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

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Lagrangian Mechanics

== =

δ= δ = δ

∑∑ ∑

() ()

11 1

ij k

WFXQX

(18.1.16)

is the virtual work of the given generalized forces, while

== =

δ= δ = δ

∑∑ ∑

() ()

11 1

Rkk

ij k

WRXRX

(18.1.16')

represents the virtual work of the constraint generalized forces.

18.1.1.3 Natural Systems

We have seen in Sects. 1.1.1.12 and in 6.1.1.2 that a system of forces

(corresponding, e.g., to a field of forces), which admits a simple potential (or

quasi-potential) or a generalized potential (or quasi-potential) is called a natural system.

A system of forces

==Fi

()

ii j

UF∇

∂

()

∇

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

the components of which

∂

== =

∂

()

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

Finj

(18.1.17)

derive from a simple potential, forms a system of conservative forces; to a simple

quasi-potential of the form

= rr r

(,,..., ;)

UU t corresponds a system of quasi-

conservative forces. Analogously, if

∂∂

=− = =

∂∂

()

() ()

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

Finj



(18.1.17')

where

= rr r rr r

12 12

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

 

is a generalized potential, then the system of forces is conservative; in case of the

generalized quasi-potential

= rr r rr r

12 12

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

UU t

 

the system of forces is quasi-conservative.

In the representative space

, the system of generalized forces is of components

∂

, 1,2,...,3

Qk n

(18.1.18)

MECHANICAL SYSTEMS, CLASSICAL MODELS

in case of a simple potential (system of conservative generalized forces)

( , ,..., )

UUXX X

or of a simple quasi-potential (system of quasi-conservative generalized forces)

(,,..., ;)

UUXX Xt

or of components

[]

∂∂

=≡− =

∂∂

, 1,2,...,3

QU k n

XtX

(18.1.18')

in case of a generalized potential (system of conservative generalized forces)

12 12

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

UUXX XXX X

 

or in case of a generalized quasi-potential (system of quasi-conservative generalized

forces)

12 12

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

UUXX XXX Xt

 

;

we have used the Euler-Lagrange derivative corresponding to the index

, given by

[]

∂∂

=− =

∂∂

, 1,2,...,3

XtX

(18.1.19)

Noting that the generalized forces do not depend explicitly on the generalized

accelerations

, 1,2,..., 3

Xkn



it results that the generalized potential (quasi-potential) is of the form

∑

UUXU



(18.1.20)

where

(,,..., )

UUXX X

is a vector potential, while

( , ,..., )

UUXX X

Lagrangian Mechanics

is a scalar potential, or where

( , ,..., ; )

UUXX Xt

is a vector quasi-potential, while

( , ,..., ; )

UUXX Xt

is a scalar quasi-potential in the space

E . Replacing in (18.1.18'), we obtain

∂

⎛⎞

=−−+=

⎜⎟

∂∂ ∂

⎝⎠

∑

,1,2,...,3

QXUkn

XX X



(18.1.21)

in case of the quasi-conservative generalized forces; if the forces are conservative, then

we have

= 0



Replacing the vector potential

U by the vector potential

∂

, =

( , ,..., ; )

XX Xtϕϕ ,

and taking into account that one can invert the order of application of the operators

∂∂

Xt tX

for a function of class

C , we get the same generalized function

Q ; hence, the vector

quasi-potential is determined leaving aside a field of gradients in the space

E . If to

this transformation of vector quasi-potential

→

UU we associate the transformation

of scalar quasi-potential →=+

000

UUUϕ , hence if we effect the gauge

transformation (of scale)

→=+

UUU

then the form (18.1.21) of the quasi-conservative generalized force remains invariant.

A field of conservative generalized forces is stationary, while a field of

quasi-conservative generalized forces is non-stationary.

The elementary work of the conservative generalized forces which derive from a

simple potential is an exact differential

=ddWU, (18.1.22)

the work depending only on the extreme positions of the representative point

()

=−

11 1 00 0

12 3 12 3

, ,..., , ,...,

WUXXXUXXX.

