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

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MECHANICAL SYSTEMS, CLASSICAL MODELS

130

′

(3.1.37)

is fulfilled, hence if the considered function is invariant to a change of orthogonal

Cartesian co-ordinates, then we say that

U is a scalar (or a tensor of zeroth order).

In the case of a scalar considered as a constant function (or of a scalar defined only

at one point) we obtain a mathematical entity characterized by modulus and sign.

Let be a vector

()=VVr (see Fig.2.2); by means of the two systems of co-

ordinates, we may write

′

=Vi i,

(3.1.38)

where

(

)

123

VVxxx

(

)

123

VVxxx

′

, ,1,2,3jk

. Taking into account

the relations (2.1.8) and (2.1.8'), respectively, we find the relations with the aid of

which one may pass from the components of the vector

V in the basis B to its

components with respect to the basis

B ' and inversely, in the form

kkj

VVα

′

kkj

VVα

′

,1,2,3jk

(3.1.39)

In general, three functions

V , 1, 2, 3i

, which are transformed accordingly to the

formulae (3.1.39) by a change of orthogonal Cartesian co-ordinates of the form

(2.1.11), are the components of a vector (or of a tensor of first order) with respect to the

considered frame of reference

B. A tensor of first order can be represented by its

components with respect to a certain basis in the form of a row or column matrix

[]

123 2

VVVV V

⎡

⎤

⎢

⎥

⎡⎤

≡=

⎢

⎥

⎣⎦

⎢

⎥

⎢

⎥

⎣

⎦

(3.1.40)

where T indicates the transpose matrix.

Taking into account (3.1.36), we can express the transformation relations (3.1.39)

also in the form

′

∂

′

∂

′

∂

, ,1,2,3jk

;

(3.1.41)

this form has a more general character, because it can be used also in the case of an

arbitrary (non-linear) co-ordinate transformation (not only in the case of a linear one).

Extending these results, we call tensor of second order a mathematical entity

which is represented by its components with respect to a positive orthonormed basis

in the form of a square matrix

Mass geometry. Displacements. Constraints

131

[]

11 12 13

21 22 23

31 32 33

aaa

aaaa

aaa

⎡

⎤

⎢

⎥

≡

⎢

⎥

⎢

⎥

⎣

⎦

;

(3.1.42)

the elements of this matrix form a set of 3

= 9 quantities

a , ,1,2,3ij

, which, by a

change of orthogonal Cartesian co-ordinates of the form (2.1.11), behave corresponding

to the relations

kl ki lj

aaαα

′

kl ki lj

aaαα

′

, ,,, 1,2,3ijkl

(3.1.43)

Similarly, a tensor of third order is a mathematical entity

a , represented by its

components

ijk

, ,, 1,2,3ijk= , with respect to a positive orthonormed basis B; these

components form a set of 3

= 27 quantities which are transformed by a change of

orthogonal Cartesian co-ordinates of the form (2.1.11) by means of the relations

lmn ijk li nk

aaαα α

′

ijk lmn li nk

aaαα α

′

, ,,,, , 1,2,3ijklmn

(3.1.44)

In general, a tensor of nth order is a mathematical entity

a , represented by its

components

...

ii i

1, 2, 3

, 1,2,...,kn

, with respect to a positive

orthonormed basis

B; these components form a set of 3

quantities which, by a change

of orthogonal Cartesian co-ordinates of the form (2.1.11), are transformed

corresponding to the relations

12 12 11 22

... ...

...

nn nn

jj iiijiji ji

aaαα α

′

12 1 2 11 22

... ...

...

nn nn

ii i j j j ji ji j i

aaαα α

′

,1,2,3

, 1, 2,...,kn

(3.1.45)

The tensors thus defined are Euclidean tensors.

Let

...

ii i

a≡

⎡

⎤

⎣

⎦

be a tensor of nth order; if the relation

12 12

... ... ... ... ... ...

iiiii iiiii

(3.1.46)

takes place in a basis

B, then we say that the tensor is symmetric with respect to the

indices

i and

i ; if the property (3.1.46) holds for any indices

i and

i , then the

tensor

a is totally symmetric. If the components of the tensor a verify the relation

12 12

... ... ... ... ... ...

iiiii iiiii

−

(3.1.46')

in a basis

B, we say that this tensor is antisymmetric (sqew-symmetric) with respect to

the indices

i and

i ; if the property (3.1.46') takes place for all the indices

i and

i ,

then the tensor is totally antisymmetric. A tensor

a which has not one of the properties

mentioned above is an asymmetric tensor.

