Báo cáo toán học: "A Presentation of the Elements of the Quotient Sheaves Ωk /Θk in Variational Sequences" pdf

Vietnam Journal of Mathematics 33:3 (2005) 271–281
A Presentation of the Elements of the Quotient
Sheaves
Ω
k
r
/Θ
k
r
in Variational Sequences
Nong Quoc Chinh
Thai Nguyen University, Thai Nguyen City, Vietnam
Received Ferbuary 05, 2004
Revised Ferbuary 17, 2005
Abstract In this paper we give a concrete presentation of the elements of the quotient
sheaves
Ω
k
r
/Θ
k
r
in the variational sequence
0 → R → Ω
0
r
E
0
−→ Ω
1
r

/Θ
1
r
E
1
−→ Ω
2
r
/Θ
2
r
E
2
−→
E
P −2
−→ Ω
P −1
r
/Θ
P −1
r
→
E
P −1
−→ Ω
P
r
/Θ
P

r
E
P
−→ Ω
P +1
r
E
P +1
−→ → Ω
N
r
→ 0.
1. Introduction
The notion of variational bicomplexes was introduced in studying the problem
of characterizing the kernel and image of Euler-Lagrange mapping in the calcu-
lus of variations. This problem has been considered by Anderson [1], Duchamp
[2], Dedecker [4], Tulczyvew [7], Takens [6] and Krupka [5]. The variational bi-
complex was mainly studied on the infinite jet prolongation J
∞
Y of the fibered
manifold Y with the base X and the projection π : Y → X, where dimX = n,
dimY −n = m. The variational bicomplexes contain Euler-Lagrange mapping as
one of its morphisms. Then its developments in the theory of variational bicom-
plexes have made an important role in many problems in caculus of variations
on manifolds, in diffrential geometry, in the theory of differential equations and
in mathematical physics.
Krupka [5] studied the sheaves of differential forms on finite r-jet prolonga-
tions J
r
Y . Then he constructed the sequence of quotient sheaves Ω

k
r
/Θ
k
r
.This
quotient sequence is called the variational sequence of order r over Y .Itisan
acyclic resolution of the constant sheaf R over Y
272 Nong Quoc Chinh
0 → R → Ω
0
r
E
0
−→ Ω
1
r
/Θ
1
r
E
1
−→ Ω
2
r
/Θ
2
r
E
2

−→
E
P −2
−→ Ω
P −1
r
/Θ
P −1
r
→
E
P −1
−→ Ω
P
r
/Θ
P
r
E
P
−→ Ω
P +1
r
E
P +1
−→ → Ω
N
r
→ 0. (1)
In the sequence (1), E

n
is the Euler-Lagrange mapping and E
n+1
is the Helmholtz-
Sonin mapping. In the study of variational sequence, it is very important to give
a concrete presentation of the elements [] ∈ Ω
k
r
/Θ
k
r
. It also has been shown in
[5] that []=p
k
r,0
 for all positive integers k satisfying 1 ≤ k ≤ n,  ∈ Ω
k
r
,where
p
k
r,0
 is the horizontal component of k-form . Then Krupka [5] gave a concrete
presentation of []for1≤ k ≤ n.
The main purpose of this paper is to give a concrete presentation of elements
[] ∈ Ω
k
r
/Θ
k

r
for all positive integers k satisfying n +1≤ k ≤ P. For simplicity,
we will solve this problem in the case of r =1andr = 2. Then the other cases
follow by the same method.
2. Notations and Premilinaries
Throughout this paper, the following notations will be used: (Y, π,X) is a fibered
manifold with base X and the projection π : Y → X, where dimX = n,dimY −
n = m. J
r
Y is the finite r-jet prolongation of the fibered manifold (Y,π,X).π
r
:
J
r
Y → X, π
r,s
: J
r
Y → J
s
Y are the canonical jet projections. (V,Ψ) is a fiber
chart on Y ,whereΨ=(x
i
,y
σ
), 1 ≤ i ≤ n, 1 ≤ σ ≤ m. (V
r
, Ψ
r
) is the fiber chart

on r-jet prolongation J
r
Y associated with (V,Ψ), where V
r
= π
−1
r,0
(V ), Ψ
r
=
(x
i
,y
σ
,y
σ
j
1
, , y
σ
j
1
j
2
j
r
), 1 ≤ i ≤ n, 1 ≤ σ ≤ m, 1 ≤ j
1
≤ ≤ j
r

≤ n. Ω
k
r
is the
sheave of k-forms over J
r
Y and π
∗
r+1,r
is the pull-back of the mapping π
r+1,r
.
We put
ω
0
= dx
1
∧ dx
2
∧ ∧ dx
n
,
ω
i
=(−1)
i−1
dx
1
∧ ∧ dx
i−1

