Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 26 (2010), 275–303 www.emis.de/journals ISSN 1786-0091 GENERALIZED LAGRANGE – HAMILTON SPACES OF ORDER k

26 (2010), 275–303 www.emis.de/journals ISSN 1786-0091

GENERALIZED LAGRANGE – HAMILTON SPACES OF ORDER k

IRENA ˇCOMI ´C

Abstract. In this paper the generalized Lagrange – Hamilton spaces are introduced. The group of transformation is given, further some complicated but useful relations concerning the partial derivatives of variables in new and old coordinate systems are derived. The sprays and antisprays are also studied.

Introduction

The (k + 1)n dimensional generalized Lagrange space is an Osc^kM space supplied with regular Lagrangian L(x^a, y^1a, y^2a, . . . , y^ka), where y^Aa = _dt^dAAx^a, A= 1, k. They are studied in many papers and books as [2], [18], [19], [20], [16]

and others. The K-Hamilton space is (k+ 1)n dimensional space, where some point of this space has coordinates (x^a, p_1a, . . . , pka), where pAa(A = 1, k) are independent covector fields. If k = 1 we have Hamilton space. Such spaces are studied in [1], [3–7], [10–12], [14], [15], [21–23], [25], [26] and many others. If instead of covector fields p₁, . . . , pk we take independent vector fields y¹, . . . , y^k we obtain K-Lagrange spaces.

The (k + 1)n dimensional Hamilton space of order k was introduced by R. Miron in [17]. Some point u of this space has coordinates

(x^a, y^1a, . . . , y^(k−1)a, pka),

wherey^Aa = _A!¹ _dt^dAAx^a,A= 1, k−1 and (pka) is a covector. The space is supplied with regular Hamiltonian H(x, y¹, . . . , y^k−1, pk) from which the metric tensor is derived in the usual manner. The complex structures in the above spaces are introduced in [24].

2000 Mathematics Subject Classification. 53B40, 53C60.

Key words and phrases. Lagrange-Hamilton spaces, adapted bases, theJ structure, sprays, Liouville vector fields.

275

The Hamilton spaces of higher order are introduced in [9]. A pointu of this (k+ 1)n dimensional space has coordinates (x^a, p_1a, p_2a, . . . , p_ka), where p₁ is a covector and p_Aa = _dt^dAA−₋¹¹p_1a, A = 2, k. In the transformation group the expressions _dt^dAA∂x_∂x^a′a appear, which are functions of y¹, y², . . . , y^A, but they were not written explicitly and were not treated as variables. Such spaces are studied in [10, 13].

Here the (2k + 1)· n dimensional generalized Lagrange – Hamilton spaces (GLH)^(nk) are introduced, where some point u∈(GLH)^(nk) has coordinates

(x^a, y^1a, . . . , y^ka, p1a, p2a, . . . , pka),

where y^(A+1)a = _dt^dAAy^1a, p_(A+1)a = _dt^dAAp_1a, A = 1, k−1. The group of trans- formation is given, further some complicated but useful relations concerning the partial derivatives of variables in new and old coordinate systems are derived.

The natural and special adapted bases of T(GLH)^(nk) and T^∗(GLH)^(nk) are es- tablished. Using the matrix representation, the duality of bases in T and T^∗ is proved. The name ‘special adapted’ comes from the fact that the elements of these bases are transforming as tensors and they have the property that the J structure in the natural and special adapted bases has the same components.

By action of the J structure on the vector field dr and 1-form field δr the cor- responding Liouville vector and 1-form fields are constructed.

The sprays and antisprays are also studied and interesting results are ob- tained. Here the metric tensor does not appear. If the regular Lagrangian L and Hamiltonian H is given the metric tensor can be derived by the usual man- ner. From this the metric connection, the torsion and curvature tensors and the structure equations can be obtained, which will be the subject of next papers.

The application of this theory is given in variation calculus in the papers which will be appeared.

Special cases of (GLH)^(nk) are Lagrange spaces of order k, Osc^kM spaces, Finsler spaces, generalized Hamilton spaces and so on.

1. Definitions, group of coordinate transformation

Let us denote by (LH)⁽ⁿ¹⁾ the 3n dimensional C^∞ manifold in which some point (y, p) has coordinates (x^a =y^0a, y^1a, p_1a), a= 1, n.

