Teodorescu P.P. Mechanical Systems, Classical Models Volume I: Particle Mechanics

MECHANICAL SYSTEMS, CLASSICAL MODELS

704

1

sin cosxrθϕ

=

,

2

sin sinxrθϕ

=

,

3

cosxrθ

=

, 0r ≥ ,

0 θπ≤≤, 02ϕπ

≤

< ;

(A.1.41)

the element of arc is expressed in the form (

1

1H

=

,

2

Hr

=

,

3

sinHrθ

=

)

2222222222

dd d sind dddsrr r rss

ϕ

θ

θθϕ=+ + =++, (A.1.41')

while the element of volume is given by (

42

singr θ= )

2

dsindddVr rθθϕ= . (A.1.41'')

The functional determinant fulfils the condition

2

sin 0Jr θ

=

≠ if 0r > ,

0 θπ<<.

The system of cylindrical co-ordinates

(,,)rzθ is linked to the orthogonal Cartesian

co-ordinates (see Fig.1.5,b) by the relations

1

cosxrθ

=

,

2

sinxrθ=

,

3

xz

=

,

0r ≥

,

02θπ

≤

<

,

z

−

∞< <∞;

(A.1.42)

the element of arc is given by (

13

1HH

=

= ,

2

Hr

=

)

22222222

dd dddddsrr zrsz

θ

θ=+ +=++ (A.1.42')

and the element of volume is expressed in the form (

2

gr

=

)

ddddVrr zθ

=

. (A.1.42'')

As well, to have

0Jr=≠, it is necessary that 0r > .

Differentiating the formula (A.1.38) with respect of the variable

k

q , we may write

,,,

ji

ik jk ijk

g⋅+⋅ =eeee , ,, 1,2,3ijk

=

, (A.1.43)

where the index at the right to the comma indicates the differentiation with respect to

the corresponding variable; we write again this relation by circular permutations

,

,,

ji j

kkijki

g⋅+⋅ =eeee ,

,

,,

iij

kj k kij

g

⋅

+⋅ =eeee , ,, 1,2,3ijk

=

. (A.1.43')

Summing the relations (A.1.43') and subtracting the relation (A.1.43), we may express

Christoffel’s symbols of first species in the form

[]

()

,

,,,

1

,

2

ij

kij k ij k jk i ki j

ij

ij k g g g

k

Γ

⎡⎤

===⋅=−++

⎢⎥

⎣⎦

ee

,

,, 1,2,3ijk

=

,

(A.1.44)

Appendix

705

where we have introduced the most used notations; to obtain this result, we have taken

into consideration that

,,ij ji

=

ee,

,1,2,3ij

=

, due to the relation of definition

(A.1.38) and to the property of the mixed derivatives of second order of the position

vectors

2

()CD∈r of not depending on the order of differentiation. The Christoffel

symbols of second species are defined in the form

[]

,

kkl

ij

k

gijl

ij

Γ

⎧⎫

⎪⎪

==

⎨⎬

⎪⎪

⎩⎭

, ,, 1,2,3ijk

=

, (A.1.45)

where

ij

g is the normalized algebraic complement (the algebraic complement divided

by

g ) of the element

ij

g of

[

]

det

ij

g

; we have

ij ji

gg= because

ij ji

gg= . We notice

that the relations

j

kj

ik

i

gg δ= ,

ik i

j

kj

gg δ

=

, ,1,2,3ij

=

,

1

det

ij

g

=

(A.1.43'')

take place. Christoffel’s symbols are symmetric with respect to the indices

i and j , so

that

[

]

[

]

,,ij k ji k= ,

kk

ij ji

⎧

⎫⎧ ⎫

⎪

⎪⎪ ⎪

=

⎨

⎬⎨ ⎬

⎪

⎪⎪ ⎪

⎩⎭⎩⎭

, ,, 1,2,3ijk

=

; (A.1.46)

hence, there are 18 distinct symbols of each species. Multiplying the relation (A.1.45)

by

km

g and taking into account (A.1.43''), we get

[]

,

kl

l

ij k g

ij

⎧

⎫

⎪

=

⎨

⎬

⎪

⎩⎭

, ,, 1,2,3ijk

=

. (A.1.45')

Christoffel’s symbols are defined by the relations (A.1.44), (A.1.45) in the case of a

linear space

n

L too.

1.2 Exterior differential calculus

In what follows, we introduce the external product of vectors as well as differential

forms of various orders; in connection with these forms, we put then in evidence the

operator of exterior differentiation.

