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

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Lagrangian Mechanics

−+ += = =−rr

(cos) 0, ,

Pl Qh

αβ ϕ γ α β , =γ arbitrary,

(18.1.80)

representing – in general – ovals of Descartes; for

= 0γ

it results a Pascal’s limaçon.

C. Neumann considered three particles

, linked so that the area of the

triangle

123

PPP

be constant, and upon which act the given forces

respectively (Fig. 18.7). If

, r

are position vectors of the particles

and

, respectively, and if we denote

==− = ×=rrr rru

12 13

, , 1,2, 3,

jk k k

PP j k C

JJJJG

where

=uu()t is a vector normal to the plane

123

PPP , while = constC , then the

constraint relation is of the form

×==rr

12 13

() constC . Taking into account the

properties of the mixed product of these vectors, we notice that the constraint relation

can be written in the form

()

[

]

()

[

]

()

[

]

××⋅δ+ ××⋅δ+ ××⋅δ=rr r r rr r r rr r r

12 13 23 1 12 13 31 2 12 13 12 3

;

associating the principle of virtual work, given by

⋅δ + ⋅δ + ⋅δ =FrFrFr

1122 33

and introducing Lagrange’s multiplier

λ , we can write the equations of equilibrium

(corresponding to Lagrange’s equations of the first kind) in the form

=− × =− × =− ×FurFurFur

123231312

,,CCCλλλ

(18.1.81)

where we have used the notation introduced above too. It results that, for the

equilibrium, the forces

, F

and F

must be in a plane normal to the unit vector u ,

hence in the plane of the triangle

123

PPP ; they must be normal to the sides opposite to

the vertices at which they act (hence, their supports must pierce at the orthocentre

H of

the triangle, while their magnitudes must be in direct proportion to these sides).

As a simple application of Gauss's principle, let us consider Atwood’s machine (see

Sect. 11.1.2.2 and Fig 11.1), where the weights

and G

are modeled as particles

and where we neglect the influence of the wheel of weight

G . The constraint will be

()()

⎡⎤

=−+−

⎢⎥

⎣⎦

11 22

11 1

Zmgmamgma

where we assume that

mm; in this case

[

]

δ=− − + + δ=

() () 0Zmgamgaa,

MECHANICAL SYSTEMS, CLASSICAL MODELS

being thus led to

−

(18.1.82)

a formula which corresponds to (11.1.21).

18.1.2.5 General Theorems. Conservation Theorems

Let be discrete a mechanical system S of n particles

P , acted upon by the given

forces

= 1,2,...,in

; assuming that the constraints allow an arbitrary virtual rigid

displacement (with respect to an inertial frame of reference)

()

∗∗

′′ ′′

δ= δ+ δ× = − =rv rrrr, , 1,2,...,

iiii

tt i nω ,

(18.1.83)

where

O is the pole of a non-inertial frame, rigidly linked to the mechanical system S

in its rigid displacement (corresponding to the formula (18.1.3)). The principle of

virtual work is thus written in the form

()

[]

∗∗

′

⋅+δ×=

∑

ttδΦω .

Because the vectors

∗

′

and

∗

ω are arbitrary, it results

′

=×=

∑∑

iii

00ΦΦ

(18.1.84)

where we took into account the second relation (18.1.83). We can also write

{

}

′

τ 0Φ ,

(18.1.84')

the torsor of d’Alembert’s lost forces, with respect to the pole of an inertial frame of

reference, vanishing. Starting from this result, we find again the general theorems of

mechanics (the theorem of momentum and the theorem of moment of momentum).

Comparing with the relation (11.1.61), written with respect to the same pole

′

, where

the lost forces of d’Alembert equilibrate the constraint forces, we see that in this new

form of the theorem of torsor do not appear the constraint forces; this happens because

the relation (11.1.61) can be written for any mechanical system, while the relation

(18.1.84') takes place only for the mechanical systems which admit an arbitrary rigid

displacement.

In case a free rigid solid which, obviously, admits an arbitrary rigid displacement,

the theorem of torsor becomes (

V is the volume of the rigid solid, μ is the density,

while

F is a volumic force, which – eventually – can be reduced to a superficial one)

Lagrangian Mechanics

{

}

= Fr

Vτμ0



ΦΦ

=−

(18.1.85)

wherefrom

() ()

=× =

∫∫

Fr rFrd0, d

VVμμ0

 

−−

(18.1.85')

resulting the theorem of momentum and the theorem of moment of momentum,

respectively. If the solid would have a fixed point, we could admit that this one is just

the point

O , so that

′

=r0

and

∗

′

=v0

; we obtain

()

∗∗

⎡⎤

⋅δ× =×⋅δ =

⎣⎦

∫∫

rrdd0

tV tVΦω Φω .

