Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
24 (2008), 115–123
www.emis.de/journals
ISSN 1786-0091
ON NONLINEAR CONNECTIONS IN HIGHER ORDER
LAGRANGE SPACES
MARCEL ROMAN
Abstract. Considering a Lagrangian of order k, we determine a nonlinear
connection N on T k M such that the horizontal and vertical distributions
to be Lagrangian subbundles for the presymplectic structure given by the
k.
Cartan-Poincaré two-form ωL
1. Introduction
We denote by (T k M, π k , M ), k ≥ 1, the space of tangent bundle of order
k over a smooth, real, n-dimensional manifold M , [5]. Local coordinates (xi )
on M induce local coordinates (xi , y (1)i , . . . , y (k)i ) on T k M , where for a k-jet
j0k ρ ∈ T k M , the coordinate functions are defined as follows
¯
1 dα (xi ◦ ρ) ¯¯
(α)i k
y
(j0 ρ) =
, α ∈ {1, . . . , k}.
¯
α!
dtα
t=0
The tangent structure of order k, J is defined as follows, [6],
J=
∂
∂
∂
⊗ dxi + (2)i ⊗ dy (1)i + · · · + (k)i ⊗ dy (k−1)i .
(1)i
∂y
∂y
∂y
The foliated structure of T k M allows for k regular, integrable, vertical distributions, Vk−α+1 = Ker J α = Im J k−α+1 , α ∈ {1, . . . , k}.
2000 Mathematics Subject Classification. 53B40, 53C20, 53C60.
Key words and phrases. higher order Lagrange spaces, nonlinear connection, CartanPoincaré forms.
This work has been supported by CEEX Grant 31742/2006 and by Grant CNCSIS
1158/2006 from the Romanian Ministry of Education.
115
116
MARCEL ROMAN
The following k vertical vector fields are globally defined on T k M and they
are called Liouville vector fields:
∂
∂
∂
+ 2y (2)i (2)i + · · · + ky (k)i (k)i ,
(1)i
∂y
∂y
∂y
Γα = J k−α (Γk ) , α ∈ {1, 2, . . . , k}.
Γk = y (1)i
A semispray is a globally defined vector field S on T k M that satisfies the
equation JS = Γk . Therefore, a semispray S, which is a vector field of order
k + 1, which can be expressed as follows
(1.1)
∂
∂
∂
+ 2y (2)i (1)i + · · · + ky (k)i (k−1)i −
∂xi
∂y
∂y
∂
− (k + 1)Gi (x, y (1) , . . . , y (k) ) (k)i
∂y
S = y (1)i
and it is perfectly determined by its coefficient functions Gi (x, y (1) , . . . , y (k) ).
A nonlinear connection, or a horizontal distribution on T k M is a regular
distribution N : u ∈ T k M 7→ N (u) ⊂ Tu T k M such that the following direct sum
holds true:
(1.2)
Tu (T k M ) = N (u) ⊕ N1 (u) ⊕ · · · ⊕ Nk−1 (u) ⊕ Vk (u),
where N1 = J(N ), Nα−1 = J α−1 (N ), α ∈ {3, . . . , k}. The adapted basis to this
decomposition is given by
¾
½
δ
δ
∂
δ
,
,
.
.
.
,
,
,
δxi δy (1)i
δy (k−1)i ∂y (k)i
where, [7]:
∂
δ
∂
∂
=
− N ji (1)j − · · · − N ji (k)j ,
δxi
∂xi
∂y
∂y
(k)
(1)
(1.3)
δ
∂
∂
∂
=
− N ji (2)j − · · · − N ji
,
(1)i
(1)i
(k)j
δy
∂y
(1) ∂y
(k−1) ∂y
..
.
∂
δ
∂
=
− N ji (k)j .
δy (k−1)i
∂y (k−1)i
∂y
(1)
The functions N ji , N ji , . . . , N ji are called the local coefficients of the nonlinear
(1)
connection N .
(2)
(k)
ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACE
117
©
ª
The dual basis of the previous adapted basis is given by dxi , δy (1)i , . . . , δy (k)i ,
where:
δy (1)i = dy (1)i + M ij dxj ,
(1)
δy
(2)i
= dy
(2)i
+ M ij dy (1)j + M ij dxj ,
(1)
(1.4)
(2)
..
.
δy (k)i = dy (k)i + M ij dy (k−1)j + · · · + M ij dxj .
(1)
(k)
The functions M ji , M ji , . . . , M ji are called the dual coefficients of the nonlin(1)
(2)
(k)
ear connection N .
2. Cartan-Poincaré forms for a higher order Lagrangian
Let us consider a regular Lagrangian of order k, (k > 1), L(xi , y (1)i , . . . , y (k)i ).