Some curvec in (LH)⁽ⁿ¹⁾ is given by c:t ∈[a, b]→c(t)∈ (LH)⁽ⁿ¹⁾, where in some local chart (U, ϕ) a point (y, p)∈ c(t) has coordinates

(x^a(t) =y^0a(t), y^1a(t), p_1a(t)).

If in some other chart (U^′, ϕ^′) the same point (y, p) has coordinates (x^a′(t) =y^0a′(t), y^1a′(t), p1a^′(t)),

then the allowable transformations are given by

x^a′ =x^a′(x^a)⇔x^a =x^a(x^a′), (x^a(t) =x^a(x^a′(t)), y^1a′ =B_a^a′y^1a, B_a^a′ = ∂x^a′

∂x^a =B_a^a′(t), p_1a′ =B_a^a′p_1a. (1.1)

The first two equations in (1.1) give the coordinate transformation in the Finsler space. Here, in (LH)⁽ⁿ¹⁾ the point of the space has three components:

(x) = (x^a) - the point in the base manifold M, the contravariant vector field (y⁽¹⁾) = (y^1a) and a covariant vector field (p₍₁₎) = (p_1a).

(y⁽¹⁾) can be interpreted as the velocity vector and (p₍₁₎) as the generalized momentum.

Let us denote by (LH)^(nk) the (2k+ 1)n dimensional C^∞ manifold in which a point (y, p) = (x=y⁽⁰⁾, y⁽¹⁾, y⁽²⁾, . . . , y^(k), p₍₁₎, p₍₂₎, . . . , p_(k)) has coordinates

(x^a = y^0a, y^1a, y^2a, . . . , y^ka, p_1a, p_2a, . . . , pka), a= 1, n.

We can interpret the point in (LH)^(nk) as a point (x) in the base manifold together with a contravariant vector (y⁽¹⁾), covariant vector (p₍₁₎) and their derivatives up to order k.

Some curvec∈(LH)^(nk) is given by

c:t∈[a, b]→c(t)∈(LH)^(nk). A point (y, p)∈c(t) has coordinates

(x^a(t) =y^0a(t), y^1a(t), . . . , y^ka(t), p1a(t), . . . , pka(t)), where

y^Aa(t) =d^A_t y^0a(t) A= 1, k d^A_t = d^A dt^A

pαa(t) =d^α_t⁻¹p1a(t), α= 1, k, d^α_t⁻¹ = d^α−1 dt^α−1. (1.2)

The allowable coordinate transformations are given by x^a′ =x^a′(x^a)⇔x^a =x^a(x^a′) y^1a′ = B_a^a′y^1a, B_a^a′ = ∂0ax^a′ =∂ax^a′, ∂Aa= ∂

∂y^Aa A= 0, k, y^2a′ =

1 0

(d¹_tB_a^a′)y^1a+ 1

1

B_a^a′y^2a =d¹_t(B^a_a^′y^1a), ...

y^ka′ =

k−1 0

(d^k_t⁻¹B_a^a′)y^1a+

k−1 1

(d^k_t⁻²B_a^a′)y^2a+· · · +

k−1 k−1

B^a_a^′y^ka=d^k_t⁻¹(B_a^a′y^1a), (1.3a)

p_1a′ =B_a^a′p_1a B_a^a′ =∂_0a′x^a = ∂x^a

∂x^a′ =B_a^a′(t), p2a^′ =

1 0

(d¹_tB_a^a′)p1a+ 1

1

B_a^a′p2a =d¹_t(B_a^a′p1a), ...

pka^′ =

k−1 0

(d^k_t⁻¹B_a^a′)p1a+

k−1 1

(d^k_t⁻²B_a^a′)p2a+· · · +

k−1 k−1

B_a^a′pka. (1.3b)

Theorem 1.1. The transformations of type (1.3) on the common domain form a group.

The proof is similar to those given in [8] and [9].

Definition 1.1. The generalized Lagrange – Hamilton space of order k, (GLH)^(nk) is a (LH)^(nk) space, where the allowable coordinate trans- formations are given by (1.3) and in which a differentiable Lagrangian L(x, y⁽¹⁾, y⁽²⁾, . . . , y^(k)) and a differentiable HamiltonianH(x, p₍₁₎, p₍₂₎, . . . , p_(k)) are given.

