Download A GENERALIZATION OF THE CARTAN FORM pdq − H dt

A GENERALIZATION OF THE CARTAN FORM
pdq − Hdt
VIOREL PETREHUŞ
We prove the existence and uniqueness of a Cartan form for Lagrangians on J k Y .
We restricted to fiber manifolds πX,Y : Y → X where X is unidimensional.
AMS 2000 Subject Classification: Primary 57R55, Secondary 53Z05.
Key words: fiber space, manifold of jets, Cartan form.
In this paper we follow the lines of [7], [6], [4] from Lagrangians on J 1 Y
or J 2 Y to Lagrangians on J k Y . The first part is standard and more general
computations can be found in [6], [4], [8]. Our result is that the form θLag
associated with a lagrangian Lag on J k Y is well defined on the space J 2k−1 Y ,
and is contained in Theorem 3. Many other forms θ with some properties of
θLag can be defined, see [5]. This form is important in the study of conservation
laws for Lag or in the study of conservative numerical methods for the Euler
equations associated with Lag.
The basic space of our computations is J k Y , the space of k jets of sections
of the fiber space πXY : Y → X. In the sequel X is the real field and Y is an
N + 1 dimensional manifold. The fibre of πXY : Y → X is diffeomorphic to a
given manifold Q. All manifolds are finite dimensional. A k jet over x ∈ X is
a class of equivalence of sections of Y defined in a neighborhood of x such that
two sections are equivalent iff their Taylor developments agree up to order k.
The jet corresponding to section s will be denoted by j k s(x). The definition
is independent
of the local
coordinates on X or Q. If x is a local coordinate
on X and y1 , y 2 , . . . , yN are local coordinates on Q, then on J k Y we use the
coordinates x, y A , yiA A=1,...,N, i=1,...,k , where for the jet of a section s : X →
di A
Y , s(x) = x, s1 (x), s2 (x), . . . , sN (x) we have y A = sA (x), yiA = dx
i s (x).
A
A
k
In what follows y0 is identical with y . The correspondence x → j s(x) is
a section of J k Y and is denoted by j k s. By truncation of a k development
to a l development (l < k) we get a projection πJ l Y,J k Y : J k Y → J l Y . The
projective limit of these spaces is denoted by J ∞ Y and the limit of their jets
MATH. REPORTS 9(59), 4 (2007), 357–368
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Viorel Petrehuş
2
corresponding to section s at x ∈ X is denoted j ∞ s(x). For more about jet
spaces and the change of coordinates see [10],
A projetable diffeomorphism η of Y is a diffeomorphism ηY : Y → Y
such that there exists a diffeomorphism ηX of X such that the diagram
(1)
η
Y


πX,Y
Y
−→
Y


πX,Y
X
X
−→
X
η
is commutative; η can be extended to a diffeomorphism ηJ k Y of J k Y
that the diagram
jk Y

πJ k−1 Y,J k Y
..
.
j1Y


πY,J 1Y
η k
J Y
−→
η 1
J Y
−→
ηY
Y


πX,Y
−→
X
−→
ηX
such
jk Y


πJ k−1 Y,J k Y
..
.
j1Y


πY,J 1 Y
Y


πX,Y
X
is commutative. The extension is given by
−1
(ηX (x)) .
(2)
ηJ k Y j k s(x) = j k ηY ◦ s ◦ ηX
∂
A ∂
∂x + y1 ∂y A +
space T J k−1 Y ,
Using local coordinates we define the total derivative Dx =
y2A ∂y∂A + · · · + ykA ∂y∂A
1
k−1
as a function from J k Y to the tangent
which sends a point γ ∈ J k Y to a vector tangent to J k−1 Y at πJ k−1 Y,J k Y (γ).
