Teodorescu P.P. Mechanical Systems, Classical Models Volume I: Particle Mechanics
Подождите немного. Документ загружается.
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
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
i
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
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
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
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
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
12
... ,1 ...
p
p
ii i
ii i n∧∧∧ ≤<<<≤ee e forms a basis in this space, so
that
dim
p
p
n
n
LC
∧
= , (A.1.49')
where we have introduced the combination symbol of
n things
p
at a time; in
particular,
dim 1
nn
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
...
12
1
1
... ...
!
p
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
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
j
j
f
f
x
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
12
...
d d d d ... d
p
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
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
23
ddd dd
i
ji
j
aa
a
xx xx
xxx
ω
∂∂
∂
⎛⎞
=∧=−∧
⎜⎟
∂∂∂
⎝⎠
∑
13 21
31 12
31 12
dd dd
aa aa
xx xx
xx xx
∂∂ ∂∂
⎛⎞ ⎛⎞
+− ∧+ − ∧
⎜⎟ ⎜⎟
∂∂ ∂∂
⎝⎠ ⎝⎠
(A.1.57')
allows to introduce the operator curl. As well, the exterior derivative
123
123
123
dddd
aaa
xxx
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
j
xxx Vxxx
=
Vi, (A.2.1)
where
1
()
j
VCD∈ , 1, 2, 3j = ; let us introduce the vector fields
,,
j
ijjij
ii
V
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
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
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
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
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)