Steps in Differential Geometry, Proceedings of the Colloquium
on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary
ON HAMILTON p2 –EQUATIONS IN SECOND-ORDER
FIELD THEORY
DANA SMETANOVÁ
Abstract. In the present paper recent results on regularizations of first order
variational problems are generalized to Lagrangians affine in the second derivatives. New regularity conditions are found and Legendre transformations are
studied.
1. Introduction and notation
In this paper we consider an extension of the classical Hamilton–Cartan variational theory on fibered manifolds.
It is known that in field theory to a variational problem represented by a Lagrangian one can associate different Hamilton equations corresponding to different
Lepagean equivalents of the Lagrangian (Dedecker [1], Krupka [7]). Accordingly, these Hamilton equations depend upon a Lagrangian (resp. its Poincaré
- Cartan form), and an auxiliary differential form corresponding to the at least
2-contact part of the Lepagean equivalent of the Lagrangian. This admits a new
approach to the problem of regularity (Dedecker [1], Krupková [11], [12], Krupková and Smetanová [13], [14]). Contrary to the classical calculus of variations
where regularity is a property of a single Lagrangian, in the generalized approach
regularity conditions (different from [3], [4], [8], [15]) depend upon a Lagrangian
and some “free” functions which can be considered as parameters. Within this
setting, a proper choice of a Lepagean equivalent can lead to a “regularization” of
a Lagrangian. Using this regularization procedure one can regularize some interesting traditionally singular physical fields, the Dirac field, and the electromagnetic
field (cf. Dedecker [1], Krupková and Smetanová [13], [14]).
Throughout this paper, π : Y → X is s fibered manifold, and dim X = n,
dim Y = m + n. The r-jet prolongation of π is a fibered manifold denoted by
1991 Mathematics Subject Classification. 35A15, 49L10, 49N60, 58E15.
Key words and phrases. Hamilton extremals, Hamilton p2 -equations, Lagrangian, Legendre
transformation, Lepage form, Poincaré-Cartan form, regularity.
Research supported by Grants MSM:J10/98:192400002, VS 96003 and FRVŠ 1467/2000 of the
Czech Ministry of Education, Youth and Sports, Grant 201/00/0724 of the Czech Grant Agency,
and by the Mathematical Institute of the Silesian University in Opava.
The author wishes to thank Prof. Olga Krupková for her valuable stimulation and discussions
about principal ideas of the present work.
329
330
DANA SMETANOVÁ
πr : J r Y → X and πr,s : J r Y → J s Y , 0 ≤ s ≤ r, we denote the natural jet
projections. A fibered chart on Y (resp. an associated fibered chart on J r Y ) is
denoted by (V, ψ), ψ = (xi , y σ ) (resp. (Vr , ψr ), ψr = (xi , y σ , yiσ , . . . , yiσ1 ...ir )). In
what follows, we consider r = 1 or r = 2.
Recall that every q-form η on J r Y admits a unique (canonical) decomposition
into a sum of q-forms on J r+1 Y as follows:
q
X
∗
pk (η),
πr+1,r
η = h(η) +
k=1
where h(η) is a horizontal form, called the horizontal part of η, and pk (η), 1 ≤ k ≤
q, is a k-contact form, called the k-contact part of η (see e.g. [5], [6] for review).
We use the following notations:
ω0 = dx1 ∧ dx2 ∧ · · · ∧ dxn ,
ωi = i∂/∂xi ω0 ,
ωij = i∂/∂xj ωi ,
and
σ
ω σ = dy σ − yjσ dxj , ωiσ = dyiσ − yij
dxj .
By an r-th order Lagrangian we shall mean a horizontal n-form λ on J r Y . A
form ρ is called a Lepagean equivalent of a Lagrangian λ if (up to a projection)
h(ρ) = λ, and p1 (dρ) is a πr+1,0 -horizontal form [5]. For an r-th order Lagrangian
we have all its Lepagean equivalents of order (2r − 1) characterized by the following
formula
(1.1)
ρ = Θ + ν,
where Θ is a global Poincaré–Cartan form associated to λ, and ν is an arbitrary
n-form of order of contactness ≥ 2, i.e., such that h(ν) = p1 (ν) = 0 (cf. Krupka
[5], [6]). Recall that for a Lagrangian of order 1, Θ = θλ where θλ is the classical
Poincaré–Cartan form of λ,
∂L
θλ = Lω0 + σ ω σ ∧ ωi .