More conveniently, we consider Dx as a “vector field” on J ∞ Y by extending
N
∂
∂
A
+ ∞
the preceding definition to Dx = ∂x
k=0
A=1 yk+1 ∂y A .
k
Any function f : J k Y → R can be considered as a function also denoted
abusively f , defined on any J n Y (n > k) by the composition J n Y
J kY
f
J ∞Y
πJ k Y,J n Y
−→
−→ R and consequently defined on
. Analogously, any form ω ∈
ΛJ k Y gives a form on J n Y (n > k) by the formula πJ∗ n Y,J k Y ω, so it gives a
form on J ∞ Y . We define Dx,p = Dx ◦ Dx ◦ · · · ◦ Dx (p times), which gives for
f : J k Y → R, a function Dx,p f = Dx Dx (. . . Dx (f )) defined on J k+p Y . For
A generalization of the Cartan form pdq − Hdt
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359
p = 0 we define Dx,0 = identity. For a section s of Y we have
dk sA (x)
dsA (x)
d
f x, sA (x),
,...,
=
dx
dx
dxk
dsA (x)
dk sA (x) dk+1 sA (x)
A
k+1
,
f
)
j
s(x)
.
,...,
=
(D
= (Dx f ) x, s (x),
x
dx
dxk
dxk+1
For the one parameter group of diffeomorphisms ηYε : Y → Y which
ε : X → X, its infinitesimal generator is of
covers the one parameter group ηX
the form
∂
∂
+ V A (x, y) A
(3)
V = V x (x)
∂x
∂y
(by convention the sum is taken over all values of A from 1 to N ). The
extended group ηJε k Y : J k Y → J k Y has a generator denoted by
j k (V ) = V x (x)
(4)
N
N
A=1 j=1
A=1
The components of
jk
k
∂
∂
∂
+
V A (x, y) A +
VjA A .
∂x
∂y
∂yj
(V ) are given (see [8]) by
A
,
VjA = Dx,j V vA + V x yj+1
where
V vA = V A − y1A V x .
∂
A
vA ∂
A x
=
is called the vertical
The field V v =
AV
A V − y1 V
∂y A
∂y A
k
part of V . The field j (V ) can be split as
(5)
j k (V ) = j k (V )v + j k (V )h ,
(6)
where the vertical part is
(7)
j k (V )v =
V vA
N N k
k
∂
∂
vA ∂
+
D
V
=
Dx,j V vA A
x,j
A
A
∂y
∂yj
∂yj
A=1 j=1
A
A=1 j=0
and the horizontal one is
(8)
N
k
∂
∂
A
+
V x yj+1
= V x Dx .
j (V ) = V
∂x
∂yjA
k
h
x
A=1 j=0
One sees that
k
j (V )
h
k
x ∂
.
j s(x) = j s∗ (x) V
∂x
k
The field j k (V ) is tangent to J k Y , but j k (V )v and j k (V )h have extra
A
which cancel each other; j k (V )v and j k (V )h are always
terms involving yk+1
∞
defined on J Y by extending the sumation over j to ∞. To avoid such
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Viorel Petrehuş
4
technical difficulties as (8) not being tangent to J k Y , we shall consider all
fields as functions or forms defined on J ∞ Y . Functions and forms are supposed
finitek depending on that form or function. For
depending on J k Y for some ∂
A
x is defined on
a vector field V = V x ∂x
+ A ∞
j=0 Vj , we assume that V
J m Y and VjA depends on J j+m Y for a fixed m which depends on V; Dx is an
example of such a vector field. In this case, the basic operations of calculus
with differential forms: outer product, exterior derivative, Lie derivative, etc.
make sense (see [1]).
A dx,
The differential forms θ0A = θ A = dy A − y1A dx, θjA = dyjA − yj+1
for j ≥ 1 are defined using local coordinates. For any section s we have
j k s∗ θjA = 0, and θjA (Dx ) = 0. For more on jet spaces see [1], [6], [10].
