Teodorescu P.P. Mechanical Systems, Volume III: Analytical Mechanics

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MECHANICAL SYSTEMS, CLASSICAL MODELS

472

(22.2.93) are reduced to Chaplygin’s classical equations (22.2.89''). The equations

(22.2.93) are different from the Boltzmann-Hamel equations (22.2.22), (22.2.22'),

because there are involved only the coefficients

β , nor the coefficients

γε

α .

22.2.4.2 Voronets’s Equations

P. V. Voronets considers, in 1901, non-holonomic mechanical systems acted upon

by conservative forces, which derive from the potential

. The constraint

non-holonomic relations are written in the form (22.2.88), but the coefficients

+hk

depend on all the generalized co-ordinate

, ,...,

qq q. The quantities

−

++ + +

∂∂ ∂ ∂

⎛⎞

=−+ −

⎜⎟

∂∂ ∂ ∂

⎝⎠

∑

,, , ,

h ki h kj h ki h kj

hlj hli

hl hl

cc c c

qq q q

(22.2.94)

cannot vanish simultaneously, because the system of relations is non-integrable.

Lagrange’s equations with multipliers can be written in the form

−

∂+

∂

⎛⎞

−= =

⎜⎟

∂∂

⎝⎠

∂+

∂

⎛⎞

−==−

⎜⎟

∂∂

⎝⎠

∑

()

, 1,2,..., ,

()

,1,2,..., .

khkj

hk hk

cj h

tq q

ksh

tq q



Taking into account the constraint relations (22.2.88), we can replace the function

by a function

Θ , so that

()

12 12 12 12

, ,..., , , ,..., ; , ,..., , , ,..., ;

ss s

Tqq qqq qt qq qqq qtΘ    ;

observing that

−

∂∂ ∂

∑

hkj

qq q

 

we can write

−

∂∂ ∂ ∂

⎛⎞ ⎛⎞

⎡

⎤

⎛⎞

=+ +

⎜⎟

⎜⎟ ⎜⎟

⎢

⎥

∂∂ ∂ ∂

⎝⎠

⎣

⎦

⎝⎠ ⎝⎠

∑

dd d

dd d d

hkj

hk hk

TTT

tq tq tq t q

  

Using the above Lagrange’s equations with multipliers, we obtain

−

∂+ ∂+

∂∂

⎛⎞

⎡⎤

=+ + =

⎜⎟

⎢⎥

∂∂ ∂ ∂

⎣⎦

⎝⎠

∑

() ()

, 1,2,...,

hkj

hk hk

TU TU

cjs

tq q q t q



By means of the relations

−

∂

∂∂ ∂

=+ =

∂∂ ∂ ∂

∑∑

, 1,2,...,

sh h

hkj

ii i

qi s

qq q q



Dynamics of Non-holonomic Mechanical Systems

473

we can eliminate the derivatives

∂∂/

Tq and

∂∂/

Tq ,

= 1,2,...,

=−1,2,...,ksh; taking into account the notations (22.2.94), we get Voronets’s

equations in the form

111

() ()

, 1,2,..., .

sh shh

ji i

hkj

hk hk

kki

cqjh

tq q q q

ΘΘ

−−

===

∂+ ∂+

∂∂

⎛⎞

=+ + =

⎜⎟

∂∂ ∂ ∂

⎝⎠

∑∑∑





(22.2.95)

If the co-ordinates

++12

, ,...,

qq q do not appear explicitly in the expression of the

kinetic energy, of the potential

U and of the constraint relations, then we find again

Chaplygin’s equations (22.2.89'), Voronets’s equations having thus a more general

character.

The equations (22.2.95) read

−

−−

== =

∂∂ ∂

⎛⎞

−−

⎜⎟

∂∂ ∂

⎝⎠

∂

−=+=

∂

∑

∑∑ ∑

11 1

, 1,2,..., ,

hkj

shh sh

ji i j

hkj hk

ki k

tq q q

qQ c Q j h

ΘΘ Θ



(22.2.95')

in the case in which the mechanical system is non-conservative, Voronets deduced

these equations by means of Hamilton’s variational principle too; as well, he found the

form of these equations also in quasi-co-ordinates.