MECHANICAL SYSTEMS, CLASSICAL MODELS

132

A tensor is, in general, definite at a point of the space; as well, we may consider a

tensor mapping

(

)

(

)

...

123 123

,, ,,

ii i

xxx a xxx→ , defining thus a tensor field.

If we use the upper index “prime” for the quantities which are obtained from the

relation (2.1.9'), by a change of co-ordinate axes, it follows that

lm l

′′

=⋅

ii,

,1,2,3lm= , the relation of definition of these cosines remaining the same; indeed, the

scalar products (2.1.9') are invariant with respect to any frame of reference, in particular

with respect to the frames

B and B '. We see that a relation of the form (3.1.43) holds,

because

lm kj lk lk km lm

α ααα αδ α

′

===

, ,1,2,3lm

where we have introduced Kronecker’s symbol. Hence

α , ,1,2,3ij

, are the

components of a tensor of second order

[]

11 12 13

21 22 23

31 32 33

ααα

αααα

ααα

⎡

⎤

⎢

⎥

≡

⎢

⎥

⎢

⎥

⎣

⎦

(3.1.47)

which makes clear the position of the orthonormed basis

B ' with respect to the basis B

and inversely.

We can define Kronecker’s symbol with respect to the basis

B in the form

0, ,

1, ,

≠

⎧

⎪

′

⎨

⎪

⎩

,1,2,3kl

;

also in this case, a relation of the form (3.1.43) is verified, because

kl ki lj ki li kl

δδααααδ

′

===, ,1,2,3kl

Hence, Kronecker’s symbols are the components of a tensor of second order too

[]

100

010

001

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

(3.1.48)

which is just the unit tensor (

); this tensor is symmetric (

ij ji

δδ

, ,1,2,3ij= ).

Let us consider also the permutation symbol, defined by the formula (2.1.29). Taking

into account the relations

(

)

[

]

123

,, det 1

′

′′

=iii and

(

)

llmn

′′ ′ ′

=∈

ii i ,

lmn= 1, 2, 3 , and the relations (2.1.37), it follows

Mass geometry. Displacements. Constraints

133

lmn ijk li nk

αα α

′

∈=∈ ,

,, 1,2,3lmn

Hence, the permutation symbol is a totally antisymmetric tensor of third order

∈, for

which the relations (2.1.45)-(2.1.46'') take place.

Two tensors

a and b of the same order n are equal (

ab) if they have the same

components

12 12

... ...

ii i ii i

ab= , 1, 2, 3

, 1,2,...,jn

, (3.1.49)

in an arbitrary frame

B; this relation has the well known properties of reflexivity,

symmetry and transitivity.

The sum of two tensors (

a and b ) of order n is a tensor

+cab of the same

order; hence, in a frame

B we have

12 12 12

... ... ...

nn n

ii i ii i ii i

abc

= . (3.1.50)

The addition of tensors is commutative and associative.

Multiplying all the components of a tensor

a of nth order in a frame B by the same

scalar

λ , one obtains a new tensor λ

ba, of the same order and of components

...

ii i

aλ ; this operation is distributive with respect to the addition of tensors, as well as

of scalars. In particular, we can have

1λ

− ; hence, by subtracting two tensors of nth

order one obtains a tensor of the same order. If

b0 (all the components of a null

tensor in an arbitrary basis

B vanish), then we have 0λ

a0.

Starting from the observations made at the previous subsection, one can show that an

asymmetric tensor of nth order can be univocally decomposed in a sum of two tensors,

the first one symmetric with respect to the indices

i and

i and the second one

antisymmetric with respect to the same indices; we can thus write

()

12 12 12

... ... ... ... ... ... ... ... ...

nnn

jjj

kkk

ii i i i ii i i i ii i i i

aaa=+

()

12 12

... ... ... ... ... ...

iiiii iiiii

aa+−.