The metric tensor is given by the symmetric d-tensor field:
(2.1)
gij =
1
∂2L
,
(k)i
2 ∂y ∂y (k)j
which has maximal rank on T k M, rank ||gij || = n.
For a regular Lagrangian of order k, we consider the following Cartan-Poincaré
one-forms, [4]:
(2.2)
α
θL
= dJ α L =
∂L
∂L
dxi + · · · + (k−α)i dy (α)i , α ∈ {1, . . . , k}.
∂y (α)i
∂y
We consider also the following Cartan-Poincaré two-forms:
µ
¶
µ
¶
∂L
∂L
α
α
i
=d
= dθL
ωL
∧
dx
+
·
·
·
+
d
∧ dy (α)i ,
∂y (α)i
∂y (k−α)i
(2.3)
α ∈ {1, . . . , k}.
We remark here that for k > 1, the regularity of the Lagrangian L implies the
k
k
fact that rank(ωL
) = 2n < (k + 1)n = dim(T k M ). We refer to ωL
as to the
canonical presymplectic structure of the Lagrangian L.
3. Nonlinear connection
Now, our question is: Can we find an adapted basis such that the horizontal
k
distribution to be Lagrangian subbundle for the presymplectic structure ωL
?
Taking into account (1.3) this is equivalent with the determination of the coefficients of a nonlinear connection N .
118
MARCEL ROMAN
As we know, [1], in the k = 1 case, we consider the canonical nonlinear
connection and we have ωL = 2gji δy j ∧ dxi . It follows that in the basis (dxi , δy i )
of the cotangent bundle, the matrix of ωL is
¶
µ
0 2gji
.
−2gji 0
For k > 1, we consider the Cartan-Poincaré two-form:
µ
¶
∂L
k
ωL
=d
∧ dxi
∂y (k)i
¶
¶¸
·
µ
µ
1 δ
δ
∂L
∂L
=
−
dxj ∧ dxi
2 δxj ∂y (k)i
δxi ∂y (k)j
µ
¶
δ
∂L
(3.1)
+ (1)j
δy (1)j ∧ dxi + · · ·
δy
∂y (k)i
µ
¶
δ
∂L
δy (k−1)j ∧ dxi
+ (k−1)j
δy
∂y (k)i
+ 2gji δy (k)j ∧ dxi .
We are looking for a nonlinear connection on T k M such that the presymplectic
k
structure of the lagrangian L to be: ωL
= 2gji δy (k)j ∧ dxi , i.e. to have the
following matrix:


0 0 · · · 0 2gji
 0



.
 ..

 .

O
kn
−2gji
(k+1)n×(k+1)n
This is equivalent with the vanishes of all coefficients from (3.1), except the last
one. We will obtain the coefficients of the nonlinear connection N.
We have:
µ
¶
∂L
∂2L
δ
=
0
=⇒
− N mj 2gmi = 0
δy (k−1)j ∂y (k)i
∂y (k−1)j ∂y (k)i
(1)
Therefore, we find the first coefficient for the nonlinear connection:
(3.2)
∂2L
1 mi
g
.
2
∂y (k−1)j ∂y (k)i
(1)
µ
¶
∂L
= 0 and we obtain the second coefficient of
∂y (k)i
N mj =
δ
δy (k−2)j
the nonlinear connection:
∂2L
1
∂2L
1
(3.3)
N mj = g mi (k−2)j (k)i − N rj g mi (k−1)r (k)i .
2
2
∂y
∂y
∂y
∂y
(1)
(2)
Now, we take
ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACE
δ
Finally, from
µ
δy (1)j
nonlinear connection N :
N mj =
(k−1)
(3.4)
∂L
∂y (k)i
119
¶
= 0 we obtain the (k − 1)th coefficient of the
1 mi
∂2L
∂2L
r 1 mi
g
−
N
g
j
2
2
∂y (1)j ∂y (k)i
∂y (2)r ∂y (k)i
(1)
1 mi
∂2L
g
.
2
∂y (k−1)r ∂y (k)i
(k−2)
− · · · − N rj
Consequently, the coefficients N mj , N mj , . . . , N mj are unique determined.
(1)
(2)
(k−1)
We have:
Theorem
respect
of the local coordinates
¡ i (1)i 3.1.(k)iWith
¢
¡ i (1)ito a transformation
¢
x ,y ,...,y
−→ x
e , ye , . . . , ye(k)i on T k M, the coefficient of the nonlinear connection N are transformed by the rule:
xm
∂e
xi m ∂e
y (1)i
e im ∂e
N
=
Nj −
,
j
m
∂x (1)
∂xj
(1) ∂x
(3.5)
xm
∂e
xi m ∂e
y (1)i m ∂e
y (2)i
e im ∂e
N
=
Nj +
Nj −
,
j
m
m
∂x (2)
∂x (1)
∂xj
(2) ∂x
..