22.2.4.3 Volterra’s Equations

V. Volterra introduced, in 1898,

s parameters

, ,..., , 3

pp p s n≤ , which he

called characteristics of a discrete mechanical system of

n particles of masses

M , in

the representative space

E (see Sect. 18.1.1.2); these characteristics are linked to the

generalized velocities

=, 1,2,...,3

Xk n



, by the relations (Volterra, V., 1893, 1899)

==, 1,2,..., 3

kkj

Xpk nξ



(22.2.96)

where

12 3

( , ,..., )

kj kj

XX Xξξ and where we have used the summation convention

of the dummy indices from

1tos . If = 3sn, the relations (22.2.96) being

non-integrable, then the parameters

p become quasi-co-ordinates. We denote by

, 1,2,...,

wj s= , the characteristics of the virtual displacements, so that

δ=δ =, 1,2,...,3

kkj

Xwk nξ ;

(22.2.96')

hence, it results

∂

δ= δ

∂



MECHANICAL SYSTEMS, CLASSICAL MODELS

474

The kinetic energy of the mechanical system can be expressed in the form

∑

ij i j

TMXEpp



(22.2.97)

where

∑

ij ji

kkikj

EE Mξξ

(22.2.97')

are known functions of the co-ordinates

12 3

, ,...,

XX X . We introduce the notations

∂

⎡⎤

=− =δ

⎣⎦

∑∑

()

( ) () ()

kkl

mll

qr qr rq

lm lm lr

aebbeE

ξξ

(22.2.98)

where δ

is Kronecker’s symbol and =, , , 1,2,...,qrlm s. Let

∑

, 1,2,...,

kkj

PXj sξ ,

(22.2.98')

be the corresponding given generalized forces; we introduce the notation

∂∂

== =

∂∂

∑

, 1,2,...,

Tjs

(22.2.98'')

By a non accurate demonstration, which he corrected afterwards (incorrect too!),

Volterra obtained the equations

∂∂

⎛⎞

=++=

⎜⎟

∂∂

⎝⎠

()

, 1,2,...,

apTPj s

tp p

(22.2.99)

which we call Volterra’s equations; but these equations are correct as it has been shown

by Yu. I. Neĭmark and N. A. Fufaev.

We can write these equations also in the form

∂

⎛⎞

=+=

⎜⎟

∂

⎝⎠

()

,1,2,...,

bpp Pj s

(22.2.99')

22.2.4.4 Maggi’ Equations

G. A. Maggi admitted, in 1896, that the virtual generalized displacements

, 1,2,...,

qj sδ= , which verify the m constraint relations (22.1.2), can be expressed

in the linear form

Dynamics of Non-holonomic Mechanical Systems

475

−

δ= =

∑

, 1,2,...,

jiji

qEj sε ,

(22.2.100)

where

−

, ,...,

εε ε are independent parameters. The d’Alembert–Lagrange equation,

written in the form (18.2.27'), leads to

−

⎧⎫

∂∂

⎡⎛ ⎞ ⎤

−−=

⎨⎬

⎜⎟

⎢⎥

∂∂

⎣⎝ ⎠ ⎦

⎩⎭

∑∑

sm s

ij i i

tq q



with the notation

==−

∑

, 1,2,...,

ijij

EQEi sm.

(22.2.101)

Taking into account that the parameters

−

, ,...,

εε ε are independent, we obtain

Maggi’s equations

∂∂

⎡⎛ ⎞ ⎤

−==−

⎜⎟

⎢⎥

∂∂

⎣⎝ ⎠ ⎦

∑

, 1,2,...,

ij i

EEi sm

tq q



(22.2.102)

Maggi showed in 1901 that, in case of a discrete mechanical system of

s particles,

these equations are reduced, in particular, to Volterra’s equations if conditions of the

form (22.2.90) are fulfilled.

22.2.4.5 Canonical Form of the Equations of Motion in Quasi-co-ordinates

S. A. Chaplygin and J. Quangel have shown in 1906 how the equations of motion of

a non-holonomic mechanical system can be obtained in quasi-co-ordinates, in a form

analogue to that of Hamilton’s equations for holonomic mechanical system; one can

thus extend known results obtained in case of holonomic systems to non-holonomic

ones.