∧ dx
i+1
∧···∧dx
n
, 1 ≤ i ≤ n,
ω
σ
j
1
j
2
j
k
= dy
σ
j
1
j
2
j
k
− y
σ
j
1
j
2
j
k
i

dx
i
, 1 ≤ k ≤ r − 1.
Note that the forms
dx
i
,ω
σ
j
1
j
2
j
k
,dy
σ
j
1
j
2
j
r
,
for 0 ≤ k ≤ r − 1, define a basis of the space of linear forms on V
r
.
Obviously we have
dω
σ
j

1
j
2
j
k
∧ ω
i
= −ω
σ
j
1
j
2
j
k
i
∧ ω
0
, for 0 ≤ k ≤ r − 2,
dω
σ
j
1
j
2
j
r−1
∧ ω
i
= −dy

σ
j
1
j
2
j
r−1
i
∧ ω
0
.
Let N = n + m

n + r
n

,M= m

n + r − 1
n

,P = m

n + r − 1
n

+2n −1.
It is clear that N =dimJ
r
Y and M is the number of linear independent

forms ω
σ
j
1
j
2
j
k
, where 0 ≤ k ≤ r − 1.
A Presentation of the Elements of the Quotient Sheaves Ω
k
r
/Θ
k
r
273
For any function f :V
r
→ R we have h(df )=

n
i=1
d
i
f.dx
i
, where
d
i
f =

∂f
∂x
i
+
m

σ=1
r

k=0
∂f
∂y
σ
j
1
j
k
y
σ
j
1
j
k
.
For any k-form ρ ∈ Ω
k
r
,wedenotebyρ
0
the horizontal component of ρ,and

ρ
q
, 1 ≤ q ≤ k, is the q-contact component of ρ. Then for any ρ ∈ Ω
k
r
there exists
a unique decomposition
π
∗
r+1,r
ρ = ρ
0
+ ρ
1
+ ···+ ρ
k
.
We denote by p
k
r,q
:Ω
k
r
→ Ω
k
r+1
the morphism of sheaves defined by p
k
r,q
ρ = ρ

q
,
for 0 ≤ q ≤ k.Ω
k
r(c)
= ker p
k
r,0
,1≤ k ≤ n, is the sheave of contact k-forms.
Ω
k
r(c)
= ker p
k
r,k−n
, n +1≤ k ≤ N, is the sheave of strongly contact k-forms.
Let Θ
k
r
= dΩ
k−1
r(c)
+Ω
k
r(c)
. In [3] and [5] the authours proved the softness of
the sheaves Θ
k
r
, considered the quotient sheaves Ω

k
r
/Θ
k
r
, and they obtained the
following short exact sequence
0 → Θ
k
r
i
k
r
→Ω
k
r
τ
k
r
→Ω
k
r
/Θ
k
r
→ 0,
where i
k
r
is the canonical injective, τ

k
r
is the canonical quotient mapping.
Especially, Krupka [5] constructed the following variational sequence of order
r over Y
0 → R → Ω
0
r
E
0
−→ Ω
1
r
/Θ
1
r
E
1
−→ Ω
2
r
/Θ
2
r
E
2
−→···
E
P −2
−→ Ω

P −1
r
/Θ
P −1
r
→
E
P −1
−→ Ω
P
r
/Θ
P
r
E
P
−→ Ω
P +1
r
E
P +1
−→ ···→Ω
N
r
→ 0,
where the sheaf morphism E
k
:Ω
k
r

/Θ
k
r
→ Ω
k+1
r
/Θ
k+1
r
is defined by the formula
E
k
([]) = [d].
He proved that, the variational sequence of order r is an acyclic resolution
of the constant sheaf R over Y ,and[]=p
k
r,0
 for all 1 ≤ k ≤ n and for all
 ∈ Ω
k
r
. This is the horizontal component of k-form . Let 1 ≤ k ≤ n − 1and
 ∈ Ω
k
r
,then
[]=
1
k!
f

i
1
···i
k
dx
i
1
∧···∧dx
i
k
, (2)
where f
i
1
i
k
are some functions on V
r
.
Let k = n and  ∈ Ω
n
r
.Then
[]=fω
0
, (3)
where f is some function on V
r
.Inthiscase[] is called the Lagrange class of .
Let n +1≤ k ≤ P. For every s>r,Krupka [5] proved that