∂yi
If r = 2, Θ is no more unique, however, there is an invariant decomposition
(1.2)
Θ = θλ + dφ,
where
θλ = Lω0 +
∂L
∂L
− dk σ
∂yjσ
∂yjk
!
ω σ ∧ ωj +
∂L σ
σ ωi ∧ ωj
∂yij
and φ does not depend upon λ (Krupka [6]).
With the help of Lepagean equivalents of a Lagrangian one obtains the following
intrinsic formulation of the Euler–Lagrange and Hamilton equations.
Theorem (Krupka [5]). Let λ be a Lagrangian on J r Y , ρ its Lepagean equivalent.
A section γ of π is an extremal of λ if and only if
(1.3)
J 2r−1 γ ∗ iJ 2r−1 ξ dρ = 0
for every π-vertical vector field ξ on Y .
ON HAMILTON p2 –EQUATIONS IN SECOND-ORDER FIELD THEORY
331
A section δ of the fibered manifold π2r−1 is called a Hamilton extremal of ρ
(Krupka [7]) if
δ ∗ iξ dρ = 0,
(1.4)
for every π2r−1 -vertical vector field ξ on J 2r−1 Y .
(1.3) are called the Euler–Lagrange equations and (1.4) the Hamilton equations
of ρ, respectively. Notice that while the Euler–Lagrange equations are uniquely
determined by the Lagrangian, Hamilton equations depend upon a choice of ν.
Consequently, one gets many different Hamilton theories associated to a given
variational problem.
In accordance with [13], by Hamilton p2 -equations we shall mean Hamilton
equations of a Lepagean equivalent ρ of λ where ν is a 2-contact n-form (i. e.,
h(ν) = pi (ν) = 0, i ≥ 1, i 6= 2).
The aim of this paper is to consider Hamilton p2 -equations for a class of second
σ
order Lagrangians. Namely, we study Lagragians affine in second derivatives ykl
,
such that their Lepagean equivalent is of the form ρ = θλ + ν, where ν = p2 (β),
for an n-form β defined on J 1 Y .
Recall that a section δ of the fibered manifold πr is said to be holonomic if
δ = J r γ for a section γ of π. Clearly, if γ is an extremal then J 2r−1 γ is a Hamilton
extremal; conversely, however, a Hamilton extremal need not be holonomic, and
thus a jet prolongation of some extremal. This suggests a definition of regularity
proposed by Krupka and Štěpánková [9] in consequence with a study of second
order Lagrangians with projectable Poincaré–Cartan forms: Throughout this paper
a Lepagean form is called regular if every its Hamilton extremal is holonomic.
Taking a Lepagean equivalent of λ in the form ρ = θλ + p2 (β), where β is defined
on J 1 Y , we can see that regularity conditions involve λ and β, and one can ask
about a proper choice β, such that ρ is regular. We study this question in Section
2. Section 3 is then devoted to the question on the existence of certain Legendre
coordinates for regularizable Lagrangians. In Section 4 we deal with Lagrangians,
affine with second derivatives, admitting a Lepagean equivalent projectable onto
J 1 Y . Our results are a direct generalization of techniques and results from [13],
[14] and provide, as a special case, the results of [9] and [2].
2. Regularization of variational problems for second-order
Lagrangians affine in second derivatives
We shall consider Lagrangians affine in the second derivatives and its Lepagean
forms (1.1), (1.2) where φ = 0, ν is 2-contact, and
ν = p2 (β),
1
where β is defined on J Y and such that pi (β) = 0 for all i ≥ 3.