1.2.1 External product of vectors

A (free) n-dimensional vector is a mathematical entity characterized by an ordered

set of

n numbers

i

V

,

1,2,...,in=

; using the way indicated in Chap. 1, Subsec. 1.1.2,

we may set up an n-dimensional vector space (the linear space

n

L , which has the same

properties as the linear space

3

L

). There exist, in this space, at the most n independent

MECHANICAL SYSTEMS, CLASSICAL MODELS

706

linear vectors; an ordered set of n independent linear vectors

{

}

, 1,2,...,

i

in

=

e forms

a basis, and an arbitrary vector

V may be written in the form

1

n

i

V

=

∑

Ve

, (A.1.47)

where

i

V

are the components of the vector in this basis (the contravariant components,

but – as till now – we do not use the notions of contravariance and covariance). The

external product or the bivector

12

∧

VV (it coincides with the vector product for

3n = ; we replace the sign “

×

” by the sign “

∧

”) of two vectors

12

,VV is defined by

the properties (

12

,,VV V are vectors;

12

,λλ are scalars):

i)

11 22 1 1 2 2

()()()λλ λ λ+∧=∧+ ∧VVV VV VV

11 22 1 1 2 2

()()()λλ λ λ∧+ =∧+∧VVV VV VV

(distributivity with respect to addition of vectors);

ii)

∧=VV0;

iii)

1221

∧

+∧=VVVV0 (anticommutativity).

We may write

12 1 1

22

11 11

nn nn

jj

ii

ij ij

VV VV

== ==

∧= ∧ = ∧

∑∑ ∑∑

VV e e ee; (A.1.48)

inverting

i

by j and taking into account the property iii), it results

()

12 1 2

21

11

1

2

nn

jj

ii

ij

VV VV

==

∧= − ∧

∑∑

VV ee. (A.1.48')

The properties ii) and iii) lead to the relation

()

12 1 2

21

11

nn

jj

ii

ij

VV V V

==

∧= − ∧

∑∑

VV ee, ij

<

. (A.1.48'')

We denote by

2

n

L∧ the vector space of the bivectors defined on

n

L (corresponding to

this notation

1

nn

LL∧=); noting that the set

{

}

,1

ij

ijn∧≤<≤ee forms a basis

in this space, its dimension is given by

22

(1)

dim 1 2 ... ( 1)

2

nn

Ln C

−

∧=+++−= =

. (A.1.49)

In general, a p-vector

12

...

p

∧

∧∧VV V in

n

L is defined by the properties

(

1212

, , , ,...,

p

WWVV V are vectors;

12

,λλ are scalars):

Appendix

707

i)

11 22 2 1 1 2

( ) ... ( ... )

pp

λλ λ+ ∧ ∧∧ = ∧ ∧∧WWV V WV V

22 2

(...)

p

λ+∧∧∧WV V

(and analogous relations, obtained by replacing the other vectors

i

V ,

2,3,...,ip= , which form the p-vector by linear combinations);

ii)

12

...

p

∧

∧∧ =VV V 0

if and only if

ij

=

VV, ij

≠

;

iii) The external product

12

...

p

∧

∧∧VV V changes its sign if two factors of it

permute.

Let be

p

n

L

∧

the vector space formed by p-vectors defined on

n

L , 2 pn≤≤. The

set

{

}

12

... ,1 ...

p

ii i

ii i n∧∧∧ ≤<<<≤ee e forms a basis in this space, so

that

dim

p

n

LC

∧

= , (A.1.49')

where we have introduced the combination symbol of

n things

p

at a time; in

particular,

dim 1

nn

LC∧==

.

In the case of a trivector

123

∧

∧VVV we may write

123 13

2

111

nnn

j

ik

ij

k

ijk

VVV

===

∧∧= ∧∧

∑∑∑

VVV ee e; (A.1.50)

inverting two upper indices and taking into account the property iii), we find

3! 6=

different representations of the external product (obtained by permuting the indices

,ij

and

k and by introducing the sign minus in the case of an odd permutation). Summing

these six representations, we obtain a representation of the form

123

111

1

3!

nnn

ijk

ij

k

ijk

V

===

∧∧= ∧∧

∑∑∑

VVV ee e, (A.1.50')

where the scalar

ijk

V is totally skew-symmetric with respect to the upper indices

,,ijk. In general, a p-vector

ϕ

may be represented by

12

...

12

1

... ...

!

p

k

n

ii i

p

ii i

i

V

p

=

≡∧∧∧= ∧∧∧

∑

VV V e e eϕ , (A.1.51)

where the scalar

12

...

p

ii i

V is totally skew-symmetric with respect to the upper indices

k

i , 1,2,...,kp= (the summation takes place for all upper indices). For 2p = one

obtains the representation (A.1.48').

