think about an engine-driven motor bike or electric bike): in that case
and (4) is setted
with
and
,
.
The Lagrangian fucntion (8) is written with A the same as in the previous Example 3.1, except for removing
the fourth row and the fourth column, and
,
. In the matter of (11), which has to be written for
, if one defines the quasi-velocities
,
,
one gets
and
![]()
Calculating the products in (15) gives
,
,
![]()
and the computation of (20) gives the three equations of motion
![]()
4. Conclusions
The paper aims at formulating a general scheme of equations for rheonomic mechanical systems exposed to either geometrical (1) and differential (2) constraints. We pay special attention to tell apart the different contributions due to the explicit dependence on time, deriving from the holonomous constrictions (via
and
of (8)), the nonholonomous constrictions (via
of (4)) and the definition of quasi-velocities (via
) of (11)).
Since the equations of motion are projected in the subspace of the velocities allowed by the constraints (both holonomous and nonholonomous), the Lagrange multipliers are absent from the equations.
The procedure proposed by (20) requires only calculation of the Jacobian matrix of vectors and the algebraic multiplication of matrices and vectors.
Making use of quasi-velocities renders the equations versatile to more than one formalism and, as it is known, the appropriate choice of them meets the target of facilitating the mathematical resolution of the problem.
The last point is part of the matters listed below and which will be dealt with in the future:
-Find an appropriate choice of the quasi-velocities in order to disentangle (20) from (13) as much as possible,
-Make use of the structure of the equations and of the properties of the various matrices involved in order to study the stability of the system,
-Take advantage of some peculiarity of the system in order to refine the set of equations and achieve information.
The latter subject is faced in [8] [9] for the stationary case by means of a robust and complex theory in connection with symmetries in nonholonomic systems.