In a fiber chart, a Lagrangian λ affine in the second derivatives is expressed by
(2.1)
λ = Lω0 ,
e+L
e ij y σ ,
L=L
σ ij
332
DANA SMETANOVÁ
κ
e L
e ij
e ij
where functions L,
σ do not depend on the variables ykl and the functions Lσ
e ij = L
e ji . In view of the above considerations we obtain
satisfy the condition L
σ
σ
!
e
e kl
∂
L
∂
L
ν
ν
jk
kl
ν
e
e+L
e ν ykl ω0 +
+
y − dk L
ω σ ∧ ωj
ρ= L
σ
∂yjσ
∂yjσ kl
(2.2)
e il ω σ ∧ ωj + f ij ω σ ∧ ω ν ∧ ωij + g kij ω σ ∧ ω ν ∧ ωij
+L
σ i
σν
σν
k
σ
ν
+ hklij
σν ωk ∧ ωl ∧ ωij ,
ij
kij
κ
where the functions fσν
, gσν
, hklij
σν do not depend on the ypq ’s and satisfy the
conditions
ij
ij
fσν
= −fνσ
,
(2.3)
ij
ji
fσν
= −fσν
,
ij
ji
fσν
= fνσ
;
kij
kji
gσν
= −gσν
;
lkij
hklij
σν = −hνσ ,
klji
hklij
σν = −hσν .
In general case, the Poincaré–Cartan forms of a second order Lagrangians is
defined on J 3 Y , but for Lagrangians of the forms (2.1) θλ is projectable onto J 2 Y .
Our choice of the 2-contact part ν of ρ conserves the Lepagean form (2.2), (2.3)
defined on J 2 Y .
In the following theorems necessary conditions for regularity are found, which
according to the definition of regularity in this paper, guarantee that extremals
and Hamilton extremals of λ = hρ are in bijective correspondence.
Theorem A. Let dim X ≥ 3. Let λ be a second-order Lagrangian affine in the
σ
variables yij
, the formula (2.1) be its expression in a fiber chart (V, ψ), ψ = (xi , y σ )
on Y . Let ρ be a Lepagean equivalent of λ of the form (2.2), (2.3).
Assume that the matrix
klpq
(2.5)
Aklj
,
νσ | Bνκ
with mn2 rows (resp. mn + mn(n + 1)/ 2 columns) labelled by (ν, k, l) (resp.
(σ, j, κ, p, q), where 1 ≤ p ≤ q ≤ n), where
!
!
e jk
e jl
e kl
∂L
1 ∂L
∂L
ν
σ
σ
kjl
ljk
klj
Aνσ =
−
+
− gσν − gσν ,
∂yjσ
2 ∂ylν
∂ykν
and
klpq
Bνκ
=
lpqk
hkpql
,
νκ + hνκ
has rank mn(n + 3)/ 2.
Then ρ is regular on π2 −1 (V ), i.e., every Hamilton extremal δ : π(V ) ⊃ U →
2
J Y of ρ is of the form δ = J 2 γ, where γ is an extremal of λ.
Proof. Expressing the Hamilton p2 -equations (1.4) in fiber coordinates we get along
δ the following system of first-order equations for section δ:
ON HAMILTON p2 –EQUATIONS IN SECOND-ORDER FIELD THEORY
333
mn2 equations
e kl
∂L
1
ν
−
σ
∂yj
2
(2.6)
e jk
e jl
∂L
∂L
σ
σ
+
∂ylν
∂ykν
+2
hkijl
νσ
+
hlijk
νσ
!
−
kjl
gσν
−
ljk
gσν
∂yiσ
σ
− yij
∂xj
!
∂y σ
− yjσ
∂xj
= 0,
mn equations
(2.7)
e
e pq
e kj
∂2L
∂
∂2L
κ
e jp − ∂ Lν − 4f jk + 2di g kij
+ σ κ ν ypq
− ν dp L
σ
σν
σν
σ
ν
∂yj ∂yk
∂yj ∂yk
∂yk
∂y σ
!