(3.1.51)

The tensor product (external product) of two tensors

and b of nth and mth order,

respectively, is a tensor

=⊗cab of (n + m)th order; in a frame B, we can write

12 1 2

... ...

...

ii i j j j

kk k

ab c

, (3.1.52)

where the indices

k , 1,2,...,lnm=+ are the indices

i , 1,2,...,pn

, and

j ,

1,2,...,

qm= . For instance, by the dyadic product of two tensors of first order one

obtains a tensor of second order

ij ij

ab c

, ,1,2,3ij

. (3.1.52')

MECHANICAL SYSTEMS, CLASSICAL MODELS

134

The external product of tensors is not commutative, but it is associative and distributive

with respect to their addition.

If we make

jj= in the relation of definition (3.1.45) and if we take into account

the formulae (3.1.35), then we obtain, in a basis

12 12 11 22

1111 1111

... ... ... ... ... ...

...

kkk lkl kkk lkl

jj j j j j j j j ii i ii i ii i ji ji

aaαα

−+−+ −+−+

′

11 11 11 11

... ... ...

kk kk ll ll

iji jiji ji

αα αα α

−− ++ −− ++

;

this operation is called the contraction of the tensor. Hence, by the contraction of two

indices of a tensor of nth order one obtains a tensor of (n – 2)th order. For instance, by

the contraction of a tensor

a of second order, of components

in a basis B, one

obtains a scalar, called the trace of the tensor

a and denoted

a . (3.1.53)

In particular, by the contraction of Kronecker’s tensor one obtains

tr 3

δ==1 .

The internal product (the contracted tensor product) of two tensors

a and b of nth

and mth order, respectively, is a tensor

ab of (n + m – 2p)th order, where

is the

number of effected contractions. For instance, the scalar product of two vectors

and

b , of components

a and

b , respectively, will be given by the contracted product

ab⋅= =ab ab . If =×cab is the vector product of the two vectors, we may write

ijk k

cab=∈ , 1, 2, 3i = . We notice that the triple scalar product of three vectors a ,

b and c can be expressed also by means of a contracted product in the form

(, ,)

ijk k

abc=∈abc . If a is a tensor of second order, of components

in the basis

B, while b is a vector of components

b in the same basis, then the contracted product

=cab is a vector of components

iijj

cab

; if a is a tensor of components

ijk

while

b is a tensor of components

b , then we obtain the contracted product of

components

ijl ijk kl

cab= . The sum is obtained by contracting the last index of the first

tensor by the first index of the second tensor; eventually, one can contract also the last

but one index of the first tensor by the second index of the second tensor a.s.o. By

means of two permutation tensors one obtains the external product

ijk lmn

∈

∈ , given by

the formula (2.1.45); by contraction, one is led to various internal products of the form

(2.1.46)-(2.1.46'').

We have seen that by algebraic operations of addition and of external or internal

products one obtains new tensors. To identify if a quantity is a tensor, the quotient law

is frequently used; so, if we have a relation of the form

(

)

...

, ,...,

jj j

kk k

ii i b c

(3.1.54)

where the indices

k , 1,2,...,lnm=+, are just the indices

i , 1,2,...,pn

, and

j ,

1,2,...,

qm= , then the function

is of the form

Mass geometry. Displacements. Constraints

135

(

)

...

, ,...,

ii i

ii i a

(3.1.54')

obtaining thus the components of a tensor of nth order in a basis

Let

(

)

123

,,UUxxx

be a scalar field, defined on a domain D , with

()UCD∈ , and let be /

∂

∂ , 1, 2, 3i

, the components of the conservative

vector field which is thus obtained; by a change of co-ordinates, we obtain

(

)

123

,,UUxxx

′′′′′

= and we may write

jiji

UU U

xxxx

′

∂

∂∂ ∂

′′

∂∂∂∂

, 1, 2, 3j

where we took into account the relations (3.1.36), (3.1.37). Hence, the derivatives of a

scalar field with respect to the independent variables lead to a tensor field of first order;

we denote

ii i

Ux UU∂∂=∂=, 1, 2, 3i

. Analogously, starting from the vector

field

(

)

123

VVxxx= ,

1, 2, 3

, with

()

VCD∈

, and by a change of co-

ordinates of the form (2.1.11), we can write

(

)

lklkik

jiji

VV V

xxxx

αα

′

∂∂ ∂

∂

′′

∂∂∂∂

,1,2,3lj

;

hence, the derivatives of first order of a tensor field of first order are the components of

a tensor field of second order. We write

ij jiij

Vx VV

∂

∂=∂ =

, ,1,2,3ij= . In

general, the derivatives of first order of a tensor field of nth order are the components of

a tensor field of (n + 1)th order.