.
xm
∂e
xi
∂e
y (1)i m
∂e
y (k−2)i m ∂e
y (k−1)i
e im ∂e
N
=
N mj +
N j + ··· +
Nj −
.
j
m
m
m
∂x (k−1)
∂x (k−2)
∂x
∂xj
(k−1) ∂x
(1)
Proof. A transformation of local coordinates
³
´
³
´
xi , y (1)i , . . . , y (k)i −→ x
ei , ye(1)i , . . . , ye(k)i
on the manifold T k M is given by, [7]:
x
ei = x
ei (x1 , . . . , xn ), rank k
∂e
xi
k = n,
∂xj
∂e
xi (1)j
y ,
∂xj
∂e
y (1)i (1)j
∂e
y (1)i
=
y
+ 2 (1)j y (2)j ,
j
∂x
∂y
ye(1)i =
2e
y (2)i
..
.
ky (k)i =
∂e
y (k−1)i (1)j
∂e
y (k−1)i (2)j
∂e
y (k−1)i
y
+2
y
+ · · · + k (k−1)j y (k)j .
j
(1)j
∂x
∂y
∂y
120
MARCEL ROMAN
Also, we have the identities, [7]:
∂e
y (α)i
∂e
y (α+1)i
∂e
y (k)i
=
=
·
·
·
=
, (α = 0, . . . , k − 1; y (0)i = xi ).
∂xj
∂y (1)j
∂y (k−α)j
With respect to the previous transformation of local coordinates, the natural
basis is changed by the following rule:
∂
∂e
xj ∂
∂e
y (1)j ∂
∂e
y (k)j ∂
=
+
+
·
·
·
+
,
∂xi
∂xi ∂e
xj
∂xi ∂e
∂xi ∂e
y (1)j
y (k)j
∂
∂e
y (1)j ∂
∂e
y (k)j ∂
=
+
·
·
·
+
,
∂y (1)i
∂y (1)i ∂e
y (1)j
∂y (1)i ∂e
y (k)j
..
.
∂e
y (k)j ∂
∂
=
.
∂y (k)i
∂y (k)i ∂e
y (k)j
Taking into account last formulas, we have
∂L
∂e
y (k)j ∂L
=
∂y (k)i
∂y (k)i ∂e
y (k)j
and consequently, we obtain
∂2L
∂e
y (k)j ∂e
y (k−1)r
∂2L
∂e
y (k)r ∂e
y (k)j
∂2L
=
+
.
∂y (k)i ∂y (k−1)m
∂y (k)i ∂y (k−1)m ∂e
y (k)j ∂e
y (k−1)r ∂y (k−1)m ∂y (k)i ∂e
y (k)j ∂e
y (k)r
But,
∂e
y (k)r
∂e
y (1)r
∂2L
=
and
2g
=
.
jr
∂xm
∂y (k−1)m
∂e
y (k)j ∂e
y (k)r
1
Now, contracting by g ij , we obtain
2
∂e
y (1)r ∂e
∂e
xj ∂e
xr e i
xj
Nr+
N jm =
i
m
m
∂x ∂x (1)
∂x ∂xr
(1)
and finally,
xi
∂e
xr
∂e
y (1)r
e ri ∂e
N
= N jm j −
.
m
∂xm
(1) ∂x
(1) ∂x
Similarly, in order to check the second relation from (3.5), we calculate
2
∂2L
, for the third relation we calculate ∂y(k)i∂∂yL(k−3)m and so on, for
∂y (k)i ∂y (k−2)m
the last relation we calculate
∂2L
.
∂y (k)i ∂y (1)m
Now, considering the following notations:
(3.6)
M rj =
(α)
∂2L
1 ri
g
, α ∈ {1, . . . , k − 1},
2 ∂y (k−α)j ∂y (k)i
¤
ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACE
121
we obtain:
Proposition 3.1. The relationship between the coefficients N mi , N mi , . . . , N mi
(1)
(2)
(k−1)
of the nonlinear connection N and the coefficents given in (3.6) is expressed by:
N mi = M mi
(1)
(1)
N mi = M mi − M mr N ri
(2)
(3.7)
(2)
(1) (1)
..
.
N mi
(k−1)
= M mi − M mr N ri − · · · − M mr N ri .
(k−1)
(k−2) (1)
(1) (k−2)
Indeed, replacing (3.6) in (3.2),(3.3)
and (3.4) we)have the conclusion. There(
fore, the system of functions
M mi , M mi , . . . , M mi
(1)
(2)
is the system of dual coef-
(k−1)
ficients of the nonlinear connection N .