!
e ij
e kj
∂L
∂L
σ
ν
kij
ikj
kilj
+
−
+ 2gσν − 2gνσ − 4dl hνσ
×
∂ykν
∂yiσ
σ
∂yilσ
∂yi
σ
ikjl
lkji
σ
×
− yij + 2 hσν + hσν
− yilj
∂xj
∂xj
σ
κ
ij
kij
kij
∂fσκ
∂gκν
∂gσν
∂y
∂y
σ
κ
− yj
− yi
+ 2 ν +
−
∂yk
∂y σ
∂y κ
∂xj
∂xi
σ
κ
lij
kij
∂hlkij
∂gσκ
∂gσν
∂y
∂yl
κν
σ
κ
+2 2
+
−
− yi
− ylj
∂y σ
∂ykν
∂ylκ
∂xi
∂xj
κ
σ
lkij
∂yp
∂hkpij
∂hplij
∂yl
∂hσν
νκ
κσ
κ
σ
+
+
− ypi
− ylj = 0,
+2
∂ypκ
∂ylσ
∂ykν
∂xi
∂xj
∂y σ
− yjσ
∂xj
and m equations
(2.8)
e
e pq
∂L
∂L
κ κ
+
y − dj
∂y ν
∂y ν pq
e
e pq
∂L
∂L
κ κ
+
y
ν
∂yj
∂yjν pq
!
!
e jk
+ dj dk L
ν
e
e pq
e
e pq
∂2L
∂2L
∂2L
∂8L
∂
κ
κ
κ
κ
e jk
+
y
−
−
ypq
− ν dk L
pq
σ
σ
σ
ν
ν
ν
ν
σ
σ
∂yj ∂y
∂yj ∂y
∂y ∂yj
∂y ∂yj
∂y
σ
∂
∂y
ij
σ
e jk
+ σ dk L
+
2d
f
−
y
i σν
ν
j
∂y
∂xj
+
e kj
e
e pq
∂L
∂2L
∂2L
∂
σ
κ
e jp + 4f jk − 2di g kij
+
−
−
yκ +
dp L
ν
νσ
νσ
∂y ν
∂yjσ ∂ykν
∂yjσ ∂ykν pq ∂ykσ
σ
∂yk
σ
×
−
y
kj
∂xj
!
!
jk
jl
σ
e kl
e
e
∂L
1
∂
L
∂
L
∂ykl
σ
ν
ν
kjl
ljk
σ
+
−
+
− gνσ − gνσ
− yklj
∂yjν
2 ∂ylσ
∂ykσ
∂xj
!
334
DANA SMETANOVÁ
σ
∂y κ
∂y
κ
σ
+2
− yi
− yj
∂xi
∂xj
κ
σ
ij
kij
kij
∂fσν
∂gνκ
∂gσκ
∂yk
∂y
κ
σ
+2 2 κ +
−
− yki
− yj
∂yk
∂y σ
∂y ν
∂xi
∂xj
lij
kij
∂gνκ
∂gνσ
∂ylκ
∂ykσ
∂hlkij
κσ
κ
σ
+
−
− xli
− ykj = 0.
+ 2
∂y ν
∂ykσ
∂ylκ
∂xi
∂xj
ij
ij
ij
∂fσν
∂fκσ
∂fνκ
+
+
∂y κ
∂y ν
∂y σ
The system (2.6) can be viewed as a system of mn2 (algebraic) linear homogeneous equations for
n+1
n+3
mn + mn
= mn
2
2
unknowns
∂y σ
σ
− yi ,
∂xi
∂yjσ
σ
−
y
ij ,
∂xi
and
j ≤ i.
According to the (algebraic) Frobenius theorem, this system has a unique
(zero)
klpq
solution if and only if the rank of the matrix of system, i. e., Aklj
|
B
is equal
νσ
νκ
to the number of unknowns, i. e., mn ((n + 3)/2). Let dim X = n ≥ 3, then
n n
n+3
2
+
≥ mn
,
mn = mn
2
2
2
as desired. Since rank of matrix (2.5) is maximal, by assumption, we obtain
∂yjσ
∂y σ
σ
σ
◦
δ
=
y
◦
δ,
◦ δ = yij
◦ δ, j ≤ i,
i
∂xi
∂xi
proving that δ = J 2 γ. Substituting this into (2.8) we get
!