By a Lagrangian on J k Y we understand a differential
of degree 1
1form
∗
X
. Using local
Λ
which with any jet γ ∈ J k Y associates a value in πX,J
kY
coordinates, a Lagrangian is a differential form
(9)
Lag = L x, y A , y1A , . . . , ykA dx.
Consider now a C ∞ -family of sections sε : [a (ε) , b(ε)] → Y defined for ε
d
|ε=0 sε . Then if all fits into a coordinate
in a neighborhood of 0, and let ζ = dε
0
chart, integration by parts gives (s equals s):
b(ε) d k ε
(10a)
L
j
s
(x)
dx =
dε ε=0 a(e)
b(ε) dk sεA (x)
dsεA (x)
d εA
,...,
L x, s (x),
dx =
=
dε ε=0 a(e)
dx
dxk
N k b
d
∂L
(10b)
(−1)j
· ζ A (x)dx+
j k s (x)
=
dxj ∂yjA
a
A=1 j=0
(10c)
+ L j k s (b) b (0) − L j k s (a) a (0)+
j−1
N k dj−m−1 ζ A b
d
∂L
m
k
(−1)
·
=
j s(x)
+
dxm ∂yjA
dxj−m−1 a
m=0
A=1 j=1
(10d)
(10e)
N k ∂L 2k
s(x)
· ζ A (x)dx+
j
A
∂y
j
A=1 j=0 a
+ L j k s (b) b (0) − L j k s (a) a (0)+
=
b
(−1)j Dx.j
dj−m−1 ζ A b
∂L 2k−1
.
(−1) Dx,m A j
s (x) ·
+
dxj−m−1 a
∂y
j
m=0
A=1 j=1
j−1
N k m
A generalization of the Cartan form pdq − Hdt
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361
The variation splits into an integral over [a, b] and a boundary part.
We shall express the variation in terms of the differential form below defined
locally on J 2k−1 Y as
θLag = Ldx +
(11)
j−1
N k (−1)m Dx,m
A=1 j=1 m=0
= Ldx +
N k−1 k−b−1
∂L
A
· θj−m−1
=
A
∂yj
(−1)m Dx,m
A=1 b=0 m=0
∂L
A
∂ym+b+1
· θbA .
Let us look at this variation in a different way, as in [6], [4]. Let M
the space of C ∞ -applications ϕ : [0, 1] → Y and ϕX : [0, 1] → X with ϕX
injective such that ϕX = πX,Y ◦ ϕ. Then s = ϕ ◦ ϕ−1
X is a section of Y defined
on [a, b] = ϕX ([0, 1]) ⊂ X and we define the action S by
S (ϕ) =
(12)
b
a
=
a
b
b k
L j s(x) dx =
L j k ϕ ◦ ϕ−1
(x)
dx =
X
∗
j k ϕ ◦ ϕ−1
Lag.
X
a
Let ηYε be a one-parameter group of diffeomorphisms of Y which covers
ε of X, and let V be its infinitesimal generator. If
the one parameter group ηX
def
def
ε ◦ ϕ , then we have ϕε = π
ε
ϕε = ηYε ◦ ϕ and ϕεX = ηX
X
X,Y ◦ ϕ . In this case,
X
ε−1
sε = ϕε ◦ ϕX is a section of Y defined on [a(ε), b (ε)] = ϕεX ([a, b]) and there
is defined the action
ε
S (ϕ ) =
b(ε)
a(ε)
b(ε)
=
a(ε)
L j s (x) dx =
b(ε)
k ε
a(ε)
ε−1
L j k ϕε ◦ ϕX
(x) dx =
ε−1 ∗
j k ϕε ◦ ϕX
Lag.