The virtual angular velocity

∗

ω being arbitrary, it results

×=

∫

r d

V 0Φ ,

(18.1.86)

which leads to the theorem of moment of momentum. In the case in which the rigid

solid has two fixed points, hence a fixed axis, we can take this axis as the

′′

Ox -axis,

′

≡OO

, after the vector

∗

ω ; in projection on the axis

′′

≡

Ox Ox , we can write

()

∗

−δ=

∫

12 21

xx tVΦΦω ,

wherefrom

()

−=

∫

12 21

xx VΦΦ ,

(18.1.87)

being thus led to the projection of the theorem of moment of momentum on a fixed

axis.

If, for a discrete mechanical system

S , the constraints admit as virtual displacement

a translation of constant direction, specified by the constant unit vector

u and of

arbitrary magnitude

′

δ= δrukt

∗

′

=vu

k , and if

∗

= 0ω , then we get

⋅=

∫

u d0

VΦ ,

(18.1.88)

hence the theorem of momentum projected on the direction of unit vector

; this allows

to write a scalar first integral, component of a conservation theorem of momentum

(corresponding to the theorem expressed by the first formula (18.1.84)).

As well, if the mechanical system

S has constraints which allow a rotation about an

axis of fixed direction, specified by the unit vector

u , then we can write

()

∗

′

δ=ωδ×rurt

′

=v0

; we obtain thus (if we take the point O on the fixed axis,

then we have

′

=r0

′

≡OO

, hence

′

=rr

)

MECHANICAL SYSTEMS, CLASSICAL MODELS

()

′

×⋅ =

∫

rud0

VΦ

(18.1.89)

that is the theorem of moment of momentum with respect to the pole

′

, projected on

the fixed axis of unit vector

u . We obtain thus a scalar first integral too, component of

a first integral of moment of momentum (corresponding to the theorem expressed by the

second formula (18.1.84)).

As in case of the theorem (18.1.84), in these results intervene only given forces; in

exchange, they take place only for the mentioned particular motions of the rigid solid.

One can make analogous considerations concerning the theorem of kinetic energy

for the conservation of the mechanical energy, assuming that the constraints of the

mechanical system

S are catastatic.

18.1.2.6 Influence of the Forces of Friction

Let us consider a particle P subjected to the action of a given force F and to

holonomic constraints, which lead to a constraint force

R . In case of a single

holonomic an rheonomous constraint

123

(,, ;) 0xxxtϕ , we can write =+RNT,

where

N is the normal constraint force and T is the tangential constraint force. If the

constraint would be scleronomic, hence the force

would correspond to the sliding

friction on the fixed rigid surface

S (of equation =

123

(,, ) 0xxxϕ ). We can

consider, e.g., a hydrodynamic friction, of the form

=−Tvk , =>rr(,;) 0kk t



which leads to the replacement of the given force

=FFrr(,;)t



by a new force

=−FF vk , the problem remaining the same from a mathematical point of view. In

case of a Coulombian friction, of the form

()

/, ,;fN v f f t==



Tv rr, the problem is

put in a different from, due to the unknown factor

N , corresponding to the normal

constraint force; we remain now to this second case.

We assume a constraint

=r(;) 0tϕ , which may correspond to a family of rigid

surfaces

S in motion, depending on the parameter t . The velocity of the particle is

=+vv v

, where v

is the velocity of transportation and v

is the sliding velocity

on the surface

(on the nature of a relative velocity); in this case, in the expression of

the constraint force

T , the velocity

is replaced by the velocity

. The constraints

are no more ideal, so that the condition of ideal constraint takes no more place; this

condition is replaced by an analogous condition of the form

×=RW 0,

(18.1.90)

where the vector W plays the rôle of the virtual displacement δr .

The principle of virtual work is thus replaced by

⋅=Φ W 0 .