Ω
k
r
/Θ
k
r
≈ Im(τ
k
s
.π
∗
s,r+1
.p
k
r,k−n
),
and this implies that for every  ∈ Ω
k
r
[]=τ
k
s
(p
k
s−1,k−n
.π
∗
s−1,r
). (4)
274 Nong Quoc Chinh

Let k = n + 1. For each element  ∈ Ω
n+1
r
,[] is called the Euler-Lagran ge
class of (n +1)-forms.
Let k = n + 2. For each element  ∈ Ω
n+2
r
,[] is called the Helmiholtz-Sonin
class of (n +2)-forms.
Below we give a concrete presentation of the elements in Ω
k
r
/Θ
k
r
for all pos-
itive integer k satisfying n +1≤ k ≤ P in the case of r =1andr =2.
3. TheCaseofr =1
Theorem 1.
a) Let  ∈ Ω
n+1
1
be a germ. Suppose that in the fiber chart (V,ψ),ψ=(x
i
,y
σ
),
p
n+1

1,1
 = f
σ
w
σ
∧ w
0
+ f
i
σ
w
σ
i
∧ w
0
, (5)
where 1 ≤ σ ≤ m, 1 ≤ i ≤ n ,andf
σ
, f
i
σ
are functions defined on V
2
⊂ J
2
Y.
Then we have
[]=(f
σ
− d

i
f
i
σ
)w
σ
∧ w
0
. (6)
b) Let  ∈ Ω
n+2
1
be a germ. Suppose that in the fiber chart (V,ψ),ψ=(x
i
,y
σ
),
p
n+2
1,2
 = f
σν
w
σ
∧ w
ν
∧ w
0
+ f
i

σν
w
σ
i
∧ w
ν
∧ w
0
+ f
ij
σν
w
σ
i
∧ w
ν
j
∧ w
0
, (7)
where 1 ≤ σ, ν ≤ m, 1 ≤ i, j ≤ n,andf
σν
,f
i
σν
,f
ij
σν
are functions defined on
V

2
⊂ J
2
Y .Then
[]=((f
σν
w
σ
+(f
i
σν
− d
i
f
ij
σν
)w
σ
i
) − f
ij
σν
w
σ
ij
) ∧ w
ν
∧ w
0
. (8)

c) Let n +3≤ k ≤ P and  ∈ Ω
k
1
be a germ. Suppose that in the fiber chart
(V,ψ),ψ =(x
i
,y
σ
),
p
k
1,k−n
 = f
σ
1
σ
k−n
w
σ
1
∧ w
σ
2
∧ ∧ ∧ w
σ
k−n
∧ w
0
+ f
i

1
σ
1
σ
k−n
w
σ
1
i
1
∧ w
σ
2
∧ ∧ ∧ w
σ
k−n
∧ w
0
+
+ f
i
1
i
k−n−1
σ
1
σ
k−n
w
σ

1
i
1
∧ w
σ
2
i
2
∧ ∧ w
σ
k−n−1
i
k−n−1
∧ w
σ
k−n
∧ w
0
+ f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1

i
1
∧ w
σ
2
i
2
∧ ∧ w
σ
k−n−1
i
k−n−1
∧ w
σ
k−n
i
k−n
∧ w
0
,
(9)
where 1 ≤ σ
1
, , σ
k−n
≤ m, 1 ≤ i
1
, i
k−n
≤ n and each function defined

on V
2
⊂ J
2
Y. Then we have
A Presentation of the Elements of the Quotient Sheaves Ω
k
r
/Θ
k
r
275
[]=

k−n−2

h=0
f
i
1
i
h
σ
1
σ
k−n
w
σ
1
i

1
∧ ∧ w
σ
h
i
h
∧ w
σ
h+1
∧ ∧ w
σ
k−n−1
+(f
i
1
i
k−n−1
σ
1
σ
k−n
− d
i
k−n
f
i
1
i
k−n
σ

1
σ
k−n
)w
σ
1
i
1
∧ ∧ w
σ
k−n−1
i
k−n−1
−
k−n−1

t=1
f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i

1
∧ ∧ w
σ
t−1
i
t−1
∧ w
σ
t
i
t
i
k−n
∧ w
σ
t+1
i
t+1
∧
∧ w
σ
k−n−1
i
k−n−1

∧w
σ
k−n
∧ w
0

,
(10)
where each function is well defined on V
3
⊂ J
3
Y .
Proof.
a) Considering all the factors in (5) which contains w
σ
i
,weget
f
i
σ
w
σ
i
∧ w
0
= −f
i
σ
d(w
σ
∧ w
i
)=−d(f
i
σ

w
σ
∧ w
i
)+df
i
σ
∧ w
σ
∧ w
i
. (11)
Since f
i
σ
w
σ
∧ w
i
∈ Θ
n
2
and
π
∗
3,2
(df
i
σ
∧ w