From (1.3) it is not obvious that pαa^′, α = 1, k are functions of y^Aa, A = 0, α−1. This can be seen if we write:

d¹_tB_a^a′ =∂a1B_a^a′y^1a1, ∂a1 = ∂

∂x^a1

d²_tB_a^a′ = (∂²a₂a₁B_a^a′)y^1a1y^1a2 + (∂a1B_a^a′)y^2a1, ∂_a²_2a1 = ∂²

∂y^0a2∂y^0a1, d³_tB_a^a′ = (∂_a³_3a2a1B^a_a^′)y^1a1y^1a2y^1a3 + 3(∂_a²_2a1B_a^a′)y^1a1y^2a2+ (∂a1B_a^a′)y^3a1, d⁴_tB_a^a′ = (∂_a⁴_4a3a2a1B_a^a′)y^1a1y^1a2y^1a3y^1a4+ 6(∂_a³_3a2a1B_a^a′)y^1a1y^1a2y^2a3

+ 3(∂_a²_2a1B_a^a′)y^2a1y^2a2 + 4(∂_a²_2a1B_a^a′)y^1a1y^3a2+ (∂a1B_a^a′)y^4a1, ...

(1.4)

From (1.4) we can obtain another set of formulae if we make the changes (a^′, a, a₁, a₂, a₃, a₄)→(a, a^′, a^′₁, a^′₂, a^′₃, a^′₄).

From (1.3) and (1.4) it follows y^0a′ =y^0a′(y^0a)

y^1a′ =y^1a′(y^0a, y^1a), . . . , y^ka′ =y^ka′(y^0a, y^1a, . . . , y^ka), p_1a^′ = p_1a^′(y^0a, p_1a),

p2a^′ = p2a^′(y^0a, y^1a, p1a, p2a), . . . ,

pka^′ =pka^′(y^0a, y^1a, . . . , y^(k−1)a, p1a, p2a, . . . , pka).

(1.5)

Theorem 1.2. The following relation is valid:

d^A_t B_a^a′ = (∂ad^A_t ⁻¹B_b^a′)y^1b+

A−1 1

(∂ad^A_t⁻²B_b^a′)y^2b+· · · +

A−1 A−2

(∂ad¹_tB_b^a′)y^(A−1)b+

A−1 A−1

(∂aB^a_b^′)y^Ab =∂ay^Aa′ (1.6)

for A= 1, k.

Proof. From the relations

(d¹_tB_a^a′) = (∂bB^a_a^′)y^1b = (∂aB_b^a′)y^1b, d^A_t B_a^a′ =d^A_t⁻¹(d¹_tB_a^a′) =d^A_t⁻¹[(∂aB^a_b^′)y^1b]

and the Leibniz rule for differentiation the first part of (1.6) follows. As y^0b = x^b, y^1b, . . . , y^Ab are independent variables, from the right hand side of (1.6) we can take out ∂a and the comparison with the obtained equation with y^Aa′ from

(1.3) results in the second part of (1.6).

Theorem 1.3. The partial derivatives of the variables, d^A_t B_a^a′, d^α_tB_a^a′ are con- nected by the following formulae:

∂0ay^0a′ =∂1ay^1a′ =· · ·=∂kay^ka′ =B_a^a′

∂ay^Aa′ =∂0ay^Aa′ =d^A_t B^a_a^′ A= 1, k,

∂Aay^(A+B)a′ = A+B

A ∂_(A−1)ay^{(A+B−1)a′} =· · ·

=

A+B A

d^B_t B_a^a′ =

A+B A

∂_0ay^Ba′,

∂^1ap1a^′ =∂^2ap2a^′ =· · ·∂^kapka^′ =B^a_a′,

∂^αa = ∂

∂pαa

α= 1, k,

∂^αap_(α+β)a′ =

α+β−1 α−1

∂^1ap_(β+1)a′ =

α+β−1 α−1

d^β_tB_a^a′. (1.7)

Proof. From (1.4) it is obvious that d^A_t B^a_a^′ A = 0, k are functions only of y^0a, y^1a, . . . , y^Aa so d^A_t B_a^a′ are functions only of y^0a′, y^1a′, . . . , y^Aa′. From this and the second part of (1.3) we can conclude that pαa^′ are linear functions of p_1a, p_2a, . . . , pαa α= 1, k. This fact results in the following equations:

∂^1ap_1a′ =B^a_a′,

∂^1ap_2a′ = 1

0

d¹_tB_a^a′, ∂^2ap_2a′ = 1

1

B_a^a′,

∂^1ap3a^′ = 2

0

d²_tB_a^a′, ∂^2ap3a^′ = 2

1

d¹_tB_a^a′, ∂^3ap3a^′ = 2

2

B^a_a′, . . . ,

∂^1apαa^′ =

α−1 0

d^α_t⁻¹B_a^a′, ∂^2apαa^′ =

α−1 1

d^α_t⁻²B^a_a′, . . . ,

∂^αapαa^′ =

α−1 α−1

B_a^a′, . . . .

The above equations are the last three equations from (1.7).

Using (1.2) and (1.5) we can write (1.3a) in the form y^1a′ = (∂_0ay^0a′)y^1a =B_a^a′y^1a =d¹_ty^0a′

y^2a′ = (∂_0ay^1a′)y^1a+ (∂_1ay^1a′)y^2a =d¹_ty^1a′

y^3a′ = (∂0ay^2a′)y^1a+ (∂1ay^2a′)y^2a+ (∂2ay^2a′)y^3a =d¹_ty^2a′, . . . ,

y^Aa′ = (∂0ay^(A−1)a′)y^1a+ (∂1ay^(A−1)a′)y^2a+· · ·+ (∂_(A−1)ay^(A−1)a′)y^Aa

= d¹_ty^(A−1)a′, . . . ,

y^ka′ = (∂0ay^(k−1)a′)y^1a+ (∂1ay^(k−1)a′)y^2a+· · ·+ (∂_(k−1)ay^(k−1)a′)y^ka

= d¹_ty^(k−1)a′. (1.8)

If we compare y^1a′, y^2a′, . . . , y^Aa′, . . . , y^ka′ from (1.3a) and (1.8) we get y^1a′ :∂_0ay^0a′ =B_a^a′

y^2a′ : (∂0ay^1a′) = 1

0

d¹_tB_a^a′,

∂1ay^1a′ = 1

1

B_a^a′,

y^Aa′ : ∂_0ay^(A−1)a′ =

A−1 0

d^A_t ⁻¹B_a^a′,

∂_1ay^(A−1)a′ =

A−1 1

d^A_t ⁻²B_a^a′, . . .

∂_(A−1)ay^(A−1)a =

A−1 A−1

B_a^a′, . . .

y^ka′ : ∂_0ay^(k−1)a′ =

k−1 0

d^k_t⁻¹B_a^a′,

∂_1ay^(k−1)a′ =

k−1 1

d^k_t⁻²B_a^a′, . . .

∂_(k−1)ay^(k−1)a =

k−1 k−1

B^a_a^′.

The above equations are the explicit form of the first three equations from

(1.7).

From (1.5) it can be seen that it is reasonable to calculate∂Aapαa forA−1≤ α. We have

Theorem 1.4. The following relation is valid:

(1.9) ∂αap_(α+β)a′ =

α+β−1 β−1

∂_0apβa^′. Proof. From (1.5) we have p_1a^′ =p_1a^′(y^0a, p_1a) and

p_2a′ =d¹_tp_1a′ = (∂0ap_1a′)y^1a+ (∂^1ap_1a′)p2a =∂_0a(p1a^′y^1a) +B_a^a′p_2a, p3a^′ =d¹_tp2a^′ =∂0a

1 1

p2a^′y^1a+ 1

0

p1a^′y^2a

+ 1

0

d¹_tB^a_a′p_2a+ 1

1

B^a_a′p_3a, and from

p_2a^′ =p_2a^′(y^0a, y^1a, p_1a, p_2a) we get

p_3a′ = (∂_0ap_2a′)y^1a+ (∂_1ap_2a′)y^2a+ (∂^1ap_2a′)p_2a+ (∂^2ap_2a′)p_3a. The comparison of the two expressions forp_3a′ gives

y^2a :∂_1ap_2a′ = 1

0

∂_0ap_1a; p_2a :∂^1ap_2a′ = 1

0

d¹_tB_a^a′. Further we have

p_4a′ =d²_tp_2a′ =∂_0a 2

2

p_3a′y^1a+ 2

1

p_2a′y^2a+ 2

0

p_1a′y^3a

+ 2

0

(d²_tB_a^a)p2a+ 2

1

(d²_tB_a^a′)p3a + 2

2

B_a^a′p_4a, p_4a′ =d¹_tp_3a′ = (∂0ap_3a′)y^1a+ (∂1ap_3a′)y^2a+ (∂2ap_3a′)y^3a+

(∂^1ap_3a′)p2a+ (∂^2ap_3a′)p3a+ (∂^3ap_3a′)p4a.