The definition of ηJε k Y as
ε
ε−1
(x)
= j k ηYε ◦ ϕ ◦ ϕ−1
(ηX (x)) =
ηJε k Y j k ϕ ◦ ϕ−1
X
X ◦ ηX
ε−1
ε
(x))
(ηX
= j k ϕε ◦ ϕX
implies
∗
ε−1 ∗
ε∗
◦ ηJε∗k Y = ηX
◦ j k ϕε ◦ ϕX
.
j k ϕ ◦ ϕ−1
X
362
Viorel Petrehuş
6
Using this, we get (L is the Lie derivative)
b(ε)
∂ ∂ ε−1 ∗
j k ϕε ◦ ϕX
Lag =
S (ϕε ) =
∂ε ε=0
∂ε ε=0 a(ε)
b
∂ ε−1 ∗
ε∗ k
ηX
j ϕε ◦ ϕX
Lag =
=
∂ε ε=0 a
b
∗ ε∗
∂ j k ϕ ◦ ϕ−1
ηJ k Y Lag =
=
X
∂ε ε=0 a
b
∗
j k ϕ ◦ ϕ−1
Lj k (V ) Lag.
=
X
a
Now, we split the Lie derivative as Lj k (V ) = Lj k (V )h + Lj k (V )v and obtain
(13a)
[a,b]
(13b)
∗
j ϕ ◦ ϕ−1
Lj k (V )h Lag =
X
k
=
[a,b]
=
(13c)
[a,b]
[a,b]
(13d)
(13e)
LV x
∂
∂x
diV x
[a,b]
=
[a,b]
∗
j k ϕ ◦ ϕ−1
LV x Dx Lag =
X
∗
j k ϕ ◦ ϕ−1
Lj k (ϕ◦ϕ−1 ) (V x ∂ ) Lag =
X
X ∗
∂x
=
diV x
∗
j k ϕ ◦ ϕ−1
Lag =
X
∂
∂x
∂
∂x
+ iV x
∂
∂x
∗
d j k ϕ ◦ ϕ−1
Lag =
X
L j k ϕ ◦ ϕ−1
(x)
dx =
X
b
x
(x)
·
V
= L j k ϕ ◦ ϕ−1
= L j k s(b) b (0) − L j k s(a) a (0) =
X
a
b
−1 ∗
2k−1
ϕ ◦ ϕX ij 2k−1 (V )h θLag =
=j
a
−1 ∗
2k−1
j
ϕ ◦ ϕX ij k (V )h θLag .
=
∂[a,b]
So, the first two terms of the boundary part of the variation (10e) come
b ∗
·
from j 2k−1 (V )h . Necessarily, the third term of (10e) is equal to a j k ϕ ◦ ϕ−1
X
A generalization of the Cartan form pdq − Hdt
7
363
Lj k (V )v Lag. To connect the third term in (10e) with θLag , we look at the equations
ε
d d d ε−1
ε−1
(x) =
ηY ◦ ϕ ◦ ϕ−1
(x) =
ζ(x) =
sε =
ϕε ◦ ϕX
X ◦ ηX
dε ε=0
dε ε=0
dε ε=0 ε
d
∂
ε−1
=
=
ηY ◦ s ◦ ηX
(x) = V (s(x)) − ds V x (x)
dε ε=0
∂x
∂
dsA (x)
= V v (s(x)).
= V A (s(x)) − V x (x)
dx
∂y A
Hence
j−m−1
dj−m−1 vA
dj−m−1 ζ A (x)
vA
=
V
(s(x))
=
D
V
s(x)
=
j
x,j−m−1
dxj−m−1
dxj−m−1
= j 2k−1 (V )vA
j−m−1 .
Then the third term of the boundary part of the variation (10e) is
b
∂L 2k−1
vA
2k−1
(−1) Dx,m A j
s (x) · j
(V )j−m−1 a
∂yj
A=1 j=1 m=0
j−1
N k =
j−1
N k m
(−1)m Dx,m
A=1 j=1 m=0
(14)
(15)
b
∂L 2k−1
v A
2k−1
s
(x)
·
θ
(V
)
j
j
j−m−1
a
∂yjA
b
= j 2k−1 s∗ ij 2k−1 (V )v θLag a
j 2k−1 (s)∗ ij 2k−1 (V )v θLag .