σ
∧ w
i
)=h(df
i
σ
) ∧ w
σ
∧ w
i
+ p(df
i
σ
) ∧ w
σ
∧ w
i
,
we have
[]=τ
n+1
3
.p
n+1
2,1
.π
∗
2,1
()=f
σ

w
σ
∧ w
0
+ h(df
i
σ
) ∧ w
σ
∧ w
i
=(f
σ
− d
i
f
i
σ
)w
σ
∧ w
0
.
b) Considering all the factors in (7) which contains w
σ
i
∧ w
ν
j
, we get

f
ij
σν
w
σ
i
∧ w
ν
j
∧ w
0
= −f
ij
σν
w
σ
i
∧ d(w
ν
∧ w
j
)=
= f
ij
σν
d(w
σ
i
∧ w
ν

∧ w
j
) − f
ij
σν
dw
σ
i
∧ w
ν
∧ w
j
= d(f
ij
σν
w
σ
i
∧ w
ν
∧ w
j
) − df
ij
σν
∧ w
σ
i
∧ w
ν

∧ w
j
+
+
n

l=1
f
ij
σν
dy
σ
il
∧ dx
l
∧ w
ν
∧ w
j
.
(12)
Since f
ij
σν
w
σ
i
∧ w
ν
∧ w

j
∈ Θ
n+1
2
, we have
[]=τ
n+2
3
.p
n+2
2,2
.π
∗
2,1
()
= f
σν
w
σ
∧ w
ν
∧ w
0
+ f
i
σν
w
σ
i
∧ w

ν
∧ w
0
− d
j
f
ij
σν
w
σ
i
∧ w
ν
∧ w
0
− f
ij
σν
w
σ
ij
∧ w
ν
∧ w
0
.
(13)
Therefore we have
[]=((f
σν

w
σ
+(f
i
σν
− d
i
f
ij
σν
)w
σ
i
) − f
ij
σν
w
σ
ij
) ∧ w
ν
∧ w
0
.
c) In the formula (9), we consider all the factors containing w
σ
k−n
i
k−n
.Theyare

f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w
σ
k−n
i
k−n
∧ w
0
. We get
276 Nong Quoc Chinh
f
i
1
i
k−n
σ
1

σ
k−n
w
σ
1
i
1
∧ ∧ w
σ
k−n
i
k−n
∧ w
0
= − f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w
σ

k−n−1
i
k−n−1
∧ d(w
σ
k−n
∧ w
i
k−n
)
=(−1)
k−n
f
i
1
i
k−n
σ
1
σ
k−n
d(w
σ
1
i
1
∧ ∧ w
σ
k−n−1
i

k−n−1
∧ w
σ
k−n
∧ w
i
k−n
)
+
k−n−1

t=1
(−1)
k−n+t
f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w
σ

t−1
i
t−1
∧ dw
σ
t
i
t
∧
∧ w
σ
t+1
i
t+1
∧ ∧ w
σ
k−n−1
i
k−n−1
∧ w
σ
k−n
∧ w
i
k−n
=(−1)
k−n
d(f
i
1

i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w
σ
k−n−1
i
k−n−1
∧ w
σ
k−n
∧ w
i
k−n
)
− (−1)
k−n
df
i
1
i
k−n

σ
1
σ
k−n
∧ w
σ
1
i
1
∧ ∧ w
σ
k−n−1
i
k−n−1
∧ w
σ
k−n
∧ w
i
k−n
+
k−n−1

t=1
n

l=1
(−1)
k−n+t+1
f

i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w
σ
t−1
i
t−1
∧ dy
σ
t
i
t
l
∧ dx
l
∧
∧ w
σ
t+1

i
t+1
∧ ∧ w
σ
k−n−1
i
k−n−1
∧ w
σ
k−n
∧ w
i
k−n
.
(14)
Since f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w

σ
k−n−1
i
k−n−1
∧ w
σ
k−n
∧ w
i
k−n
∈ Θ
k−1
2
, we have
τ
k
3
.p
k
2,k−n
(f
i
1
i
k−n
σ
1
σ
k−n
w

σ
1
i
1
∧ ∧ w
σ
k−n
i
k−n
∧ w
0
)
= − d
i
k−n
f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w