(18.1.22')

MECHANICAL SYSTEMS, CLASSICAL MODELS

If the trajectory of the point

P is a closed hypercurve C in

E , then the

corresponding work vanishes (

= 0

W ). If these generalized forces result from a

generalized potential, then the elementary work is a total differential

ddWU

(18.1.22'')

too, the scalar potential

U playing the rôle of the simple potential in the frame of the

generalized potential (18.1.20), while the vector potential has no contribution

concerning this elementary work; all the considerations made in case of the simple

potential remain thus valid. In case of quasi-conservative generalized forces, which

derive from a simple quasi-potential, the elementary work is no more a total differential,

being of the form

=−dd dWUUt



(18.1.23)

where we put in evidence the partial derivatives with respect to time. If the generalized

forces result from a generalized quasi-potential, then we get

=−−

∑

dd d d

WUUt UX



∂

⎛⎞

=− + = −

⎜⎟

∂

⎝⎠

∑∑

ddd

kk k k

UUVUt UX

 

(18.1.23')

We see, easily, that the transformations mentioned above for the generalized

potential have no influence on this elementary work.

We notice that a system of conservative generalized forces derives always from a

generalized potential or from a simple potential as it depends or not explicitly on the

generalized velocities of the representative point

P in

E .

Introducing the potential energy

=−VU

, in case of a simple potential, or the

potential energy

=−

VU, in case of a generalized potential, we can write

=−=− =− +

01 01

PP PP

WVVVVWV

(18.1.24)

where

ΔV is the variation of the potential energy. Consequently, the potential energy

of a discrete mechanical system, acted upon by conservative generalized forces (natural

mechanical system), is equal to the work with changed sign, effected by the generalized

forces, beginning with the initial moment (excepting an arbitrary constant

, which

represents the potential energy at the initial moment; often, one chooses

V so that the

minimum of the potential energy be equal to zero).

The generalized forces which do not derive from a simple quasi-potential can be

expressed in the from

()

∂

=+ =

∂

, , ,..., ;

QQUUXXXt

(18.1.25)

the elementary work being given by

Lagrangian Mechanics

⎛⎞

=+ −

⎜⎟

⎝⎠

∑

dd d

WU QVUt



(18.1.25')

In the case in which

= 0U



( =

( , ,..., )

UUXX X is a potential), the quantity

∑

of the nature of a power, represents the power of the non-potential generalized forces.

The non-potential generalized forces

Q of a vanishing power

∑

are called gyroscopic generalized forces, depending – obviously – on the distribution of

the generalized velocities. The generalized forces

Q are, in this case, conservative; if

the power of the non-potential generalized forces is non-zero, then these forces are

non-conservative. The non-conservative generalized forces of negative power

∑

are called dissipative generalized forces, because – in this case – the energy diminishes

(we take into account (18.1.25') and notice that

=−UV). The forces (18.1.21) which

admit a generalized quasi-potential are, as well, of the form (18.1.25), where we take

UU and where

∂

⎛⎞

=− −=

⎜⎟

∂∂

⎝⎠

∑

, 1,2,...,3

QXUkn



(18.1.26)

If neither the vector potential does not depend explicitly on time (

= 0



), then

it results

∑



the non-potential generalized forces being thus gyroscopic. Hence, any generalized

force which admits a generalized potential is the sum of a generalized force which

admits a simple potential and a gyroscopic force. If upon a mechanical system

S act

only gyroscopic generalized forces, then we have

⋅= =

∑∑

(18.1.27)

and reciprocally. As well, the condition of dissipation becomes

MECHANICAL SYSTEMS, CLASSICAL MODELS

⋅= <

∑∑

QV .

(18.1.27')

18.1.1.4 Case of a Single Particle

In case of a single particle P we have = 1n , the representative space being just the

space

E , of co-ordinates

123

,,xxx. Using the notations in Sect. 18.1.1.2, we can write

the constraint relations, in general, in the form

≡+==

dd0,1,2

bx bt

αα

ωα,

(18.1.28)

=,1,2

ωα

, being differential forms of first degree; we consider, in general, that

123

(,, ;), 0,1,2,3

bbxxxtj

αα

. We may write

+= =

0, 1,2

bv b

αα

α ,

(18.1.28')

too. If these forms are integrable (see Sect. 3.2.2.6 too), then the constraints are

holonomic, so that

123

(,, ;) 0, 1,2

fxxxt k ,

(18.1.29)

relations which can be written also in the form

+==

dd0,1,2

kj k

fx ft k



(18.1.29')

or in the form

+= =

0, 1,2

kj k

fv f k



(18.1.29'')

0, 1,2b

α , then the constraints are catastatic, while if = 0



, = 1, 2α ,

= 1, 2, 3

, too, then the constraints are scleronomic; otherwise, they are rheonomous.