!
e
e pq
e
e pq
∂L
∂L
∂L
∂L
κ κ
κ κ
jk
e
+
y − dj
+
y
+ dj dk Lν ◦ J 3 γ
∂y ν
∂y ν pq
∂yjν
∂yjν pq
!
∂L
∂L
∂L
=
− dj ν + dj dk ν
◦ J 3 γ = 0,
∂y ν
∂yj
∂yjk
i. e., γ is an extremal of λ.
σ
Theorem B. Let λ be a second-order Lagrangian affine in the variables yij
, the
i σ
formula (2.1) be its expression in a fiber chart (V, ψ), ψ = (x , y ) on Y . Let ρ
be a Lepagean equivalent of λ of the form (2.2), (2.3). Suppose that ρ satisfies the
conditions
(2.9)
hklij
= 0.
σν
ON HAMILTON p2 –EQUATIONS IN SECOND-ORDER FIELD THEORY
335
Assume that the matrix
(2.10)
Aklj
νσ
=
e kl
∂L
1
ν
−
∂yjσ
2
e jk
e jl
∂L
∂L
σ
σ
+
ν
∂yl
∂ykν
!
!
−
kjl
gσν
−
ljk
gσν
with mn2 rows (resp. mn columns) labelled by (ν, k, l) (resp. (σ, j)), has the maximal rank (i.e. rank Aklj
νσ = mn).
Then every Hamilton extremal δ : π(V ) ⊃ U → J 2 Y of ρ is of the form π2,1 ◦δ =
J 1 γ, where γ is an extremal of λ.
Proof. Substituting (2.9) into Hamilton p2 -equations (2.6), and using the condition
rank Aklj
νσ = mn we obtain
∂y σ ◦ δ
= yjσ ◦ δ.
∂xj
The previous condition means π2,1 ◦ δ = J 1 γ. However, the last equations (2.8)
now mean that γ is an extremal of λ.
3. Legendre transformation on J 2 Y for second order Lagrangians
affine in second derivatives
Writing the Lepagean equivalent (2.2), (2.3) in the form of a noninvariant deσ
composition in the canonical basis (dxi , dy σ , dyiσ , dyij
) of 1-forms we get
σ
ρ = −Hω0 + piσ dy σ ∧ ωi + pij
σ dyi ∧ ωj
ij
kij
σ
ν
+ fσν
dy σ ∧ dy ν ∧ ωij + gσν
dy σ ∧ dykν ∧ ωij + hklij
σν dyk ∧ dyl ∧ ωij ,
where
σ ν
∂L
ij
σ e ij σ
ij σ ν
kij
jik
e
−
d
L
j σ yi +Lσ yij +2fσν yi yj − gσν +gσν yi yjk
σ
∂yi
σ ν
1 klij
kjil
ijkl
−
hσν + hilkj
yik yjl ,
σν + hσν + hσν
2
ν
∂L
ij ν
kij
jik
e ij
− dj L
piσ =
σ − 4fσν yj − gσν + gσν yjk ,
σ
∂yi
ν
ν
ikj
jki
kilj
likj
e ij
pij
ykl .