(18.1.91)

Noting that

=± − = ± −Rn n()NfNN fττ, where τ and n are unit vectors in

the frame of Darboux, we can write the condition (18.1.90) also in the form

Lagrangian Mechanics

()

±⋅=nW0f τ

(18.1.90')

neglecting the scalar

≠ 0N . To express the unit vectors τ and n as functions of the

data of the problem, we use the curvilinear co-ordinates u and v on the surface

that

′′

=++

vr r r



 ; putting in evidence the partial derivatives with respect to these

co-ordinates, it results

′′

=+vr r

ru v



 and then = vvers

τ . As well,

′′

==×nrrvers grad vers ( )

(see Sect. 4.1.1.4 too). Eliminating

W between the relation (18.1.90'), (18.1.91), which

do not contain the unknown scalar N , by means of Lagrange’s multiplier λ , we obtain

()

+±=n fλ 0Φτ

(18.1.92)

If we associate also the scalar constraint relation

= 0ϕ to this vector equation, then

we can determine the unknowns

=rr()t and = ()tλλ.

In case of two holonomic constraints

=r(;) 0

tϕ , = 1, 2k , which can correspond

to a family of rigid curves

C in motion, depending on the parameter t , we proceed as

above and obtain the equations

()

⋅= ± ⋅= =Wn W0, 0, 1,2

fkΦτ .

(18.1.93)

We are thus led to Lagrange’s equation of the first kind

()()

+±+±=nn

11 22

ffλλ 0Φτ τ.

(18.1.94)

In the scleronomic case, this equation – to which we associate the constraint relations

– is sufficient to solve the problem. In the rheonomous case, we use a representation in

curvilinear co-ordinates

=rr(;)qt (see Sect. 4.1.1.3 too) and representations

=rr(, ;)

qq t , = 1, 2k , of the surface, so that the position of an arbitrary point of

these ones be specified by the same values of the parameters

q and

q , no matter its

position in space. We have

′′

=×

nrrvers ( )

and

′

vrvers vers

qτ , this

being the unit vector of the tangent to the curve

C ; in this case too, the problem is

entirely determined.

18.2 Lagrange’s Equations

A decisive step ahead has been made by Lagrange, by eliminating the holonomic

constraints from the computation and by introducing independent generalized

co-ordinates, in case of mechanical systems subjected only to such constraints; thus one

passes from the representative space

E to a new representative space

, called the

MECHANICAL SYSTEMS, CLASSICAL MODELS

space of configurations. The equations of motion of the representative point are thus

Lagrange’s equations of second kind, which will be studied in this paragraph.

18.2.1 Space of Configurations

By introducing the Lagrangian generalized co-ordinates, in a minimal parame-

terization, one passes from the space

E (or from the representative space

E ) to the

space of configurations

. In what follows, we put in evidence the form taken by

various mechanical quantities (displacement, velocity, acceleration, kinetic energy,

work etc.), obtaining a new formulation of the fundamental problem of mechanics in

the new representative space.

18.2.1.1 Generalized Co-ordinates. Representative Space

Let us consider a discrete mechanical system

S of n a particles

P , = 1,2,...,in,

subjected to

p holonomic constraints of the form (3.2.8), the corresponding

representative point

P in the space

E being subjected to p constraints of the form

(18.1.9). In this case, the system

S has

−=

3nps

degrees of geometric freedom, its

position being thus specified, at a moment

t , by means of s independent parameters.

To have motion, it is necessary that

≥ 1s ; if = 1s , then we say that the mechanical

system is with complete constraints. Let be

, ,...,

qq q a set of such parameters, which

will be called Lagrangian generalized co-ordinates (shortly, generalized co-ordinates).

The position of the mechanical system

S will be thus given by relations of the form

=≡=rr r

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

ii ij

qq qt qt i n;

(18.2.1)

correspondingly, the position of the representative point

P in the space

E is

specified by

≡=( ; ), 1,2,...,3

XXqtk n.

(18.2.2)

Obviously, the relations (18.2.1) verify identically the constraint relations (3.2.8),

while the relations (18.2.2) verify identically the constraint relations (18.1.9). If the

constraints are scleronomic, hence of the form (3.2.33'), then the relations (18.2.1) take

the form

==rr

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

qq q i n.

(18.2.1')

As well, for constraints of the form (18.1.12), the relations (18.2.2) become

≡=

( , ,..., ), 1,2,...,3

XXqq qk n.