=
∂[a,b]
∗
ij 2k−1 (V ) θLag for the
Now, (13e) and (15) yield ∂[a,b] j 2k−1 ϕ ◦ ϕ−1
X
boundary part of the variation. These results may be summarized as
Lemma 1. Let Lag = L x, y A , y1A , . . . , ykA dx be a Lagrangian defined
on J k Y and ηYε : Y → Y a one parameter group of projetable diffeomrphisms
of Y with infinitesimal generator V . Let ηJε k Y be the extension of ηYε to J k Y
and j k (V ) its infinitesimal generator. Then for ϕ ∈ M, ϕX ([0, 1]) = [a, b],
such that the image of ϕ is contained into a coordinate chart, the variation of
the action functional (12) is given by
k b
N ∂L ∂ ε
(−1)j Dx,j A j 2k ϕ ◦ ϕ−1
(x) ·V vA (s(x)) dx+
S (ϕ ) =
X
∂ε ε=0
∂yj
A=1 j=0 a
∗
j 2k−1 ϕ ◦ ϕ−1
ij 2k−1 (V ) θLag . +
X
∂[a,b]
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Viorel Petrehuş
8
The exterior derivative of θLag is given by
dθLag
−
j−1
N k ∂L
=
(−1)m d Dx,m A
∂yj
A=1 j=1 m=0
j−1
N k (−1)m Dx,m
A=1 j=1 m=0
A
∧ θj−m−1
−
∂L
A
dyj−m
∧ dx + dL ∧ dx.
A
∂yj
∗
iW dθLag for a vertical
Now, we are going to compute j 2k−1 ϕ ◦ ϕ−1
X
2k−1 A ∂
2k−1 Y . We have
W
on
J
vector field W = N
j ∂y A
A=1
j=0
j
∂L
A
(−1) d Dx,m A (W ) · θj−m−1
−
iW dθLag =
∂yj
A=1 j=1 m=0
j−1
N k ∂L
m
A
(−1) Wj−m−1
· d Dx,m A −
−
∂yj
A=1 j=1 m=0
j−1
N k j−1
N k m
N k
∂L
∂L A
A
(−1) Dx,m A · Wj−m · dx +
W · dx.
−
∂yj
∂yjA j
A=1 j=1 m=0
A=1 j=0
m
∗ A
θj = 0, we get
Taking into account that j 2k−1 ϕ ◦ ϕ−1
X
(16a)
(16b)
(16c)
(16d)
(16e)
∗
iW dθLag
j 2k−1 ϕ ◦ ϕ−1
X
∂L −1
(−1)
· m+1
ϕ ◦ ϕX (x) dx
=−
A
dx
∂y
j
A=1 j=1 m=0
j−1
N
k d
∂L m
A
−1
(−1) Wj−m · m
ϕ ◦ ϕX (x) · dx
−
A
dx
∂y
j
A=1 j=1 m=0
j−1
k N m
A
Wj−m−1
d
k
N ∂L −1
ϕ
◦
ϕ
(x)
· WjA · dx
+
X
A
∂yj
A=1 j=0
N
∂L −1
ϕ
◦
ϕ
(x)
· W0A +
=
X
A
∂y0
A=1
N k
∂L j d
−1
(−1)
Dx,j A ϕ ◦ ϕX (x) · W0A dx
+
dxj
∂yj
A=1 j=1
A generalization of the Cartan form pdq − Hdt
9
(16f)
=
N


A=1
k
j=0
365

∂L  · W0A dx.
Dx,j A ϕ ◦ ϕ−1
X (x)
∂yj
d
(−1)j
dxj
Now, Lemma 1 may be reformulated as
Proposition 2. Under the hypotheses of Lemma 1, the variation of the
functional S is
(17)
∂
|ε=0 S (ϕε ) =
∂ε
[a,b]
∗
j 2k−1 ϕ ◦ ϕ−1
ij 2k−1 (V ) dθLag +
X
+
∂[a,b]
∗
j 2k−1 ϕ ◦ ϕ−1
ij 2k−1 (V ) θLag .