σ = Lσ + gνσ + gνσ yk − 2 hνσ + hνσ
H = − L+
(3.1)
ji
kilj
likj
kjli
ljki
If pij
σ = pσ (i. e., hνσ + hνσ = hνσ + hνσ ) and


∂piσ
∂piσ
ν 
 ∂y ν
∂ykl
k


 6= 0,
det 


ij 
 ∂pij
∂p
σ
σ
∂ykν
ν
∂ykl
then
(3.2)
σ
ψ2 = (xi , y σ , yiσ , yij
) → (xi , y σ , piσ , pij
σ)=χ
336
DANA SMETANOVÁ
is a coordinate transformation over an open set U ⊂ V2 . We call it Legendre
transformation, and the χ (3.2) the Legendre coordinates. Accordingly, H, piσ , pij
σ
ij
kij
are called a Hamiltonian and momenta, respectively. Since the functions fσν
, gσν
,
i
ij
hlikj
(2.3)
may
depend
upon
the
momenta
p
(not
upon
p
),
the
Hamilton
p
2
νσ
σ
σ
equations (1.4) in these “Legendre coordinates” take a rather complicated form:
ij
κ
ij
ij
ij
∂H
∂piσ
∂fσν
∂y ν
∂fκν
∂fκσ
∂fνκ
∂y ∂y ν
=
−
+
4
+
2
+
+
∂y σ
∂xi
∂xj ∂xi
∂y σ
∂y ν
∂y σ ∂xi ∂xj
κ
ij
kij
kij
kij
∂fσν
∂pkκ ∂y ν
∂gσν
∂ykν
∂gκν
∂gσν
∂y ∂ykν
−4 k
+
+
2
−
∂pκ ∂xi ∂xj
∂xj ∂xi
∂y σ
∂y κ
∂xi ∂xj
−2
kij
κ
ν
∂gσν
∂plκ ∂ykν
∂hklij
κν ∂yk ∂yl
+
2
,
∂plκ ∂xi ∂xj
∂y σ ∂xi ∂xj
jk
kjl
∂H
∂y σ
∂fκν
∂y κ ∂y ν
∂gκν
∂y κ ∂ykν
∂hkljm
∂ykκ ∂ylν
κν
+
2
+
2
,
=
+
2
∂piσ
∂xi
∂piσ ∂xj ∂xk
∂piσ ∂xj ∂xl
∂piσ ∂xj ∂xm
∂yjσ
∂H
1 ∂yiσ
=
+
.
2 ∂xj
∂xi
∂pij
σ
However, if dη = 0, where
ij
kij
σ
ν
η = fσν
dy σ ∧ dy ν ∧ ωij + gσν
dy σ ∧ dykν ∧ ωij + hklij
σν dyk ∧ dyl ∧ ωij ,
then
∂H
∂piσ
=
−
,
∂y σ
∂xi
∂H
∂y σ
=
,
∂piσ
∂xi
∂H
∂pij
σ
1
=
2
∂yjσ
∂yiσ
+
∂xj
∂xi
.
In general case the regularity of the Lepagean form (2.3), (2.3) and regularity of
Legendre transformation (3.2) do not coincides. By the following Theorem C the
existence of Legendre transformation (3.2) guarantees that Theorem B holds.
σ
, the
Theorem C. Let λ be a second-order Lagrangian affine in the variables yij
i σ
formula (2.1) be its expression in a fiber chart (V, ψ), ψ = (x , y ) on Y . Let ρ
be a Lepagean equivalent of λ of the form (2.2), (2.3). Suppose that ρ satisfies the
conditions hklij
= 0. Suppose that ρ admits the Legendre transformation
σν
σ
ψ2 = (xi , y σ , yiσ , yij
) → (xi , y σ , piσ , pij
σ) = χ
defined by (3.1), (3.2).
Then π2,1 ◦ δ = J 1 γ, where γ is an extremal of λ.
Proof. Since, the functions hklij
σν vanish, the Jacobi matrix of the Legendre transformation takes the form
 i

i
∂pσ
ν
 ∂yijk
∂pσ
ν
∂yk
∂pσ
ν
∂ykl

0
ON HAMILTON p2 –EQUATIONS IN SECOND-ORDER FIELD THEORY
337
i ij ∂p
∂p
The above matrix is regular if and only if the matrices ∂yνσ , and ∂yσν have
kl
k
the maximal rank. Explicit computations lead to
!
e ik
e kl
e il
∂piσ
∂L
1 ∂L
∂L
ν
σ
σ
kil
lik
+
− gσν
− gσν
,
ν = ∂y σ − 2
∂ykl
∂ylν
∂ykν
i
i T
∂p
i.e. in the notation (2.10), ∂yνσ = Aklj
.
νσ
kl
Accordingly, from Theorem B we obtain π2,1 ◦ δ = J 1 γ, where γ is an extremal
of λ.
For a deeper discussion on Legendre transformations and their geometric meaning we refer to [11], [12].