(18.2.2')

Indeed, if for the relations (18.1.12), e.g., we assume that

⎡⎤

∂∂ ≠

⎣⎦

det 0

fX ,

=, 1,2,...,lk p (the matrix

⎡

⎤

∂∂

⎣

⎦

fX is of rank

), then it results, from the theorem

of implicit functions, that

≡=

12 3

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

pp n

XXXX X k p. We take

Lagrangian Mechanics

== ==−

11 22 3

, ,..., , 3

pp n

XqXqXqsnp

; one obtains thus relations of the

form (18.2.2'). Obviously, in case of relations of the form (18.2.1') or of the form

(18.2.2') one can have only holonomic and scleronomic constraints.

In particular, if no one holonomic constraint takes place (the mechanical system

is free) or if we ignore these constraints, we can take

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

Xqk n,

corresponding to the representative space

E . But if we take into account all p

holonomic constraints, the mechanical system

S having only =−3snp degrees of

freedom, then the parameterization

=,1,2,...,

qk s considered above is minimal.

Otherwise, if we do not take into account all the holonomic constraints or if, during the

motion, appear supplementary holonomic constraints of the form

≡==

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

lk l

fq t fqq qt l h,

(18.2.3)

then the mechanical system

S remains with −sh degrees of freedom. Obviously,

these constraints are independent too, so that the matrix

⎡

⎤

∂∂

⎣

⎦

fq is of rank h ; as

well, we suppose that no one of these constraints is a consequence of the equations of

motion (it is not a first integral of these equations). The constraints of the form (18.2.3)

are rheonomous; we can consider, in particular, also scleronomic constraints of the form

≡=

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

fq fqq q l p.

(18.2.3')

We mention that one can have also unilateral constraints, expressed by means of

inequalities; but, in what follows, we will consider only bilateral constraints.

The space of generalized co-ordinates

, ,...,

qq q is, as well, a representative space

with

s dimensions, called the space of configurations (figurative space or Lagrange’s

space), which will be denoted by

Λ (in the honour of Lagrange). A point

( , ,..., )

Pq q q is a representative point, its position in the space

Λ specifying

univocally the position of the representative point

P in the space

E , as well as the

position of the discrete mechanical system

S in the space

E ; obviously, we must

assume that the matrix

⎡⎤

∂∂

⎣⎦

Xq is of rank s , so that – on the basis of the theorem

of implicit functions – to a given discrete mechanical system

S corresponds only one

representative point

P (we notice that one can have several non-zero determinants of

order

s , so that we can express the minimal parameterization

q in various modes,

depending on the generalized co-ordinates

; if one takes into account the constraint

relations (18.1.9) too, one sees that the respective expression are equivalent). The

succession of value taken by the generalized co-ordinates during time leads to the

functions

=∈ =

( ), [ , ], 1,2,...,

qqttttk s, which represent the parametric

equations of the trajectory of the representative point

, characterizing – at the same

time – the motion of the mechanical system

S .

MECHANICAL SYSTEMS, CLASSICAL MODELS

In case of a particle constrained to stay on a curve one uses only one generalized

co-ordinates

q ( ==(,), 1,2,3

xxqti ; see Sect. 6.2.2.1) in the space

Λ ; e.g., in

case of the simple pendulum this co-ordinate is the angle at the centre

θ (see Sect.

7.1.3.1). In case of a particle constrained to stay on a surface we use two generalized

co-ordinates (

=rr

(, ;)qqt; see Sect. 6.2.2.2) in the space

Λ ; e.g., in case of the

spherical pendulum these co-ordinates are the applicate

z and the angle at the centre θ

(see Sect. 7.1.3.7). We notice that the particles of the mechanical system

S can be

replaced by rigid solids. E.g., in case of a physical pendulum we use also a generalized

co-ordinate (the angle θ ; see Sect. 14.2.1.2), in the space

Λ , while in case of a double

pendulum one introduces two generalized co-ordinates (the angles

θ and

θ ; see Sect.

17.1.1.2), in the space

Λ . As it can be seen, the generalized co-ordinates are not

necessarily lengths; they can be angles too. Moreover, in the frame of the same

Lagrangian space, some co-ordinates can be lengths, the other ones can be angles or

even other quantities; it is not necessary that all generalized co-ordinates have all the

same physical dimension. Practically, one chooses as generalized co-ordinates the

parameters which can specify the configuration (position) of the discrete mechanical

system

S at any moment t .

18.2.1.2 Generalized Displacements. Generalized Velocities. Constraints.

Generalized Accelerations

Starting from the relation (18.2.1), we can express the real displacements in the form

(we denote

=∂ ∂rr



)

∂

=+=

∂

rrd d d , 1,2,...,

iji

qti n



(18.2.4)

putting thus in evidence the real generalized displacements

q .