One can define a p-vector

12

...

p

≡

∧∧∧VV V

ϕ

and a q-vector

1

≡∧Wψ

2

...

q

∧∧WW on

n

L ; their external product is a p+q-vector, pq n+≤,

defined in the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

708

12 1 2

( ... ) ( ... )

pq

∧

≡ ∧ ∧∧ ∧ ∧ ∧∧VV V WW Wϕψ

11

... ...

pq

=∧∧∧ ∧∧VVWW.

(A.1.52)

We mention following properties:

i)

11 22 1 1 2 2

()λλ λ λ∧+ =∧+∧ϕψψ ϕψϕψ (distributivity with respect to

addition;

12

,ψψ are p-vectors,

12

,λλ

are scalars);

ii)

() ()

∧

∧=∧ ∧ϕψ χϕ ψχ (associativity;

χ

is r-vector; pqr n

+

+≤);

iii)

(1)

pq

∧=− ∧ϕψ ψϕ.

If one of the numbers

,pq is even, then the external product is commutative,

otherwise it is anticommutative.

1.2.2 Differential forms. Exterior derivative

We call differential form of first degree in

12

( , ,..., )

nn

xxx x E

≡

∈ the expression

1

d

n

ii

i

axω

=

∑

, const

i

a

=

. (A.1.53)

If between the basis

{

}

i

e of the linear space

n

L and

{

}

d

i

x , 1,2,...,in= , is

established an isomorphism, then

ω is an element of the space

1

n

L∧ , introduced in the

preceding subsection. Analogously, a form of pth degree in

x is an element of the

space

p

n

L∧ , being expressed by

12 1 2

...

d d ... d

pp

ii i i i i

axx xω =∧∧∧

∑

,

12

...

const

p

ii i

a

=

. (A.1.53')

In the case of a form of pth degree on

n

DE⊂ ,

12

( , ,..., )

n

ii

aaxx x

=

may be smooth

functions (differentiable as much as it is necessary) on

D . We denote by ()

p

FD the

set of the forms of pth degree on

D ; in this case,

0

()FD

represents the set of smooth

functions.

Let be the forms of first degree

1123

(,, )d

ii

axxx xω = ,

2123

(,, )d

j

bxxx xω

=

; (A.1.54)

the external product introduced in the previous subsection is calculated in the form

(

d

i

x play the rôle of vectors

i

e )

12 23322 3 31133 1

()dd()ddab ab x x ab ab x xωω∧= − ∧ + − ∧

12 21 1 2

()ddab ab x x+− ∧.

(A.1.54')

One obtains thus a form of second degree.

The operator

1

d: () ()

pp

FD F D

+

→ , called (after Cartan) operator of exterior

differentiation, exists and is unique, being defined by the properties (

1

,ωω and

2

ω are

p-forms):

Appendix

709

i)

12 1 2

d( ) d dωω ω ω+=+;

ii)

1

grad

12 12 1 2

d( ) d ( 1) d

ω

ωω ωω ω ω∧=∧+− ∧;

iii)

d(d ) 0ω = (Poincaré’s lemma);

iv)

1

dd

n

j

f

x

=

∂

=

∂

∑

for any function

f

.

These properties are independent of the system of co-ordinates. One may show that the

exterior derivative of the form (A.1.53') is expressed by

12

...

d d d d ... d

p

ii i

j

ii i

j

a

xx x x

x

ω

∂

=∧∧∧∧

∂

∑

. (A.1.55)

In the particular case

3n =

one may introduce the gradient operator in the form

123

12 3

dd d d

f

ff

f

xxx

∂

∂∂

=++

∂∂∂

, (A.1.56)

where

f

is a function defined on

3

DE⊂ . For a form of first degree on D

1123 1 2123 2 3123 3

(, , )d (, , )d (, , )daxxx x axxx x axxx xω

=

++ (A.1.57)

the exterior derivative

32

23

ddd dd

i

ji

j

aa

a

xx xx

xxx

ω

∂∂

∂

⎛⎞

=∧=−∧

⎜⎟

∂∂∂

⎝⎠

∑

13 21

31 12

dd dd

aa aa

xx xx

∂∂ ∂∂

⎛⎞ ⎛⎞

+− ∧+ − ∧

⎜⎟ ⎜⎟

∂∂ ∂∂

⎝⎠ ⎝⎠

(A.1.57')

allows to introduce the operator curl. As well, the exterior derivative

123

dddd

aaa

xxx

ω

∂∂∂

⎛⎞

=++ ∧∧

⎜⎟

∂∂∂

⎝⎠

, (A.1.58)

corresponding to the form of second degree on

D

12 3 23 1 31 2

dd dd ddax x ax x ax xω =∧+∧+∧, (A.1.58')

introduces the operator divergence.