X
Proof. Because
∗
∗
ij 2k−1 (V )h dθLag = j 2k−1 ϕ ◦ ϕ−1
ij 2k−1 (ϕ◦ϕ−1 ) (V x ∂ ) dθLag
j 2k−1 ϕ ◦ ϕ−1
X
X
X ∗
∂x
−1 ∗
2k−1
= iV x ∂ dj
ϕ ◦ ϕX θLag = iV x ∂ 0 = 0,
∂x
∂x
by (16) we have
∗
∗
ij 2k−1 (V ) dθLag = j 2k−1 ϕ ◦ ϕ−1
ij 2k−1 (V )v dθLag
j 2k−1 ϕ ◦ ϕ−1
X
X
∂L 2k −1
=
(−1) Dx,j A j
ϕ ◦ ϕX (x) · VjvA ϕ ◦ ϕ−1
X dx.
∂yj
A=1 j=0
N k
j
The result now follows from Lemma 1.
Now, we shall prove that the definition (11) of θLag is independent
of any coordinate chart on the space J 2k−1 Y of jets and formula (17) of the
variation holds for any ϕ ∈ M.
Theorem 3. Let the fibration πX,Y : Y → X, with X a one-dimensional
k
manifold. Let Lag be a Lagrangian
defined on
J Y , which εis given in a co
A
A
A
ordinate chart by Lag = L x, y , y1 , . . . , yk dx and let ηY : Y → Y be a
one-parameter group (1) of projetable diffeomrphisms of Y with infinitesimal
generator V given by (3). Let ηJε k Y be the extension (2) of ηYε to J k Y and
j k (V ) its infinitesimal generator given by (4), (6), (7), (8). Let on J 2k−1 Y
the form θLag given locally by (11). Then
a) θLag is well defined on J 2k−1 Y ;
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Viorel Petrehuş
10
b) for ϕ ∈ M, ϕX ([0, 1]) = [a, b], the variation of the action functional
(12) is given by
∗
∂
|ε=0 S (ϕε ) =
j 2k−1 ϕ ◦ ϕ−1
ij 2k−1 (V ) dθLag +
X
∂ε
[a,b]
∗
j 2k−1 ϕ ◦ ϕ−1
ij 2k−1 (V ) θLag ;
+
X
∂[a,b]
c) we have the formula
∗
∗
−1
2k−1
j k ϕ ◦ ϕ−1
Lag
=
j
θLag ;
ϕ
◦
ϕ
X
X
d) in a standard coordinate chart for J 2k−1 Y , θLag only depends at each
point j 2k−1 (s)(x) on dx and dyjA , for j ≤ k − 1;
e) if W is a vector field on J 2k−1 Y , tangent to the fibers of πY,J 2k−1 Y ,
∗
(iW dθLag ) = 0;
then j 2k ϕ ◦ ϕ−1
X
f) θLag is uniquely defined by b), c), d), e);
g) the Euler-Lagrange equations
d
∂L j
−1
(−1)
ϕ ◦ ϕX
= 0, A = 1, 2, . . . , N
(18)
dxj ∂yjA
0≤j≤k
are equivalent to
(19)
∗
(iW dθLag ) = 0
j 2k−1 ϕ ◦ ϕ−1
X
for any vector field W on J 2k−1 Y .