σ
k−n−1
i
k−n−1
∧ w
σ
k−n
∧ w
0
−
k−n−1

t=1
f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w
σ
t−1

i
t−1
∧ w
σ
t
i
t
i
k−n
∧ w
σ
t+1
i
t+1
∧ ∧ w
σ
k−n−1
i
k−n−1
∧ w
σ
k−n
∧ w
0
.
(15)
Then [] can be presented in the following form
[]=

k−n−2


h=0
f
i
1
i
h
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w
σ
h
i
h
∧ w
σ
k+1
∧ ∧ w
σ
k−n−1
+(f
i
1

i
k−n−1
σ
1
σ
k−n
− d
i
k−n
f
i
1
i
k−n
σ
1
σ
k−n
)w
σ
1
i
1
∧ ∧ w
σ
k−n−1
i
k−n−1
−
k−n−1


t=1
f
i
1
i
k−n
σ
1
σ
k−n
w
σ
1
i
1
∧ ∧ w
σ
t−1
i
t−1
∧ w
σ
t
i
t
i
t−1
∧ w
σ

t+1
i
t+1
∧
∧ w
σ
k−n−1
i
k−n−1

∧w
σ
k−n
∧ w
0
,
where each function is defined on V
3
⊂ J
3
Y .

4. TheCaseofr =2
Theorem 2.
a) Let  ∈ Ω
n+1
2
be a germ. Suppose that in the fiber chart (V,ψ),ψ=(x
i
,y

σ
),
p
n+1
2,1
 = f
σ
w
σ
∧ w
0
+ f
i
σ
w
σ
i
∧ w
0
+ f
ij
σ
w
σ
ij
∧ w
0
, (16)
A Presentation of the Elements of the Quotient Sheaves Ω
k

r
/Θ
k
r
277
where 1 ≤ σ ≤ m, 1 ≤ i, j ≤ m, f
σ
,f
i
σ
,f
ij
σ
are functions well defined on V
3
⊂
J
3
Y . Then we have
[]=(f
σ
− d
i
f
i
σ
+ d
i
d
j

f
ij
σ
)w
σ
∧ w
0
, (17)
where each function is well defined on V
5
⊂ J
5
Y .
b) Let  ∈ Ω
n+2
2
be a germ. Suppose that in the fiber chart (V,ψ),ψ=(x
i
,y
σ
),
p
n+2
2,2
 =(f
J
σν
w
σ
J

∧ w
ν
∧ w
0
+ f
Ji
σν
w
σ
J
∧ w
ν
i
∧ w
0
+ f
Jij
σν
w
σ
J
∧ w
ν
ij
∧ w
0
, (18)
where 0 ≤|J|≤2, 1 ≤ σ, ν ≤ m, 1 ≤ i, j ≤ n, and functions f
J
σν

,f
Ji
σν
,f
Jij
σν
are
wel l defined on V
3
⊂ J
3
Y . Then we have
[]=((f
J
σν
− d
i
f
Ji
σν
+ d
i
d
j
f
Jij
σν
)w
σ
J

+(−f
Ji
σν
+2d
j
f
Jij
σν
)w
σ
Ji
+ f
Jij
σν
w
σ
Jij
) ∧ w
ν
∧ w
0
, (19)
where each function is well defined on V
5
⊂ J
5
Y .
c) Let n +3≤ k ≤ P and  ∈ Ω
k
2

be a germ. Suppose that in the fiber chart
(V,ψ),ψ =(x
i
,y
σ
) we have
p
k
2,k−n
 =
2

q=0
f
J
1
J
k−n−1
j
1
j
q
σ
1
σ
k−n
w
σ
1
J

1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
j
1
j
q
∧ w
0
, (20)
where 0 ≤|J
1
|, , |J
k−n−1
|≤2, 1 ≤ σ
1
, , σ
k−n
≤ m, 1 ≤ j
1
, , j
q
≤ n,
and every function f

J
1
J
k−n−1
j
1
j
q
σ
1
σ
k−n
is well defined on V
3
⊂ J
3
Y .Thenwe
have
[]=

(f
J
1
J
k−n−1
σ
1
σ
k−n
− d

j
1
f
J
1
J
k−n−1
j
1
σ
1
σ
k−n
+ d
j
1
d
j
2
f
J
1
J
k−n−1
j
1
j
2
σ
1

σ
k−n
)w
σ
1
J
1
∧
∧ w
σ
k−n−1
J
k−n−1
+
k−n−1

t=1
(−f
J
1
J
k−n−1
j
1
σ
1
σ
k−n
+2d
j

2
f
J
1
J
k−n−1
j
1
j
2
σ
1
σ
k−n
)w
σ
1
J
1
∧
∧ w
σ
t−1
J
t−1
∧ w
σ
t
J
t

j
1
∧ w
σ
t+1
J
t+1
∧ ∧ w
σ
k−n−1
J
k−n−1
(21)
+
k−n−1

h=1
h=t
k−n−1

t=1
f
J
1
J
k−n−1
j
1
j
2

σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
h−1
J
h−1
∧ w
σ
h
J
h
j
1
∧ w
σ
h+1
J
h+1
∧
∧ w
σ
t−1