2. Notions of field theory

In what follows, we deal with conservative vectors, with the operator gradient, as

well as with the introduction of the curl and the divergence of a vector; differential

operators of second order are considered too and some integral formulae are given. We

mention also the absolute and the relative derivatives.

MECHANICAL SYSTEMS, CLASSICAL MODELS

710

2.1 Conservative vectors. Gradient

Let be a point

3

123

(, , )Px x x D∈⊂\ . In what follows, the vector mapping

123 123

(, , ) (, , )xxx xxx→ V defines a vector field ( : DL→V ); the respective

vectors are bound vectors, the points

P being their points of application. This field is a

steady one; if

123

(, , ;)xxxt=VV , then the vector field defined on

[

]

0

1

,Dtt× is

non-stationary. We express the vector

123

(, , ) ()xxx

=

=VV Vr in the form

123 123

(, , ) (, , )

j

xxx Vxxx

=

Vi, (A.2.1)

where

1

()

j

VCD∈ , 1, 2, 3j = ; let us introduce the vector fields

,,

j

ijjij

ii

V

xx

∂

== =

∂∂

V

Vii

, 1, 2, 3i

=

, (A.2.2)

and the differential

,

dd

ii

x

=

VV , (A.2.3)

where the index at the right of the comma specifies the derivative with respect to the

corresponding variable. The curves for which the tangents at each point

P are directed

along the vectors

()=VVr of the field are called vector lines (or field lines); the lines

form a congruence of curves. Because the differential

dr is tangent to these lines, their

vector equation is of the form

() d

×

=Vr r 0; (A.2.4)

scalarly, these lines are given by a system of differential equations of first order

123

dddxxx

VVV

==

. (A.2.4')

Let be

C

a curve which is not a field line; on the basis of the theorem of existence

and uniqueness for a system of differential equations of the form (A.2.4'), through each

point of the curve

C

passes a field line (the integral curve of the system (A.2.4')). The

surface generated by these curves is called field surface.

In the case of a non-steady field, the differential is of the form

,

ddd

ii

xt=+



VV V, (A.2.3')

where

/VVt

=

∂∂



, and the total derivative is given by (we consider the mapping

()tt→ r )

,

d

ii

x

t

=

+



V

VV

, (A.2.3'')

Appendix

711

being the sum of the space and time derivatives; in the case of a steady field remains

only the space derivative.

In what follows we introduce some particular fields of vectors.

2.1.1 Conservative vectors. The nabla operator

Let us consider a scalar function

→

123 123

(,,) (,,)xxx Uxxx ,

123

(, , )xxx D∈

3

⊂ \ ; the function

1

()UCD∈ defines a scalar field (

3

:UD→ \ ), because to each

point

P

, which is of position vector

123

(, , )xxxr one can associate the scalar

()UUP= . This scalar field is steady; we may consider also non-steady fields of the

form

123

(, , ;)Ux x x t , defined on

[

]

0

1

,Dtt

×

. If it is necessary, the function

U

may

have also continuous derivatives of higher order. Let be a vector field

V , defined by

the relations

,ii

VU

=

,

1, 2, 3i

=

. (A.2.5)

Considering a unit vector

()

i

nn , we notice that

,ii

U

Un

n

∂

⋅= =

∂

Vn ; (A.2.6)

hence, the components of

V with respect to a new three-orthogonal trihedron of

reference

123

Ox x x

′′′

are /

i

Ux

′

∂∂, 1, 2, 3i

=

. Thus, the definition given to the vector

field does not depend on the chosen co-ordinate system. Such a field is called a

conservative field (which derives from the potential

U ); the corresponding vectors are

called conservative vectors and the field

U is called potential. Assuming that

(;)UU t= r , one obtains a vector field defined by the same formulae (A.2.5); this is a

quasi-conservative field and the corresponding vectors are quasi-conservative vectors.

Analogously, the function

U is called quasi-potential.

We notice that one can formally write

(

)

,

/

jj j j

UxU==∂∂Vii

; applying the

vector differential operator

j

jj

j

x

∂

=

=∂

∂

∇ ii, (A.2.7)

which is called nabla (or del) and was introduced by Hamilton, we get

U

=

∇

V . (A.2.8)

In the case of a quasi-potential, the differential is of the form

,

ddd dd

jj

UUx Ut U Ut= +=⋅+



∇ r , (A.2.9)

where

/UUt

=

∂∂



, while the total (substantial) derivative is given by (we consider

the mapping

()tt→ r )

MECHANICAL SYSTEMS, CLASSICAL MODELS

712

,

d

jj

U

Ux U U U

t

=

+= ⋅+





∇ r (A.2.9')

and is the sum of the space derivative and the time derivative; in the case of a potential

remains only the space derivative.