The comparison of two above equations gives:

y^2a :∂_1ap_3a′ = 2

1

∂_0ap_2a′ y^3a :∂_2ap_3a′ = 2

0

∂_0ap_1a′

p₂:∂^1ap_3a′ = 2

0

d²_tB_a^a′ p_3a :∂^2ap_3a′ = 2

1

d¹_tB_a^a′a^′, p_4a :∂^3ap_3a^′ =

2 2

B_a^a′.

In the similar way comparing the relations p_αa′ = ∂_0ad^α_t⁻²(p1a^′y^1a) +d^α_t⁻²(B^a_a′p_2a),

pαa^′ = d¹_tp_(α−1)a^′(y^0a, y^1a, . . . , y^(α−2)a, p_1a, p_2a, . . . , p_(α−1)a)

we obtain (1.9).

2. The natural and special adapted bases in T(GLH)^(nk) and T^∗(GLH)^(nk)

The natural basis, ¯B_LH ofT(GLH)^(nk) as usual consists of partial derivatives of variables, i.e. ¯B_LH= {∂0a, ∂_1a, . . . , ∂ka, ∂^1a, ∂^2a, . . . , ∂^ka},

(2.1) ∂_0a = ∂_a = ∂

∂x^a = ∂

∂y^0a, ∂_Aa= ∂

∂y^Aa, A= 1, k, ∂^αa = ∂

∂pαa

, α= 1, k.

Theorem 2.1. The elements of B¯_LH are transforming in the following way:

∂_0a = (∂0ay^0a′)∂0a^′ + (∂0ay^1a′)∂1a^′+· · ·+ (∂0ay^ka′)∂ka^′

+ (∂0ap_1a′)∂^1a′ + (∂0ap_2a′)∂^2a′+· · ·+ (∂0ap_ka′)∂^ka′,

∂1a = (∂1ay^1a′)∂1a^′ + (∂1ay^2a′)∂2a^′+· · ·+ (∂1ay^ka′)∂ka^′

+ (∂1ap2a^′)∂^2a′ + (∂1ap3a^′)∂^3a′+· · ·+ (∂1apka^′)∂^ka′,

∂2a = (∂2ay^2a′)∂2a^′ + (∂2ay^3a′)∂3a^′+· · ·+ (∂2ay^ka′)∂ka^′

+ (∂_2ap_3a^′)∂^3a′ +· · ·+ (∂_2apka^′)∂^ka′, . . .

∂ka = (∂kay^ka′)∂ka^′

(2.2a)

∂^1a = (∂^1ap_1a′)∂^1a′+ (∂^1ap_2a′)∂^2a′ +· · ·+ (∂^1ap_ka′)∂^ka′,

∂^2a = (∂^2ap_2a′)∂^2a′+ (∂^2ap_3a′)∂^3a′ +· · ·+ (∂^2ap_ka′)∂^ka′,

∂^3a = (∂^3ap_3a′)∂^3a′+· · ·+ (∂^3ap_ka′)∂^ka′, . . .

∂^ka= (∂^kapka^′)∂^ka′. (2.2b)

The proof follows from (1.5).

Let us introduce the notations

(2.3) [∂Aay]1,k+1 = [∂0a∂_1a. . . ∂_ka], [∂^αap]1,k = [∂^1a∂^2a. . . ∂^ka]

(2.4) [A^a_a^′]k+1,k+1 =















∂_0ay^0a′ 0 0 · · · 0

∂0ay^1a′ ∂1ay^1a′ 0 · · · 0

∂_0ay^2a′ ∂_1ay^2a′ ∂_2ay^2a′ · · · 0 ...

∂0ay^ka′ ∂1ay^ka′ ∂2ay^ka′ · · · ∂kay^ka′















(2.5) [Baa^′]k,k+1 =















∂_0ap_1a′ 0 0 · · · 0 0

∂0ap2a^′ ∂1ap2a^′ 0 · · · 0 0

∂_0ap_3a′ ∂_1ap_3a′ ∂_2ap_3a′ · · · 0 0 ...