Example 3.1. Let R = (M, gij (x)) be a Riemannian space and Prol2 Rn its
prolongation of order 2, [8].
We consider the Liouville d-vector fields, [7]:
z (1)m = y (1)m ,
i
1 h (1)m
m (1)i (1)j
z (2)m =
Γz
+ γij
z z
,
2
(3.8)
m
(x) are the Christoffel symbols and the operator Γ is given by:
where γij
Γ = y (1)i
∂
∂
+ 2y (2)i (1)i .
∂xi
∂y
Considering the Lagrangian L = gij (x)z (k)i z (k)j , we remark that the first
coefficient N mi of our nonlinear connection is the same with the first coefficient
(1)
from nonlinear connections determined by I. Bucataru, [2], M. de Leon, [4], and
R. Miron, [7], i.e. is expressed by:
i
N ij = M ij = γjh
(x)y (1)h .
(1)
(1)
Now, for the last coefficient of the nonlinear connection N ij , we have:
(k)
122
MARCEL ROMAN
Theorem 3.2. The skew symmetric part of the coefficient N ij is expressed as
(k)
follows:
N
[ji]
(k)
(3.9)
¶
∂2L
∂2L
−
∂xi ∂y (k)j
∂xj ∂y (k)i
!
Ã
k−1
1 sm X
N js M im − N is M jm .
− g
2
(α) (k−α)
(α) (k−α)
α=1
1
=
4
µ
= gim M mj .
where M
ji
(k)
(k)
Proof. For the last coefficient of the nonlinear connection N ij we have to consider
(k)
the first coefficient from (3.1):
¶
¶¸
·
µ
µ
1 δ
∂L
δ
∂L
(3.10)
−
=0
2 δxj ∂y (k)i
δxi ∂y (k)j
We obtain:
N
[ji]
(k)
1
=
2
where N
ji
(k)
!
N
ji
(k)
−N
ij
(k)
¶
∂2L
∂2L
−
∂xi ∂y (k)j
∂xj ∂y (k)i
Ã
!
1
∂2L
∂2L
m
m
−
N
− N i (1)m (k)j
2 (1)j ∂y (1)m ∂y (k)i
∂y
(1) ∂y
Ã
!
∂2L
1
∂2L
m
m
Nj
,
− ··· −
− Ni
2 (k−1)
∂y (k−1)m ∂y (k)i (k−1)
∂y (k−1)m ∂y (k)j
1
=
4
(3.11)
Ã
µ
= gim N mj .
(k)
The condition (3.10) determines uniquely the skew symmetric part of the
coefficient N mj , only.
¤
(k)
For the symmetric part we need a supplementary condition. In the k = 1
case, I. Bucataru proved that the symmetric part is uniquely determined by a
metric condition, [3].
So far, we have a whole family of nonlinear connections that are determined
k
by the presymplectic structure ωL
. These connections are derived directly from
the Lagrangian L and does not use the canonical semispray.
ON NONLINEAR CONNECTIONS IN HIGHER ORDER LAGRANGE SPACE
123
References
[1] M. Anastasiei. Symplectic structures and Lagrange geometry. In Finsler and Lagrange
geometries (Iaşi, 2001), pages 9–16. Kluwer Acad. Publ., Dordrecht, 2003.
[2] I. Bucătaru. Sprays and homogeneous connections in the higher order geometry. Stud.
Cerc. Mat., 50(5-6):307–315, 1998.
[3] I. Bucataru. Metric nonlinear connections. Differential Geom. Appl., 25(3):335–343, 2007.
[4] M. de León and P. R. Rodrigues. Generalized classical mechanics and field theory, volume
112 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam,
1985. A geometrical approach of Lagrangian and Hamiltonian formalisms involving higher
order derivatives, Notes on Pure Mathematics, 102.
[5] C. Ehresmann. Les prolongements d’une variété différentiable. I. Calcul des jets, prolongement principal. C. R. Acad. Sci. Paris, 233:598–600, 1951.
[6] H. A. Eliopoulos. On the general theory of differentiable manifolds with almost tangent
structure. Canad. Math. Bull., 8:721–748, 1965.
[7] R. Miron. The geometry of higher-order Lagrange spaces, volume 82 of Fundamental Theories of Physics. Kluwer Academic Publishers Group, Dordrecht, 1997. Applications to
mechanics and physics.
[8] R. Miron and G. Atanasiu. Prolongation of Riemannian, Finslerian and Lagrangian structures. Rev. Roumaine Math. Pures Appl., 41(3-4):237–249, 1996.
[9] W. M. Tulczyjew. The Lagrange differential. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 24(12):1089–1096, 1976.
Marcel Roman,
Technical University ”Gh. Asachi”,
Department of Mathematics,
Iaşi, 700506, Romania
E-mail address: marcelroman@gmail.com