One can define, analogously, derivatives of higher order with respect to the

independent variables of a tensor field. If

()

aCD∈ , the we can write

,,ijk ikj

, ,, 1,2,3ijk

, (3.1.55)

the derivatives of second order being immaterial of the order of differentiation

(Schwartz’s theorem); we notice thus that the tensor

,ijk

is symmetric with respect to

the indices

and k .

Besides the operators grad, div and curl introduced in App. §2, for a vector field, we

can introduce also the operators Grad, Div and Curl for the tensors of higher order.

1.2.3 Tensors of second order

Let

a be an asymmetric tensor of second order, expressed by its components

a in

the form of a square matrix (3.1.42). If we denote the symmetric part of this tensor by

()

=+aaa,

()

ij ji

aaa=+,

(3.1.56)

and the antisymmetric one by

MECHANICAL SYSTEMS, CLASSICAL MODELS

136

()

=−aaa,

[]

()

ij ji

aaa=−,

(3.1.56')

then we can write univocally

=+aa a,

()

[]

aa a

. (3.1.56'')

Here

a is the transpose tensor, represented by the transpose matrix

[]

11 21 31

12 22 32

13 23 33

aaa

aaaa

aaa

⎡

⎤

⎢

⎥

≡

⎢

⎥

⎢

⎥

⎣

⎦

;

(3.1.57)

we have

()

=aa,

=aa

and

−aa

A symmetric tensor of second order is represented by the matrix

[]

11 12 13

12 22 23

13 23 33

aaa

aaaa

aaa

⎡

⎤

⎢

⎥

≡

⎢

⎥

⎢

⎥

⎣

⎦

(3.1.58)

which has only six distinct components, while an antisymmetric tensor of second order

is represented in the form

[]

12 13

12 23

13 23

aa a

⎡

⎤

−

⎢

⎥

≡−

⎢

⎥

⎢

⎥

−

⎢

⎥

⎣

⎦

(3.1.58')

and has only three distinct components.

Let us consider the contraction

11 22 33

aaaa

=++a

(3.1.59)

of the symmetric tensor of components

a ; it is an invariant by the transformations of

co-ordinates, being a scalar. The tensor

(tr / 3)=aa1

of components /3

a δ ,

,1,2,3ij= , and matrix

100

010

001

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

(3.1.60)

Mass geometry. Displacements. Constraints

137

constitutes the spheric tensor of the symmetric tensor

a ; the tensor (tr / 3)

′

=−aa a1

of components

ij ij ij

aaaδ

′

=−

, and matrix

11 12 13

12 22 23

13 23 33

aa a a

aaaa

⎡⎤

−

⎢⎥

−

⎢⎥

−

⎢⎥

⎣⎦

(3.1.60')

is called the deviator of the considered symmetric tensor. We obtain thus the canonical

decomposition of a symmetric tensor of second order

′

+aa a.

(3.1.61)

An antisymmetric tensor

a of second order is a degenerate one, equivalent to a

tensor of first order; we can associate the axial vector

kijk

aa=∈ , 1, 2, 3k

(3.1.62)

being thus led to

123

aa= ,

231

312

. Inversely, multiplying both members

klm

∈ and taking into account (2.1.46), we find

()

11 1

22 2

ij mj mi ij

klm k klm ijk li lj lm ml

aa aaaδδ δδ∈=∈∈= − = −;

hence, one can write

ijk k

∈ ,

,1,2,3

. (3.1.62')

For instance, the vector product

c of two vectors a and b (

×cab) leads to an

antisymmetric tensor of second order

ijk k

ab c

∈ , ,1,2,3ij

. (3.1.63)