∂_0apka^′ ∂_1apka^′ ∂_2apka^′ · · · ∂_(k−1)apka^′ 0















[0] = [0]_k+1,k

(2.6) [C_a^a′]k,k =











∂^1ap_1a′ 0 · · · 0

∂^1ap_2a′ ∂^2ap_2a′ · · · 0 ...

∂^1apka^′ ∂^2apka^′ · · · ∂^kapka^′









 .

Using the above notations (2.2) can be written in the form:

(2.7) [∂Aay∂^αap]_1,2k+1 = [∂a^′y∂^a′p]_1,2k+1

A^a_a^′ 0 Baa^′ C_a^a′

2k+1,2k+1

Using (1.7) and (1.9) the elements of matrices [A^a_a^′], [Baa^′], [C_a^a′] can be written in the form

(2.8) [A^a_a^′]_k+1,k+1 =



























0 0

B^a_a^′ 0 0 0 · · · 0

1 0

d¹_tB_a^{a′ 1}₁

B_a^a′ 0 0 · · · 0

2 0

d²_tB_a^{a′ 2}₁

d¹_tB_a^{a′ 2}₂

B_a^a′ 0 · · · 0

3 0

d³_tB_a^{a′ 3}₁

d²_tB_a^{a′ 3}₂

d¹_tB_a^{a′ 3}₃

B_a^a′ · · · 0 ...

k 0

d^k_tB^a_a^{′ k}₁

d^k_t⁻¹B_a^{a′ k}₂

d^k_t⁻²B_a^a′ · · · ^k_k B_a^a′



























[Baa^′]k,k+1 =

=





















∂0ap1a^′ 0 0 0· · · 0 0

∂0ap2a^′ 1 0

∂0ap1a^′ 0 0 0 0

∂0ap3a^′ 2 1

∂0ap2a^′ 2 0

∂0ap1a^′ 0 0 0 ...

∂_0apka^′ k−1 k−2

∂_0ap_(k−1)a^′ k−1 k−3

∂_0ap_(k−2)a^′ · · · ^k−₀¹

∂_0ap_1a^′ 0



















 (2.9)

(2.10) [C_a^a′]k,k =





















0 0

B^a_a′ 0 0 · · · 0

1 0

d¹_tB_a^a′

1 1

B^a_a′ 0 · · · 0

2 0

d²_tB_a^a′

2 1

d¹_tB_a^a′

2 2

B^a_a′ 0 ...

k−1 0

d^k_t⁻¹B_a^a′

k−1 1

d^k_t⁻²B_a^a′

k−1 2

d^k_t⁻³B_a^a′

k−1 k−1

B_a^a′





















The natural basis of T^∗(GLH)^(nk) is

B¯^∗_LH={dy^0a, dy^1a, . . . , dy^ka, dp_1a, dp_2a, . . . , dp_ka}.

Theorem 2.2. The elements of B¯_LH^∗ are transforming in the following way:

dy^0a′ = (∂0ay^0a′)dy^0a

dy^1a′ = (∂0ay^1a′)dy^0a+ (∂1ay^1a′)dy^1a, . . . ,

dy^ka′ = (∂0ay^ka′)dy^0a+ (∂1ay^ka′)dy^1a+· · ·+ (∂kay^ka′)dy^ka, dp_1a′ = (∂0ap_1a′)dy^0a + (∂^1ap_1a′)dp1a,

dp_2a′ = (∂0ap_2a′)dy^0a + (∂1ap_2a′)dy^1a+ (∂^1ap_2a′)dp1a+ (∂^2ap_2a′)dp2a, ...

dp_ka′ = (∂0ap_ka′)dy^0a+ (∂1ap_ka′)dy^1a +· · ·+ (∂_(k−1)ap_ka′)dy^(k−1)a+ (∂^1apka^′)dp1a+· · ·+ (∂^kapka^′)dpka.

(2.11)

Let us introduce the notations

(2.12) [dy^a]k+1,1=









 dy^0a dy^1a ... dy^ka











, [dpa]k,1 =









 dp1a

dp_2a ... dpka









 .

Using notations (2.8), (2.9) and (2.10) we can write (2.11) in the form (2.13)

dy^a′ dp_a′

2k+1,1

=

A^a_a^′ 0 B_aa′ C_a^a′

2k+1,2k+1

dy^a dpa

.