In case of a holonomic constraint, the conditions to be a scleronomic one too are

reduced to

==0, 1,2



. As in the general case, the constraint relations can be

expressed also by means of the virtual displacements in the form

δ= =0, 1,2

α ,

(18.1.30)

or in the form

∗

==0, 1,2

α .

(18.1.30')

The relations (18.1.30) and (18.1.30') coincide with the relations (18.1.28) and

(18.1.28'), respectively, in case of catastatic constraints (the set of virtual displacements

Lagrangian Mechanics

coincides with the set of possible constraints, while the real displacements belong to

these sets).

A simple example of particle subjected to a non-holonomic constraint is represented

by a particle

P which has a curvilinear and uniform motion of velocity v

, collinear

with the vector

JJJG

, tangent to the trajectory, the point

P having a rectilinear and

uniform motion of velocity

(the problem of meeting; see Sect. 10.2.1.2). Starting

from the relations

′

xx,

′

21 2

xvtx, which allow to pass from a movable frame

of reference to a fixed one, and from the relation

112

()

xvx

,

one can write the constraint relation

()

′′′′

−−=

21 112

dd0xvtxxx ,

(18.1.31)

which is non-holonomic.

In case of scleronomic constraints

== ==

123

d 0, ( , , ; ), 1,2, 1,2,3

jj j j

bx b bxxxt j

ααα

α ,

(18.1.32)

the conditions of integrability are of the form (3.2.30'). The integrant factors

123

(,, ), 1,2xxx

αα

λλ α , must exist, so that b

λ and b

λ , =bi

= 1, 2α

, be gradients, hence curl ( ) curl grad

αα α α α α

λλ λ=+×=0bb b; a scalar

product by

leads to the conditions of integrability

curl⋅=∈ ==bb

0, 1,2

ijk k j

αα

α .

(18.1.32')

If, in particular,

==bcurl , 1,2

α0 , the conditions of integrability are fulfilled

and we can take

= const.

λ Pfaff showed that, by a convenient transformation of co-

ordinates, the non-integrable constraint relation (18.1.32) can be expressed in the form

−=

31 2

dd 0xx x .

(18.1.33)

We can see, easily, that the relation (18.1.32') does not hold.

==b grad , 1,2f

αα α

λα, where

λ is a scalar field of class

C , if f

is a

scalar field of class

C and if

λ and f

are functional independent, then the vector

field

is called biscalar; in particular, if

λ and

are functional dependent, then

is a potential field. The necessary and sufficient condition that the vector field b

be locally biscalar is (18.1.32'); in this case, the scleronomic relations (18.1.32) are

completely integrable (are locally exact or admit a locally integrant factor). J. Bertrand

considered also the field lines

MECHANICAL SYSTEMS, CLASSICAL MODELS

′

123

(,, )xxx c

αα

ϕ ,

′′

123

(,, )xxx c

αα

ψ ,

′′′

==,const,1,2cc

αα

α ,

of the

bcurl

(vortex lines), expressed by means of the first integrals

ϕ and

ψ ; we

notice that

⋅=bgrad curl 0

αα

ϕ , ⋅=bgrad curl 0

αα

ψ ,

= 1, 2α

. Associating the

relations (18.1.32') too, one sees that the vectors

, grad

ϕ , grad

ψ are coplanar,

so that

=+ +≠=b

grad grad , 0, 1,2

αα αα ααα

λϕμψλμ α .

In this case,

⋅= = ⋅+ ⋅= + =br r rdd graddgraddd d,1,2

ααααααααα

λϕ μψ λϕμψα

and one can show that the ratio

≠,0

depends only on the first integral

ϕ and

ψ .

Let be a particle

P subjected to a holonomic and scleronomic constraint

123

(,, ) constfx x x ; in this case, starting from a given point (let be from the origin,

to fix the ideas), one can reach only the points on a two-dimensional manifold (the

points of the surface

123

( , , ) (0,0,0)fx x x f ). If the constraint is non-holonomic and

scleronomic (e.g., of the form (18.1.33)), then we can find a path which leads from the

origin to an arbitrary point

123

(,, )xxx . Let be thus the path

()xfx,

′

()xfx,

∈

()fx C

;

the constraint relation (18.1.33) is satisfied and we must choose the function

()fx

with the properties:

=(0) 0f ,

′

=(0) 0f

, =

()fx x,

′

()fx x.