Proof. If the image of ϕ is contained into a coordinate chart, then b) is
Proposition 2, c) follows from (11), d) is evident from the definition of θLag ,
e) follows from (16) taking into account that W0A = 0. Assuming again that
the image of ϕ is contained into a coordinate chart, let θ be any form with
properties b), c), d), e) above. It follows from c) anc d) that θ = Ldx +
k−1 j A
N k−1 j A
FA θj . Let ∆ = θ − θLag . Then ∆ = N
A=1
j=0
A=1
j=0 GA θj . From
2k−1
∗
(s) ij 2k−1 (V ) d∆ = 0 for any projetable vector field V on
b) we get [a,b] j
Y and any section s = ϕ ◦ ϕ−1
X . From e) we have iW d∆ = 0 for any vertical
vector field πY,J 2k−1 Y , whence
j 2k−1 (s)∗ iW d∆ = 0
(20)
[a,b]
for any section s and any vector field
W = W x (x)
N 2k−1
∂
∂
∂
+ W A (x, y) A +
WjA A ,
∂x
∂y
∂yj
A=1 j=1
A generalization of the Cartan form pdq − Hdt
11
367
where WjA are arbitrary functions on J 2k−1 Y . A simple calculation shows that
for W x (x) = 0 we have
j
2k−1
∗
(s) (iW d∆) =
N k−1 A=1 j=1
−
d j
j−1
− GA (s (x)) − GA (s(x)) WjA (s(x))dx−
dx
N
N
d 0
A
Gk−1
GA (s(x))W0A (s(x))dx −
A (s(x))Wk (s(x)) dx.
dx
A=1
A=1
Now, condition (20) implies
−
d j
j−1
G (s(x)) − GA
(s(x)) = 0, j = 1, 2, . . . , k − 1, A = 1, 2, . . . , N
dx A
Gk−1
A (s(x)) = 0, A = 1, 2, . . . , N
whence GjA (s(x)) = 0, j = k − 1, k − 2, k − 3, . . . , 1, 0. Consequently, ∆ = 0
and θ = θLag , whence f).
We proved b)–f) locally, that is, in the case where the image of ϕ is
contained into a coordinate chart. Let η : Y1 = (a1 , b1 ) × D1 → Y2 = (a2 , b2 ) ×
D2 , D1 , D2 ⊂ RN , be a change of coordinates for the fiber bundle Y , η(x, y) =
ofcoordinates for J k Y , k ∈ N,
(ηX (x), ηRN (x, y)). Then the associated
k change
−1
k
k
(ηX (x)), see
is given by ηk : J Y1 → J Y2 , ηk j s(x) = j k ηRN ◦ s ◦ ηX
k
k
(2). Let Lag1 and Lag2 be the Lagrangians on J Y1 , and on J Y2 , respectively,
and let θ1 , θ2 be the Cartan forms (11) using local coordinates on J 2k−1 Y1
∗
and on J 2k−1 Y2 respectively. Then on J 2k−1 Y1 the forms θ1 and η2k−1
θ2 fulfill
b), c), d), e) of the theorem with respect to the Lagrangian Lag1 = ηk∗ Lag2 ,
∗
θ2 , which proves that θLag
whence, by the local form of f) we have θ1 = η2k−1
2k−1
is well defined on J
Y , that is, a). Now b)–f) follow from a) and the local
forms of b)–f).
g) By (16), the Euler-Lagrange equations (18) are equivalent to (19) for
the vertical part πX,J 2k−1 Y of W . For the horizontal vector field Dx we have
∗
∗
(iDx dθLag ) = i ∂ dj 2k−1 ϕ ◦ ϕ−1
θLag = i
j 2k−1 ϕ ◦ ϕ−1
X
X
∂x
∂
∂x
0=0
∗
θLag is a two form on R. Since any vector field is the sum
as dj 2k−1 ϕ ◦ ϕ−1
X
of Dx and a vertical vector field πX,J 2k−1 Y , g) follows. As many of the equations of movement are equations of the EulerLagrange type (see [2]), the form θLag is important in the study of such
equations. Applications of the Cartan form to the numerical integration of
the Euler-Lagrange equations can be found for example, in [6], [7], [4], [9].
368
Viorel Petrehuş
12
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Received 20 March 2007
Technical University of Civil Engineering
Department of Mathematics
Bd. Lacul Tei 124
020396 Bucharest, Romania