We assume that the considered mechanical system are conservative, so that we

introduce the kinetic potential

=+TUL ; using the constraint relations (22.1.1'), we

can write Lagrange’s equations with multipliers

∂∂

⎛⎞

−= =

⎜⎟

∂∂

⎝⎠

∑

, 1,2,...,

kkj

aj s

tq q



(22.2.103)

Introducing Hamilton’s function (19.1.11), we get Hamilton’s equations

∂∂

==−+ =

∂∂

∑

, , 1,2,...,

kkj

qp ajs

 ,

(22.2.104)

corresponding to conservative non-holonomic mechanical systems. Together with the

constraint relations (22.2.1'), one obtains a complete system of

+2sm differential

equations for the unknowns

, , 1,2,..., , 1,2,...,

qp j sk m==, and

, 1,2,...,

kmλ = , as functions of the time t .

MECHANICAL SYSTEMS, CLASSICAL MODELS

476

Having to do with conservative forces, we can introduce the kinetic potential

L ;

hence, the equations (22.2.35) read

⎛⎞

∂∂ ∂

−+ == −

⎜⎟

∂∂ ∂

⎝⎠

** *

0, 1,2,...,

kj k

jj j

jsm

γπ

ππ π





LL L

(22.2.105)

In general,

L depends on all quasi-velocities =, 1,2,...,

isπ , which are linked to the

generalized velocities

, 1,2,...,

qj s= , by the relations (22.2.17). We assume that m

non-holonomic and catastatic constraint relations (22.2.31) take place; the

quasi-co-ordinates

π will be chosen so that the last m quasi-velocities do vanish

−+

==0, 1,2,...,

smk

kmπ ,

(22.2.106)

if the constraint relations (22.2.31) hold. As. well, corresponding to the relations (22.2.33),

we assume that

−

∑

,1,2,...,

jk k

qjsβπ

(22.2.107)

Thus, the equations (22.2.105), (22.2.107), in which we take into account the conditions

(22.2.106), form a complete system of

−2sm

differential equations for the unknowns

( ), 1,2,...,

qt j s= and ( ), 1,2,...,

tk smπ =− .

Introducing the quasi-momenta (22.2.23), we define the function

=−

Kpπ L ,

(22.2.108)

analogue to Hamilton’s functions. Taking into account

−−

∂∂

δ= δ = =

∂∂

∑∑

, , 1,2,...,

sm sm

jk k jk

qjs

βπ β

we can write

−

∂∂

δ=δ+δ− δ− δ

∂∂

=δ− δ=δ− δ

∂∂

∑

jj jj j j

jjjj

jk k

Kpp q

ππ π

πβπππ

 





the canonical quasi-co-ordinates being thus

π and

, 1,2,...,

pj s= . After eliminating

the quasi-velocities, the variation of the function

** *

12 1 2

( , ,..., , , ,..., ; )

KKqq qpp pt is

of the form

Dynamics of Non-holonomic Mechanical Systems

477

∂∂ ∂∂

δ= δ+ δ= δ+ δ

∂∂

jjj

KK KK

Kq p p

so that it results

∂∂ ∂

=−=

∂∂

∂

ππ



;

the equations of motion (22.2.105) are thus rewritten in the form

∂∂∂

==−− =−

∂

∂∂

, , 1,2,...,

jj j

KKK

ppjsm

πγ

 ,

(22.2.109)

obtaining the canonical equations of motion in a non-holonomic case. The conditions

(22.2.106) lead to

−+

∂

0, 1,2,...,

smk

(22.2.106')

and the relations (22.2.107) to

−

∂

===

∂

∑

0, 1,2,...,

qjs

β .

(22.2.107')

The equations (22.2.106'), (22.2.107') and (22.2.209) form a complete system of

−3sm

differential equations to determine the unknown functions

(), ()

qt pt,

= 1,2,...,js, and ( ), 1,2,...,

tk smπ =− .

Let us start now from Chaplygin’s equations (22.2.89''), where the function

12 12

( , ,..., , , ,..., )

qq qqq q LL and the coefficients

+ ,hkj

c do not depend on the

−sh last generalized co-ordinates

++12

, ,...,

qq q, having to do with a system of h

differential equations with the unknown functions

( ), 1,2,...,

qt j h= .