Here and in what follows we use the convention of summation of dummy indices

(from 1 to

s ) in the space

Λ ; analogously, we express the possible displacements in

the form

∂

Δ= Δ+Δ =

∂

rr, 1,2,...,

iji

qti n



(18.2.5)

the virtual displacements being given by

∂

δ= δ =

∂

r , 1,2,...,

qi n

(18.2.6)

One introduces thus the possible generalized displacements

q and the virtual

generalized displacements

q , which have the same properties as those corresponding

to the spaces

E and

E . E.g., the virtual generalized displacements are differences of

possible generalized displacements (as one sees from (18.2.5), (18.2.6)); as well, the

Lagrangian Mechanics

real generalized displacements belong to the set of possible generalized displacements,

which – in general – does not coincide with the set of virtual generalized displacements.

Let be two virtual displacements

δ r

and δ r

of the same particle

P , written in

the form

∂∂

δ= δ δ= δ

∂∂

1122

ijij

where

q and δ

q are corresponding virtual generalized displacements; we may

write

()

∂

δ+δ= δ +δ

∂

12 1 2

ii j j

too. Hence, by composition of two virtual displacements one obtains a new virtual

displacement (the same property for the virtual generalized displacements).

Let be the virtual generalized displacements

q ; if −δ

q is, as well, a virtual

generalized displacement of the representative point

at a given moment

, then the

displacement

q is a reversible virtual generalized displacements. From the relations

(18.2.6), it results that to reversible virtual displacements correspond reversible virtual

generalized displacements.

The velocities of the particles of the mechanical system

S will be given by (we

denote

= dd

qqt )

∂

=+=

∂

, 1,2,...,

iji

qi n



(18.2.4')

putting in evidence the real generalized velocities

q (shortly, the generalized

velocities). As well, we express the possible velocities in the form (we denote

=Δ Δ

qqt



)

∂

=+=

∂

, 1,2,...,

iji

qi n



(18.2.5')

the virtual displacements being given by (we denote

∗

=δ Δ

qqt )

∗∗

∂

, 1,2,...,

qi n



(18.2.6')

We introduce thus the possible generalized velocities



and the virtual generalized

velocities

∗

q . We mention that also these quantities can have different dimensions (can

be linear velocities, angular velocities etc.).

MECHANICAL SYSTEMS, CLASSICAL MODELS

Using the generalized displacements introduced above, we can express the constraint

relation (18.2.3) in one of the forms

∂

+==

∂

d d 0, 1,2,...,

qft l p



(18.2.7)

∂

δ= =

∂

0, 1,2,...,

ql p

(18.2.8)

= 0



, then it results

()

∂∂ ∂ ∂=∂ ∂ = =0, 1,2,...,

fq t fq j s



, too, so that the

constraints are scleronomous; in this case, the set of virtual generalized displacements

coincides with the set of possible generalized displacements, while the real generalized

displacements belong to both sets (in case of holonomic constraints). Obviously, we can

express the relations (18.2.7), (18.2.8) in one of the forms

∂

+= =

∂

0, 1,2,...,

qf l p



(18.2.7')

∗

∂

0, 1,2,...,

ql p



(18.2.8')

where we have introduced the generalized velocities and the virtual generalized

velocities, respectively.

If we replace in (3.2.13) the real displacements by the relations (18.2.4) or (18.2.4'),

then we can express – in general – the constraint relations in one of the forms

+==

d d 0, 1,2,...,

kj k

aq a t k m,

(18.2.9)

+= =

0, 1,2,...,

kj k

aq a k m ,

(18.2.9')

where

∂

=⋅ =

∂

∑

, 1,2,...,

kj kl

ajs

(18.2.9'')

=⋅+=

∑

, 1,2,...,

kklk

akmα



α .

We notice, from (18.2.4') – (18.2.6'), that a necessary condition to have catastatic

constraints is

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



; indeed, in this case, to = 0

q corresponds

(hence, rest with respect to an inertial frame of reference), the inverse implication

taking, as well, place (the matrix

⎡

⎤

∂∂

⎣

⎦

Xq is of rank s ). In case of holonomic

constraints, the respective condition in sufficient too (as a mater of fact, in this case the

constraints are scleronomic). In case of non-holonomic constraints one must have also

α ; hence, from (18.2.9'') it results that these constraints are catastatic if and only

0, 1,2,...,

ak m. We notice that