J
t−1
∧ w
σ
t
J
t
j
2
∧ w
σ
t+1
J
t+1
∧ ∧ w
σ
k−n−1
J
k−n−1
+
k−n−1

t=1
f
J
1
J
k−n−1
j
1

j
2
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
t−1
J
t−1
∧ w
σ
t
J
t
j
1
j
2
∧ w
σ
t+1
J
t+1

∧
∧ w
σ
k−n−1
J
k−n−1

∧w
σ
k−n
∧ w
0
,
where each function is well defined on V
5
⊂ J
5
Y .
278 Nong Quoc Chinh
Proof.
a) In the formula (16) we consider the factors containing w
σ
i
. They aref
i
σ
w
σ
i
∧w

0
.
We get
f
i
σ
w
σ
i
∧ w
0
= −f
i
σ
d(w
σ
∧ w
i
)=−d(f
i
σ
w
σ
∧ w
i
)+df
i
σ
∧ w
σ

∧ w
i
.
(22)
Since f
i
σ
w
σ
∧ w
i
∈ Ω
n
3
=Θ
n
3
and
π
∗
4,3
(df
i
σ
∧ w
σ
∧ w
i
)=h(df
i

σ
) ∧ w
σ
∧ w
i
+ p(df
i
σ
) ∧ w
σ
∧ w
i
,
it implies that
τ
n+1
4
.p
n+1
3,1
(f
i
σ
w
σ
i
∧ w
0
)=−d
i

f
i
σ
∧ w
σ
∧ w
0
. (23)
Now we consider all the factors containing w
σ
ij
in the formula (16). Then we
have
f
ij
σ
w
σ
ij
∧ w
0
= −d(f
ij
σ
w
σ
i
∧ w
j
)+df

ij
σ
∧ w
σ
i
∧ w
j
. (24)
This implies that
τ
n+1
4
.p
n+1
3,1
(f
ij
σ
w
σ
ij
∧ w
0
)=−d
j
f
ij
σ
w
σ

i
∧ w
0
= d(d
j
f
ij
σ
.w
σ
∧ w
i
) − d(d
j
f
ij
σ
) ∧ w
σ
∧ w
i
. (25)
Therefore we have
τ
n+1
5
.p
n+1
4,1
.π

∗
4,3
(f
ij
σ
w
σ
ij
∧ w
0
)=d
i
d
j
f
ij
σ
w
σ
∧ w
0
. (26)
Since (16), (23) and (26) we get
[]=τ
n+1
5
.p
n+1
4,1
.π

∗
4,2
()=(f
σ
− d
i
f
i
σ
+ d
i
d
j
f
ij
σ
)w
σ
∧ w
0
,
where each function is defined on V
5
⊂ J
5
Y .
b) In the formula (18) we consider all the factors containing w
ν
i
. Then we get

f
Ji
σν
w
σ
J
∧ w
ν
i
∧ w
0
= −f
Ji
σν
w
σ
J
∧ d(w
ν
∧ w
i
)
= f
Ji
σν
d(w
σ
J
∧ w
ν

∧ w
i
) − f
Ji
σν
dw
σ
J
∧ w
ν
∧ w
i
= d(f
Ji
σν
w
σ
J
∧ w
ν
∧ w
i
) − df
Ji
σν
∧ w
σ
J
∧ w
ν

∧ w
i
+
n

l=1
df
Ji
σν
dy
σ
Jl
∧ dx
l
∧ w
ν
∧ w
i
.
Therefore
τ
n+2
4
.p
n+2
3,2
(f
Ji
σν
w

σ
J
∧ w
ν
i
∧ w
0
)=− d
i
f
Ji
σν
w
σ
J
∧ w
ν
∧ w
0
− f
Ji
σν
w
σ
Ji
∧ w
ν
∧ w
0
. (27)

Now we consider all the factors containing w
ν
ij
in the formula (16). Then we
have
A Presentation of the Elements of the Quotient Sheaves Ω
k
r
/Θ
k
r
279
f
Jij
σν
w
σ
J
∧ w
ν
ij
∧ w
0
= −f
Jij
σν
w
σ
J
∧ d(w