We observe also that, in the case of a conservative vector field, the elementary work

of a conservative vector is given by

dddWU U

=

⋅=

∇

r , (A.2.10)

so that it is a total differential; it results

q

0

1

0

1

() ()

PP

WUPUP

=

− , (A.2.10')

where we started from the formula (A.1.28). Hence, in the case of a conservative

vector, the work between two points does not depend on the path, but only on the

values of the potential at its extremities; analogously, using the formula (A.1.29'), we

notice that the work of a conservative vector on a closed curve (circulation of a

conservative vector) vanishes if the corresponding domain

D is simply connected.

2.1.2 Equipotential surfaces. Gradient

Let be

123

(, , )Ux x x C

=

, constC

=

(A.2.11)

the equation of a surface, which is the locus of the points for which the scalar potential

is constant; we assume that

1

UC

∈

. This surface is called an equipotential surface; if

()U r defines an arbitrary scalar field, then the surface (A.2.11) is called also a level

surface. If

123

(, , ;)UUxxxt= , then we get an equiquasi-potential surface

123

(,, ;)Ux x x t C

=

, constC

=

, (A.2.11')

which is a variable surface in time. Let us suppose that through the point

(

)

000

123

,,xxx

pass two equipotential surfaces, so that for that point we may write

(

)

000

123 1

,,Uxxx C= ,

(

)

000

123 2

,,Ux x x C= ,

12

,constCC

=

; it follows that

12

0 CC=−. Hence, the two equipotential surfaces coincide, admitting that the

function

U is uniform. If two equipotential surfaces do not coincide, then they have not

common points.

Applying the operator

∇ to a scalar function, one obtains a vector function which is

called the gradient of the scalar function. Hence,

gradUU

=

∇

, (A.2.12)

so that the operator

∇ transforms the scalar field in a vector one. Thus, the gradient

allows to appreciate the variation of a scalar function, obtaining also its derivatives of

Appendix

713

first order. We observe thus that a field of conservative vectors is a field of gradients.

We mention the properties:

i)

12 1 2

grad( ) grad gradUU U U+= + (distributivity with respect to addition of

scalars);

ii)

grad gradCU C U

=

, constC

=

;

iii)

gradC = 0 , constC = .

The relation (A.2.6) may be written in the form

grad

U

n

∂

⋅=

∂

n , (A.2.6')

so that the gradient forms a vector field, independent on the chosen system of co-

ordinate axes;

/Un∂∂ is the derivative of the scalar field U in the direction of the unit

vector

n . Because

,

grad

j

jjj

UU U

=

=∂ii, (A.2.12')

it results that the gradient of the function

U is normal to the equipotential surface

123

(, , ) 0Ux x x C−=. As well, if we take vers gradU

=

n in the relation (A.2.12'),

then the gradient of the scalar function

U is a vector directed in the same direction for

which the value of the function is increasing. The congruence of gradient lines is thus

normal to the family of corresponding equipotential surfaces, the direction of travelling

through being that in which the value of the scalar function

U is increasing. These

properties hold also for a quasi-conservative scalar field. If

dr is along the tangent to a

curve

C , which is travelled through by a point

123

(, , )Px x x , then it results – by

equating to zero the expression (A.2.9) – that, for a quasi-potential scalar field, the

curve

C cannot stay on an equiquasi-potential surface; but – in exchange – for a

potential scalar field, the curve

C may belong to a corresponding equipotential surface.

We can verify the above mentioned properties, e.g., in the case of the scalar potential

222

123 123

(, , )

ii

Ux x x xx x x x

=

=++ (A.2.13)

and in the case of the quasi-potential scalar

2222 2

123 123

22

11

(, , ;)

ii

Ux x x t xx t x x x t

cc

= − =++− ,

constc

=

. (A.2.13')

The necessary and sufficient conditions for the scalar functions

123

(, , ;)

ii

VVxxxt

=

, 1, 2, 3i

=

(A.2.14)

to be the components of a quasi-conservative vector are of the form

,

0

j

ijk k ijk k j

VV∈∂ =∈ =, 1, 2, 3i

=

; (A.2.15)

Teodorescu P.P. Mechanical Systems, Classical Models Volume I: Particle Mechanics

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