We introduce the generalized momenta

( ), 1,2,...,

pt j h=

, by the relations

∂

, 1,2,...,

pjh



;

(22.2.110)

Hamilton’s function is given by

=−

∑

Hpq L .

(22.2.111)

MECHANICAL SYSTEMS, CLASSICAL MODELS

478

By calculations analogue to these above, we get

∂∂∂

=−=

∂∂∂

***

pqq



and the system (22.2.89'') can be replaced by

−

∂

∂∂

∂∂ ∂

⎛⎞

=− − −

⎜⎟

∂∂∂∂∂

⎝⎠

∑

, 1,2,..., ,

hkl hkj

hk l l

qjh

qqqqp



(22.2.112)

We obtained thus Chaplygin’s equations in a canonical form; they form a complete

system of

equations for the unknown functions ()

qt and ( ), 1,2,...,

ptj h= .

22.3 Gibbs–Appell equations

Starting from the acceleration energy as “central function”, P. Appell studies in detail,

in 1899, the equations of motion given by W. Gibbs in 1879; we present these

equations and apply them to various particular cases (Appell, P., 1941–1953).

22.3.1 Gibbs–Appell Equations of Motion

We introduce, in what follows, the acceleration energy, making a thorough study of it;

using this mechanical quantity, we deduce then the Gibbs–Appell equations, valid for

holonomic mechanical systems, as well as for non-holonomic ones. These equations

represent the simplest and the most extensive form of the equations of motion.

22.3.1.1 Acceleration Energy

Searching a form of the equations of motion which be unique either for the holonomic

mechanical systems or for the non-holonomic ones, the rôle of the kinetic energy will be

taken by the acceleration energy, called Appell’s function too and defined by

∑∑

ii ii

Sm m



;

(22.3.1)

expressing the accelerations

by means of the generalized co-ordinates, the generalized

velocities and the generalized accelerations in the form (18.2.14), the accelerations energy is

given by

=++

SS S S

(22.3.2)

where

∂∂

===⋅=

∂∂

∑2

, , , 1,2,...,

jk k jk kj

Sqq m jk s

γγγ 

(22.3.2')

Dynamics of Non-holonomic Mechanical Systems

479

is a quadratic form in the generalized accelerations,

∂∂ ∂∂∂

⎛⎞

=⋅+⋅+⋅=

⎜⎟

∂∂∂ ∂∂ ∂

⎝⎠

∑

rr rrr

2 , 1,2,..., ,

ii iii

ji i

jjj

kl k

mqq js

qqq qq q







(22.3.2'')

is a linear form in the generalized accelerations and

000

422 4

ii ii i i

ij ij i

kl k

jjjj

kl k k

Smqqqq

qq qq

qqq qq q

qq q q q qq q

γγ

∂∂

⎡

== ⋅

⎢

∂∂ ∂∂

⎣

∂∂ ∂∂∂ ∂

⎛⎞⎤

+⋅ +⋅+⋅+⋅+

⎜⎟

⎥

∂∂ ∂ ∂ ∂ ∂∂ ∂

⎝⎠⎦

∑



 

  

  

rr rr r r

rrr

(22.3.2''')

is a constant with respect to these accelerations.

We notice that

kjk

gγ , the coefficients being the same as these which appear in

case of the kinetic energy, which are given by the formula (18.2.15'). We can write

∂

⎛⎞

=≥

⎜⎟

∂

⎝⎠

∑

Smq

 .

(22.3.3)

This quadratic form vanishes only if each bracket vanishes, hence for

( / ) , 1,2,...,

ijj

qq i n∂∂ = =0r . Writing these linear equations in components, we

notice that the matrix

∂∂[/]

Xq of the coefficients is of rank s (the co-ordinates

are specified in Sect. 18.1.1.2); it results

0, 1,2,...,

qj s== , the quadratic form

being thus positive definite. All the properties mentioned in Sect. 18.2.1.3 for the

coefficients

g remain valid. We notice as well, that

S is a non-singular quadratic

form (eventually, excepting the (isolated) points in which the correspondence (18.2.1)

(or (18.2.2)) is not one-to-one).