4. Projectability onto J 1 Y
σ
, i. e.,
Theorem D. Let λ be a second-order Lagrangian affine in the variables yij
in fibered coordinates expressed by (2.1). Let ρ be a Lepagean equivalent of λ of the
form (2.2), (2.3). The following conditions are equivalent:
I. ρ is projectable onto J 1 Y .
II. ρ satisfies the conditions
likj
hkilj
νσ + hνσ = 0,
(4.1)
kjl
ljk
gσν
+ gσν
e kl
∂L
1
ν
=
−
∂yjσ
2
e jk
e jl
∂L
∂L
σ
σ
+
ν
∂yl
∂ykν
!
.
Proof. Taking into account
σ
ρ = −Hω0 + piσ dy σ ∧ ωi + pij
σ dyi ∧ ωj
ij
kij
σ
ν
+ fσν
dy σ ∧ dy ν ∧ ωij + gσν
dy σ ∧ dykν ∧ ωij + hklij
σν dyk ∧ dyl ∧ ωij ,
σ
it is sufficient to find conditions of the independence H, piσ , and pij
σ on yij ’s. Explicit
computations lead to
!
e kl
e ik
e il
∂piσ
∂L
1 ∂L
∂L
ν
σ
σ
kil
lik
− gσν
− gσν
= 0,
ν = ∂y σ − 2
ν + ∂y ν
∂ykl
∂y
i
l
k
∂pij
σ
likj
= −2 hkilj
= 0,
νσ + hνσ
ν
∂ykl
∂H
ν =
∂ykl
e kl
∂L
1
ν
−
∂yiσ
2
e ik
e il
∂L
∂L
σ
σ
+
ν
∂yl
∂ykν
!
!
−
kil
gσν
σ
kilj
ljki
likj
− hkjli
yij .
νσ + hνσ + hνσ + hνσ
−
lik
gσν
yiσ
338
DANA SMETANOVÁ
κ
Corollary. Every second-order Lagrangian affine in the variables yij
has a Lep1
agean equivalent projectable onto J Y .
ij
kij
Remark. If the functions fσν
, gσν
, hklij
σν (2.2), (2.3) vanish, i. e., ρ = θλ the
projectability conditions (4.1) take the form (cf. [9])
!
e kl
e jk
e jl
∂L
1 ∂L
∂L
ν
σ
σ
−
+
= 0.
∂yjσ
2 ∂ylν
∂ykν
σ
Theorem E. Let λ be a second-order Lagrangian affine in the variables yij
, the
i σ
formula (2.1) be its expression in a fiber chart (V, ψ), ψ = (x , y ) on Y . Let ρ be
a Lepagean equivalent of λ of the form (2.2), (2.3) and suppose that it is projectable
onto J 1 Y . If ρ satisfies the conditions
hklij
σν = 0,
ij
1
∂fσν
=
∂ykκ
2
(4.2)
ikj
gνσ
−
kij
gσν
kij
kij
∂gκσ
∂gκν
−
∂y ν
∂y σ
1
=
2
e ij
e kj
∂L
∂L
σ
ν
−
ν
∂yk
∂yiσ
!
lij
kij
∂gσκ
∂gσν
−
= 0.
∂ykν
∂ylκ
and the matrix
kj
Cνσ
=
e
e pq
e kj
∂2L
∂2L
∂
∂L
κ
ν
κ
jp
jk
kij
e
+
y
−
d
L
−
− 4fσν
+ 2di gσν
p
σ
∂yjσ ∂ykν
∂yjσ ∂ykν pq ∂ykν
∂y σ
!
,
with rows (resp. columns) labelled by (ν, k) (resp. (σ, j)), is regular, then ρ is
regular, i.e., every Hamilton extremal δ : π(V ) ⊃ U → J 1 Y of ρ is of the form
δ = J 1 γ, where γ is an extremal of λ.
Proof. Expressing the Hamilton p2 -equations (1.4) of a Lepagean equivalent ρ projectable onto J 1 Y in fiber coordinates and using (4.2) we get along δ the following
system of first-order equations:
mn equations
!
e
e pq
e kj
∂2L
∂2L
∂
∂L
κ
ν
κ
jp
jk
kij
e
(4.3)
+ σ ν ypq − ν dp Lσ −
− 4fσν + 2di gσν
∂yjσ ∂ykν
∂yj ∂yk
∂yk
∂y σ
σ
∂y
σ
×
= 0,
−
y
j
∂xj
m equations
!