One sees easily, geometrically, that it exists an infinity of such functions, e.g., the

functions of the family

()

1231231

()=3 2

fx x xx x xx

⎡⎤⎛⎞

−−−

⎜⎟

⎢⎥

⎣⎦⎝⎠

()

11 1 1 2

sin

Cx x x C

⎛⎞

+−+

⎜⎟

⎝⎠

(18.1.34)

Lagrangian Mechanics

where

C and

C are arbitrary constants. Hence, starting from a given point, a particle

P with −=321 degrees of freedom can reach only the points of a surface (a two-

dimensional manifold), in case of a holonomic constraint, while, in case of a

non-holonomic constraint, one can reach any point in

E (a three-dimensional

manifold). These considerations remain valid also in case of rheonomous constraints. In

case of a particle

P subjected to two constraints (hence, which has −=321 degrees

of freedom) one can make analogous considerations. If both constraints are holonomic.

then one can reach – starting from a given position – only the points of a curve (a

unidimensional manifold), while, if only one of the constraints is holonomic, one can

reach the points of a surface (a two-dimensional manifold); if both constraints are

non-holonomic, then one can reach any point

E (a three-dimensional manifold). The

results obtained above put in evidence that a non-holonomic constraint leads to a

greater possibility of motion (it is less strict) that a holonomic constraint.

If the resultant

F of the forces which act upon the particle P derive from a simple

quasi-potential

123

(,, ;)UUxxxt, then we can write

==FFgrad , curlU 0 ,

(18.1.35)

while if the force

derives from a generalized quasi-potential

=⋅+UUv

U , =UU

123

(,, ;)xxxt, =

123

(,, ;)UUxxxt,

then one has

=+×−FvU

grad curlUU



(18.1.35')

In general, a force

=+FFgradU , =

123

(,, ;)UUxxxt, is non-conservative; if

123

(,, ;)UUxxxt and ⋅=Fv 0 , then the force F is gyroscopic. E.g., a force of the

form

=×Fv Ucurl , =

123

(,, ;)UUxxxt (corresponding to the vector potential in

(18.1.35')), is a gyroscopic force. As an example of force which derives from a generalized

quasi-potential we mention the force

F exerted upon a particle P in an electromagnetic

field

EB{, }, where E is the intensity of the electric field, while B is the magnetic

induction. If

q is the electric charge of the particle P , then we can write

=+×FEvB()q ,

(18.1.36)

where

Eq is the force exerted in the electric field, while ×vBq is the force exerted in

the magnetic field (Lorentz’s force); the force

F is called the generalized force of

Lorentz. The magnetic induction is of the form

=BAcurl ,

(18.1.36')

MECHANICAL SYSTEMS, CLASSICAL MODELS

where

=AA

123

(,, ;)xxxt is the magnetic vector quasi-potential, while the intensity of

the electric field is of the form

()

=−EA+

grad A



(18.1.36'')

where

123

(,, ;)AAxxxt is a scalar quasi-potential. Replacing in (18.1.36), it results

()

=− + × −FvAA

grad curlqAq



(18.1.37)

the force deriving from the quasi-potential

()

=⋅−Av

Uq A.

(18.1.37')

We notice that Lorentz's force

×vBq is a gyroscopic one.

18.1.2 Differential Principles of Mechanics

In what follows, let us present some important differential principles of mechanics

(Newton’s, d’Alembert’s, Gauss’s and Hertz’s principles), from which derive the

equations of motion of the considered discrete mechanical system

S in

E and of the

representative point

P in the representative space

E , respectively; a special attention

is given to the principle of virtual work. The results thus obtained will be illustrated by

some applications.

18.1.2.1 Newton’s and d’Alembert’s Principles. General Theorems. Conservation

Theorems

As we have seen in Sect. 11.1.2.10, the equations of motion of a free discrete

mechanical system

S , free or subjected to constraints, are written in the form (11.1.8)

or in the form (11.1.52), respectively, corresponding to the second principle of Newton,

considered as the first differential principle of mechanics. Passing to the representative

space

E , the equation of motion of the representative point P will be of the form

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

kk k

MX Q k n



(18.1.38)

in case of a free point

P , or of the form

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

kk k k

MX Q R k n



(18.1.38')

in case of a point

P subjected to constraints, respectively. This differential principle

can be expressed also in the form