In case of catastatic constraints, we have

, 1,2,...,

in==0



r , so that

0, , 1,2,...,

jk sγγγ== = = ; hence, ===

0SSS



, so that = 0S



, (the

acceleration energy does not depend explicitly on time). As well,

∂∂

=⋅ =

∂∂∂

∑

,1,2,...,

mqqjs

qqq

γ  ,

(22.3.4)

∂∂

=⋅

∂∂ ∂∂

∑

kkl

mqqqq

qq qq

γ  ,

(22.3.4')

the respective coefficients being homogeneous forms in generalized velocities. In

general, the coefficients of the quadratic form are of the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

480

( , ,..., ; ), , 1,2,...,

jk jk jk

ggqqqtjk sγ == = , and the other coefficients are of the

form

( , ,..., ; ), 0,1,2,...,

qq qt j sγγ==

; we have

( , ,..., ), , 1,2,..., ,

( , ,..., ), 0,1,2,..., ,

jk jk

qq q jk s

qq q j s

γγ

in case of catastatic constraints. We notice also that, in fact, the conditions

==r , 1,2,...,

in0



, are only necessary conditions of catastaticity, so that the above

properties can take place even if the constraints are not catastatic (if they are

non-holonomic).

If the mechanical system

S is studied with respect to an inertial frame of reference

′

R and with respect to a non-inertial frame of Koenig type R (which does not rotate

with respect to the fixed inertial frame), then we notice that

, 1,2,...,

′′

=+ =aa a , so that

=== =

′′ ′′ ′

==+⋅+

∑∑∑ ∑

aaaaa

22 2

111 1

11 1

22 2

i i

nnn n

iiiii

iii i

Sm mm m,

wherefrom it results

′′′

=+⋅ +

∑

aa a

OO O

SM mS;

(22.3.5)

we have denoted by

∑

(22.3.5')

the acceleration energy with respect to the frame of Koenig type. In case of a Koenig

frame of reference, the pole of which coincides with the mass centre ( ≡OC), the

static moment with respect to

C vanishes, so that

∑

m 0 ;

it results

′′

∑

SS M

(22.3.5'')

and we can state

Theorem 22.3.1 (of Koenig type). The acceleration energy of a discrete mechanical

system with respect to a given frame of reference is equal to the sum of the acceleration

energy of the same system with respect to a Koenig frame of reference and the

Dynamics of Non-holonomic Mechanical Systems

481

acceleration energy of the mass centre at which we have considered that the whole

mass of the mechanical system is concentrated, with respect to the given frame.

22.3.1.2 Equations of Motion

Starting from the relations (22.2.18), which link the kinematic characteristics (and

the quasi-co-ordinates) to the generalized co-ordinates, and assuming the existence of

m non-holonomic constraint relations (22.1.19), (22.1.20), we eliminate the

corresponding kinematic characteristics, remaining with

, 1,2,..., ,

ihω = hsm=− ,

non-zero kinematic characteristics. In this case, the generalized velocities will be

expressed in the form

=+=

∑

, 1,2,...,

kjkk

qksβπ β

(22.3.6)

as functions of quasi-velocities, where

( , ,..., ; ), 0,1,2,...,

kj kj

qq qt j hββ==, it

results also that

δ= δ+ =

∑

, 1,2,...,

kjkk

qksβπ β .

(22.3.6')

Taking into account the expression (18.2.4') of the velocity, we get

=+=

∑

vBB

, 1,2,...,

iijji

inπ

(22.3.7)

δ= δ =

∑

, 1,2,...,

iijj

inπ ,

(22.3.7')

where

∂∂

==+

∂∂

BB r

ij i i

kj k

ββ



(22.3.7'')

with

( , ,..., ; ), 1,2,..., , 0,1,2,...,

ij ij

qq qt i nj h===BB . One can thus calculate

the acceleration

=+=

∑

..., 1,2,...,

iijj

inπ ,

(22.3.8)

where we have put in evidence only the terms which contain the quasi-accelerations; it

results

∂

== =

∂

, 1,2,..., , 1,2,...,

injh



(22.3.8')