!
pq
e
e pq
e
e
∂L
∂L
∂
L
∂
L
κ κ
κ κ
e jk
(4.4)
+
y − dj
+
y
+ dj dk L
ν
∂y ν
∂y ν pq
∂yjν
∂yjν pq
ON HAMILTON p2 –EQUATIONS IN SECOND-ORDER FIELD THEORY
339
e
e pq
e
e pq
∂2L
∂2L
∂2L
∂2L
∂
κ
κ
e jk
− σ ν − σ κ ν ypq
− ν dk L
+ σ κ ν ypq
σ
σ
ν
∂yj ∂y
∂yj ∂y
∂y ∂yj
∂y ∂yj
∂y
σ
∂
∂y
jk
ij
σ
e
+ σ dk Lν + 2di fσν
− yj
∂y
∂xj
+
e
e kj
e pq
∂2L
∂L
∂2L
∂
σ
κ
jk
kij
e jp
+
− σ ν − σ κ ν ypq
+ σ dp L
ν + 4fνσ − 2di gνσ
ν
∂y
∂yj ∂yk
∂yj ∂yk
∂yk
σ
∂yk
σ
×
− ykj
∂xj
ij
κ
σ
ij
ij
∂fσν
∂fκσ
∂fνκ
∂y
∂y
κ
σ
+
− yi
− yj = 0.
+2
+
∂y κ
∂y ν
∂y σ
∂xi
∂xj
!
kj
The matrix Cνσ
is regular. Hence, from equations (4.3) we obtain the formula
∂y σ ◦ δ
= yjσ ◦ δ.
∂xj
(4.5)
Substituting this into (4.4) we get
!
!
e
e pq
∂L
∂L
κ κ
jk
eν ◦ J 3 γ
+
y
+ dj dk L
∂yjν
∂yjν pq
!
∂L
∂L
∂L
− dj ν + dj dk ν
◦ J 3 γ = 0,
∂y ν
∂yj
∂yjk
e
e pq
∂L
∂L
κ κ
+
y − dj
ν
∂y
∂y ν pq
=
proving our assertion.
Remark. a) Let λ be a second-order Lagrangian (2.1), suppose that the functions
e ij
L
σ satisfy the conditions
e ki
e ki
∂L
∂L
σ
ν
=
ν
∂yj
∂yjσ
e ij
This means that L
σ take the form
e ij = 1
L
σ
2
∂f j
∂f i
+ σ
σ
∂yi
∂yj
!
and the Lagrangian equivalent with a first-order Lagrangian.
kij
We can choose the functions gσν
in a regular Lepagean equivalent (in the sense
of Theorem E) in the following form
!
ki
e kj
e
1
∂
L
∂
L
ν
ν
kij
gσν
=
−
+ tkij
σν ,
2 ∂yiσ
∂yjσ
340
DANA SMETANOVÁ
ν
where the functions tkij
σν do not depend on the variables ykl and satisfy the conditions
kji
kij
jik
kij
ikj
tkij
σν = −tσν , tσν = −tσν , tσν = −tνσ .
e ij
b) Let λ be a second-order Lagrangian (2.1) and suppose that the functions L
σ
satisfy the conditions
e kl
∂L
1
ν
−
σ
∂yj
2
e jk
e jl
∂L
∂L
σ
σ
+
∂ylν
∂ykν
!
= 0.
kij
Then we can choose the functions gσν
as follows:
!
e kj
e ij
1 ∂L
∂L
ν
σ
kij
−
+ tijk
gσν =
σν ,
4 ∂yiσ
∂ykν
where the tkij
σν ’s are as above.
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Mathematical Institute of the Silesian University in Opava, Bezručovo nám. 13, 746
01 Opava, Czech Republic
E-mail address: Dana.Smetanova@math.slu.cz