Theorem 2.3. If the bases B¯_LH^∗ and B¯LH are dual to each other, then the bases B¯_LH^∗′ and B¯^′_LH are also dual to each other.

Proof. Under duality of two bases we understand as usual the following relations hdy^Aa, ∂Bbi= δ_B^Aδ_b^a hdpAa, ∂^Bbi=δ_A^Bδ_a^b

hdy^Aa, ∂^Bbi= 0 hdpAa, ∂Bbi= 0 or

(2.14)

dy^a dpa

[∂by∂^bp] =I_2k+1,2k+1.

We want to prove that if (2.14) is valid, then the same relation is valid if a is everywhere substituted by a^′, i.e., that (2.14) is coordinate invariant. If we introduce the notation

T =

A^a_a^′ 0 Baa^′ C_a^a′

, then (2.7) and (2.13) can be written in the form

[∂by∂^bp] = [∂b^′y∂^b′p]T (2.15)

dy^a′ dpa^′

=T dy^a

dp_a

⇒ dy^a

dp_a

= T⁻¹ dy^a′

dpa^′

(2.16) .

The substitution of (2.15) and (2.16) into (2.14) results in T⁻¹

dy^a′ dp_a′

[∂b^′y∂^b′p]T =I ⇒

dy^a′ dp_a′

[∂b^′y∂^b′p] =T IT⁻¹=I.

From (2.2) and (2.11) it is obvious that under coordinate transformation (1.3) the elements of natural bases ¯B_LHand ¯B_LH^∗ are not transforming as tensors. Now we shall construct a new so-called special adapted bases BLH and B_LH^∗ , whose elements transform as tensors and in which theJ structure (which will be defined later) has the same components as in the natural bases.

Definition 2.1. The special adapted basis B_LH ofT(GLH)^(nk) (2.17) B_LH={δ0a, δ_1a, . . . , δ_ka, δ^1a, δ^2a, . . . , δ^ka}

is defined by δ_0a =

0 0

∂_0a − 1

0

N_0a^1b∂_1b− · · · − k

0

N_0a^kb∂kb

− 0

0

N_0a1b∂^1b− 1

0

N_0a2b∂^2b− · · · −

k−1 0

N_0akb∂^kb δ_1a =

1 1

∂_1a − 2

1

N_0a^1b∂_2b− · · · − k

1

N_0a^(k−1)b∂_kb

− 1

1

N0a1b∂^2b− 2

1

N0a2b∂^3b− · · · −

k−1 1

N_0a(k−1)b∂^kb δ_2a =

2 2

∂_2a − 3

2

N_0a^1b∂_3b− · · · − k

2

N_0a^(k−2)b∂kb

− 2

2

N_0a1b∂^3b− · · · −

k−1 2

N_0a(k−2)b∂^kb, . . . δ_ka =

k k

∂_ka (2.18a)

δ^1a = 0

0

∂^1a− 1

0

N_2b^0a∂^2b− 2

0

N_3b^0a∂^3b− · · · − k

0

N_kb^0a∂^kb δ^2a =

1 1

∂^2a− 2

1

N_2b^0a∂^3b− · · · −

k−1 1

N_(k^0a−1)b∂^kb. . . δ^kb=

k−1 k−1

∂^kb. (2.18b)

Let us introduce the notations

[δAa(y)]_1,k+1 = [δ_0aδ_1a. . . δka], A= 0, k [δ^αa(p)]1,k = [δ^1aδ^2a. . . δ^ka], α= 1, k

[N_0a^Bb]_k+1,k+1 =

















0 0

δ_a^b 0 · · · 0 0

− ¹₀

N_0a^{1b 1}₁

δ_a^b · · · 0 0

... ... ...

− ^k₀

N_0a^kb− ^k₁

N_0a^(k−1)b · · · _k−^k₁ N_0a^1b δ_a^b

















[N0aβb]k,k+1 =















− ⁰₀

N0a1b 0 · · · 0 0

− ¹₀

N_0a2b − ¹₁

N_0a1b · · · 0 0

... ...

− ^k−₀¹

N_0akb − ^k−₁¹

N_0a(k−1)b · · · ^k_k⁻₋¹₁

N_0a1b 0













 (2.19a)