Let

be a symmetric tensor of second order, of components

a , represented by the

matrix (3.1.58), and let be the associate quadratic form

ij i j

axxΦ

; (3.1.64)

this form being a scalar, we can write as well

ij i j

axxΦ

′

′′

(3.1.64')

MECHANICAL SYSTEMS, CLASSICAL MODELS

138

in a new system of orthonormed Cartesian co-ordinates. Considering the tensor

defined at a point, which we choose as the origin of the co-ordinate axes, the equation

1Φ

(3.1.65)

represents a quadric

Γ (one chooses the sign so as to obtain a real quadric); this one

allows us to give a geometric image to the variation of the tensor components, by a

change of the system of co-ordinate axes. To reduce the matrix (3.1.58) to a diagonal

one, corresponds to the reduction of the quadratic form (3.1.64) to a sum of squares

(eventual with a sign minus), as well as to the representation of the quadric

Γ in the

canonical form (with respect to its axes).

Let us consider the components for which the indices

k and l are equal (without

summation) in the relation (3.1.43). In this case,

α and

α are direction cosines of an

axis

Δ with respect to the frame

123

Ox x x

; denoting them by

α , 1, 2, 3i

, we may

write

ij i j

αα

, (3.1.66)

where

is a component of the principal diagonal of the tensor a in the new system

123

Ox x x

′′′

, Δ being one of the new axes of co-ordinates. The directions for which this

component has extreme values are called principal directions and correspond to the

quadric’s

Γ axes; we have 0

′

, kl

≠

, ,1,2,3kl

, in this system of co-

ordinates, and quadric’s equation can be written in the canonical form, while the

quadratic form (3.1.64') is expressed as a sum of squares.

We notice that

αα

; (3.1.67)

we have thus to solve a problem of extremum with constraints. Using the method of

Lagrange’s multipliers, we consider the function

(

)

(

)

iijij ii

Faαααλαα

+− ;

we calculate the partial derivatives

()

222

ij j i ij ij j

αλα λδα

∂

=−=−

∂

, 1, 2, 3i

and are led to the equations

(

)

ij ij j

a λδ α−=, 1, 2, 3i

(3.1.68)

This system of three equations with three unknowns is homogeneous; because, by

virtue of the relation (3.1.67), the system allows only non-zero solutions, its

determinant must vanish, so that

Mass geometry. Displacements. Constraints

139

[]

11 12 13

12 22 23

13 23 33

det 0

ij ij

aaa

aaaa

aaa

λδ λ

−

−= − =

−

(3.1.69)

Developing this equation, we get

123

0λλλ−+−=III,

(3.1.70)

where

1112233

ijk ljk il il il

aaaaaaδ=∈∈ = = = + +I ,

(3.1.70')

()

2223333111122

jm ii jj ij ij

ijk lmk il

aa aa aa a a a a a a=∈∈ = − = + +I

(

)

222

23 31 12

aaa−++

(3.1.70'')

[]

3112233

det

ijk lmn il kn

aa a a a a a=∈∈ = =I

(

)

222

11 23 22 31 33 12 23 31 12

2aa aa aa aaa−++ + .

(3.1.70''')

The equation (3.1.70) is of the third degree and has three roots, at least one of them

being real; the system (3.1.68) together with the condition (3.1.67) leads, for each root,

to a principal direction for which

ij j i

a αλα

1, 2, 3

. (3.1.71)

We may write the relations

ij j i

a αλα

′′

ij j i

a αλα

′

′′′

, 1, 2, 3i

corresponding to two roots

λ and

λ ; multiplying the first relation by

′′

and the

second one by

′

, we get

ij j i i i

a αα λαα

′′′ ′′′

ij j i i i

a αα λαα

′

′′ ′′′

Interchanging the dummy indices in the first member of the latter equation, taking into

account that tensor

a is symmetric, and subtracting the second equation from the first

one, one obtain

(

)

λλαα

′

′′

−

= .

(3.1.71')

λλ= , then the two roots are real; if

λλ

≠

, then we obtain 0

αα

′′′

; hence,

the corresponding principal directions are orthogonal. If the two roots are complex