ν
i
∧ w
j
)
= d(f
Jij
σν
w
σ
J
∧ w
ν
i
∧ w
j
) − df
Jij
σν
∧ w
σ
J
∧ w
ν
i
∧ w
j
+
n


l=1
f
Jij
σν
dy
σ
Jl
dx
l
∧ w
ν
i
∧ w
j
τ
n+2
4
.p
n+2
3,2
−→ − d
j
f
Jij
σν
w
σ
J
∧ w
ν

i
∧ w
0
− f
Jij
σν
w
σ
Jj
∧ w
ν
i
∧ w
0
τ
n+2
5
.p
n+2
4,2
−→ d
i
d
j
f
Jij
σν
w
σ
J

∧ w
ν
∧ w
0
+ d
j
f
Jij
σν
w
σ
Ji
∧ w
ν
∧ w
0
+ d
i
f
Jij
σν
w
σ
Jj
∧ w
ν
∧ w
0
+ f
Jij

σν
w
σ
Jij
∧ w
ν
∧ w
0
,
(28)
where τ
n+2
4
.p
n+2
3,2
(resp.τ
n+2
5
.p
n+2
4,2
) are morphisms of sheaves. Since the formu-
las (18), (27), (28) and the symmetry of indexes i, j we have
[]=τ
n+2
5
.p
n+2
4,2

.π
∗
4,2
()=((f
J
σν
− d
i
f
Ji
σν
+ d
i
d
j
f
Jij
σν
)w
σ
J
+(−f
Ji
σν
+2d
j
f
Jij
σν
)w

σ
Ji
+ f
Jij
σν
w
σ
Jij
) ∧ w
ν
∧ w
0
,
where each function is defined on V
5
⊂ J
5
Y .
c) We consider the factors containing w
σ
k−n
J
1
in the formula (20). We have
f
J
1
J
k−n−1
j

1
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
j
1
∧ w
0
= − f
J
1
J
k−n−1
j
1
σ

1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ d(w
σ
k−n
∧ w
j
1
)
=(−1)
k−n
f
J
1
J
k−n−1
j
1
σ

1
σ
k−n
d(w
σ
1
J
1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
∧ w
j
1
)
+
k−n−1

t=1
(−1)
k−n+t
f
J
1
J

k−n−1
j
1
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
t−1
J
t−1
∧
∧ dw
σ
t
J
t
∧ w
σ
t+1
J
t+1
∧ ∧ w
σ

k−n−1
J
k−n−1
∧ w
σ
k−n
∧ w
j
1
=(−1)
k−n
d(f
J
1
J
k−n−1
j
1
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
k−n−1

J
k−n−1
∧ w
σ
k−n
∧ w
j
1
)
− (−1)
k−n
df
J
1
J
k−n−1
j
1
σ
1
σ
k−n
∧ w
σ
1
J
1
∧ ∧ w
σ
k−n−1

J
k−n−1
∧ w
σ
k−n
∧ w
j
1
−
k−n−1

t=1
n

l=1
(−1)
k−n+t
f
J
1
J
k−n−1
j
1
σ
1
σ
k−n
w
σ

1
J
1
∧ ∧ w
σ
t−1
J
t−1
∧
∧ dy
σ
t
J
t
l
∧ dx
l
∧ w
σ
t+1
J
t+1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n

∧ w
j
1
τ
k
4
.p
k
3,k−n
−→ − d
j
1
f
J
1
J
k−n−1
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
k−n−1
J

k−n−1
∧ w
σ
k−n
∧ w
0
−
k−n−1

t=1
f
J
1
J
k−n−1
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
t−1
J
t−1
∧ w

σ
t
J
t
j
1
∧
∧ w
σ
t+1
J
t+1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
∧ w
0
.
(29)
We consider all the factors containing w
σ
k−n
j
1
j

2
in the formula (20). We have
280 Nong Quoc Chinh
f
J
1
J
k−n−1
j
1
j
2
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
j

1
j
2
∧ w
0
τ
k
4
.p
k
3,k−n
−→
− d
j
2
f
J
1
J
k−n−1
j
1
j
2
σ
1
σ
k−n
w
σ

1
J
1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
j
1
∧ w
0
−
k−n−1

t=1
f
J
1
J
k−n−1
j
1
j
2
σ
1

σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
t−1
J
t−1
∧ w
σ
t
J
t
j
2
∧ w
σ
t+1
J
t+1
∧
∧ w
σ
k−n−1
J
k−n−1

∧ w
σ
k−n
j
1
∧ w
0
τ
k
5
.p
k
4,k−n
d
j
1
−→ d
j
2
f
J
1
J
k−n−1
j
1
j
2
σ
1

σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
∧ w
0
+
k−n−1

h=1
d
j
2
f
J
1
J
k−n−1
j

1
j
2
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
h−1
J
h−1
∧ w
σ
h
J
h
j
1
∧ w
σ
h+1
J
h+1
∧

∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
∧ w
0
+
k−n−1

t=1
d
j
1
f
J
1
J
k−n−1
j
1
j
2
σ
1
σ
k−n

w
σ
1
J
1
∧ ∧ w
σ
t−1
J
t−1
∧ w
σ
t
J
t
j
2
∧ w
σ
t+1
J
t+1
∧
∧ w
σ
k−n−1
J
k−n−1
∧ w
σ

k−n
∧ w
0
+
k−n−1

h=1
h=t
k−n−1

t=1
f
J
1
J
k−n−1
j
1
j
2
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w

σ
h−1
J
h−1
∧ w
σ
h
J
h
j
1
∧ w
σ
h+1
J
h+1
∧
∧ w
σ
t−1
J
t−1
∧ w
σ
t
J
t
j
2
∧ w

σ
t+1
J
t+1
∧ ∧ w
σ
k−n−1
J
k−n−1
∧ w
σ
k−n
∧ w
0
+
k−n−1

t=1
f
J
1
J
k−n−1
j
1
j
2
σ
1
σ

k−n
w
σ
1
J
1
∧ ∧ w
σ
t−1
J
t−1
∧ w
σ
t
J
t
j
1
j
2
∧ w
σ
t+1
J
t+1
∧
∧ w
σ
k−n−1
J

k−n−1
∧ w
σ
k−n
∧ w
0
.
(30)
Since the formulas (20), (29), (30) and the symmetry of indexes j
1
,j
2
we have
[]=τ
k
5
.p
k
4,k−n
.π
∗
4,2
()
=

(f
J
1
J
k−n−1

σ
1
σ
k−n
− d
j
1
f
J
1
J
k−n−1
j
1
σ
1
σ
k−n
+ d
j
1
d
j
2
f
J
1
J
k−n−1
j

1
j
2
σ
1
σ
k−n
)w
σ
1
J
1
∧
∧ w
σ
k−n−1
J
k−n−1
+
k−n−1

t=1
(−f
J
1
J
k−n−1
j
1
σ

1
σ
k−n
+2d
j
2
f
J
1
J
k−n−1
j
1
j
2
σ
1
σ
k−n
)w
σ
1
J
1
∧
∧ w
σ
t−1
J
t−1

∧ w
σ
t
J
t
j
1
∧ w
σ
t+1
J
t+1
∧ ∧ w
σ
k−n−1
J
k−n−1
+
k−n−1

h=1
h=t
k−n−1

t=1
f
J
1
J
k−n−1

j
1
j
2
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
h−1
J
h−1
∧ w
σ
h
J
h
j
1
∧ w
σ
h+1
J
h+1

∧
∧ w
σ
t−1
J
t−1
∧ w
σ
t
J
t
j
2
∧ w
σ
t+1
J
t+1
∧ ∧ w
σ
k−n−1
J
k−n−1
A Presentation of the Elements of the Quotient Sheaves Ω
k
r
/Θ
k
r
281

+
k−n−1

t=1
f
J
1
J
k−n−1
j
1
j
2
σ
1
σ
k−n
w
σ
1
J
1
∧ ∧ w
σ
t−1
J
t−1
∧ w
σ
t

J
t
j
1
j
2
∧ w
σ
t+1
J
t+1
∧
∧ w
σ
k−n−1
J
k−n−1

∧w
σ
k−n
∧ w
0
,
where each function is defined on V
5
⊂ J
5
Y .


References
1. I. M. Anderson, Aspect of the inverse problem to the calculus of variation, Arch.
Math. 24 (1988) 181–202.
2. I.M. Anderson and T. Duchamp, On the existence of global variational principles,
Amer. Math. J. 102 (1980) 781–868.
3. N. Q. Chinh, Sheaf of contact forms , East-West J. Math. 4 (2002) 41–55.
4. P. Dedecker and W. M. Tulczyjew, Spectral Sequences and the Inverse Problem
of the Calculus of Variations, Internat. Coll. on Diff. Geom. Methods in Math.
Physics, Sept., 1979.
5. D. Krupka, Variational Sequences on Finite Order Jet Spaces, World Scientific,
Singapore, (1990) 236–254.
6. F. Takens, A global version of the inverse problem of the calculus of variations,
J. Diff. Geom. 14 (1979) 543–562.
7. W. M. Tulczyjew, The Euler-Lagrange Resolution, Internat. Coll. on Diff. Geom.
Methods in Math. Physics, Sept., 1979.