ZERMELLO’S CONDITIONS AND ENERGIES OF HIGHER ORDER IN GENERALIZED LAGRANGE-HAMILTON SPACES

Vol. 40, No. 1, 2010, 57-76

ZERMELLO’S CONDITIONS AND ENERGIES OF HIGHER ORDER IN GENERALIZED

LAGRANGE-HAMILTON SPACES

Irena ˇComi´c¹

Abstract. Many significant geometers contributed to the generalization of Riemann spaces in different directions. An almost complete list of them can be found in Miron’s books. Here are mention [1], [2], [17] and [18]

in which Hamilton and Finsler spaces are examined, further [3–9], where generalized Hamilton spaces are studied; [10] is most connected with the subject, and in [11–16] this problem also appears. Zermello’s condition in Miron’sOsc^kM was examined in [6]. Here, the Zermello’s conditions are given in Lagrange-Hamilton spaces, introduced in [9] and presented at the Workshop on Finsler Geometry 2009, Debrecen. It is proved that for a fundamental function for which the Zermello’s conditions are satisfied all energies of higher order are equal to zero.

AMS Mathematics Subject Classification (2000): 53B40, 53C60

Key words and phrases: Lagrange-Hamilton spaces, Zermello’s conditions

1. Some invariants in generalized Lagrange-Hamilton spaces

Group of coordinate transformations. Generalized Lagrange-Hamilton spaces are introduced in [9]. We shall recall only those parts of which are necessary for the understanding of Zermello’s condition in these spaces.

Let us denote by (LH)^(nk)the (2k+ 1)ndimensionalC^∞manifold in which a point (y, p) = (x=y⁽⁰⁾, y⁽¹⁾, y⁽²⁾, . . . , y^(k), p₍₁₎, p₍₂₎, . . . , p_(k)) has the coordi- nates

(x^a =y^0a, y^1a, y^2a, . . . , y^ka, p1a, p2a, . . . , pka), a= 1, n.

Some curve c ∈ (LH)^(nk) is given by c : t ∈ [a, b] → c(t) ∈ (LH)^(nk). A point (y, p)∈c(t) has the coordinates (x^a(t) =y^0a(t), y^1a(t), . . . , y^ka(t), p_1a(t), . . . , pka(t)), where

y^Aa(t) =d^A_ty^0a(t) A= 1, k, d^A_t = d^A dt^A, (1.1)

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

1Faculty of Technical Sciences, 21000 Novi Sad, Serbia, e-mail: [email protected], http://imft.ftn.uns.ac.rs/~irena/

The allowable coordinate transformations are given by x^a0 =x^a0(x^a)⇔x^a=x^a(x^a0)

(1.2)

y^1a0 =B^aa⁰y^1a, Ba^a0 =∂0ax^a0=∂ax^a0, ∂Aa= ∂

∂y^Aa A= 0, k,

y^2a0 = Ã

1 0

!

(d¹tBa^a0)y^1a+ Ã

1 1

!

Ba^a0y^2a=d¹t(B^aa⁰y^1a),

y^3a0 = Ã

2 0

!

(d²tBa^a0)y^1a+ Ã

2 1

!

(d¹tB^aa⁰)y^2a+ Ã

2 2

!

Ba^a0y^3a=d²t(Ba^a0y^1a), . . . ,

y^Aa0 = Ã

A−1 0

!

(d^A−1t B^aa⁰)y^1a+ Ã

A−1 1

!

(d^A−2t Ba^a0)y^2a+· · ·

· · ·+ Ã

A−1 A−1

!

Ba^a0y^Aa=d^A−1_t (B^aa⁰y^1a), . . . ,

y^ka0 = Ã

k−1 0

!

(d^k−1_t B^aa⁰)y^1a+ Ã

k−1 1

!

(d^k−2_t Ba^a0)y^2a+· · ·

· · ·+ Ã

k−1 k−1

!

Ba^a0y^ka=d^k−1t (Ba^a0y^1a),

p1a⁰ =B^a_a0p1a B_a^a0 =∂0a⁰x^a= ∂x^a

∂x^a0 =B_a^a0(t),

p2a⁰ = Ã

1 0

!

(d¹tBa^a0)p1a+ Ã

1 1

!

Ba^a0p2a=d¹t(B^aa⁰p1a),

p3a⁰ = Ã

2 0

!

(d²tBa^a0)p1a+ Ã

2 1

!

(d¹tB^aa⁰)p2a+ Ã

2 2

!

Ba^a0p3a=d²t(Ba^a0p1a),

p4a⁰ = Ã

3 0

!

(d³tBa^c0)p1c+ Ã

3 1

!

(d²tB^ca⁰)p2c+ Ã

3 2

!

(d¹tBa^c0)p3c+ Ã

3 3

!

B^ca⁰p4c, . . . ,

pαa⁰ = Ã

α−1 0

!

(d^α−1t Ba^a0)p1a+ Ã

α−1 1

!

(d^α−2t Ba^a0)p2a+· · ·

· · ·+ Ã

α−1 α−1

!

B^a_a0pαa, . . . ,

pka⁰ = Ã

k−1 0

!

(d^k−1t B^aa⁰)p1a+ Ã

k−1 1

!

(d^k−2t Ba^a0)p2a+· · ·

· · ·+ Ã

k−1 k−1

! B^a_a0pka.

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

Definition 1.1. The generalized Lagrange-Hamilton space of order k, (GLH)^(nk), is an(LH)^(nk)space, where the group of allowable transformations is given by (1.2) and in which a fundamental function F(x, y⁽¹⁾, y⁽²⁾, . . . , y^(k), p₍₁₎, p₍₂₎, . . . , p_(k))is given, whereF :U →Ris differentiable onU˜ (where rank [y^1a] = 1, rank [p1a] = 1) and continuous in those points of U, where y^1a and p1a are equal to zero, U is a domain in (GLH)^(nk).

The natural and special adapted bases in T(GLH)^(nk) and T^∗(GLH)^(nk). The natural basis, ¯BLH of T(GLH)^(nk) as usual consists of partial derivatives of variables, i.e.

B¯_LH ={∂_0a, ∂_1a, . . . , ∂_ka, ∂^1a, ∂^2a, . . . , ∂^ka}, ∂_0a =∂_a= ∂

∂x^a = ∂

∂y^0a, (1.3)

∂Aa = ∂

∂y^Aa A= 1, k, ∂^αa= ∂

∂pαa, α= 1, k.

Theorem 1.2. The elements of B¯_LH transform in the following way:

(1.4)

∂0a=(∂0ay^0a0)∂0a⁰+(∂0ay^1a0)∂1a⁰+(∂0ay^2a0)∂2a⁰+(∂0ay^3a0)∂3a⁰+· · ·+(∂0ay^ka0)∂ka⁰+ (∂0ap1a⁰)∂^1a0+(∂0ap2a⁰)∂^2a0+(∂0ap3a⁰)∂^3a0+· · ·+(∂0apka⁰)∂^ka0,

∂1a= (∂1ay^1a0)∂1a⁰+(∂1ay^2a0)∂2a⁰+(∂1ay^3a0)∂3a⁰+· · ·+(∂1ay^ka0)∂ka⁰+ (∂1ap2a⁰)∂^2a0+(∂1ap3a⁰)∂^3a0+· · ·+(∂1apka⁰)∂^ka0,

∂2a= (∂2ay^2a0)∂2a⁰+(∂2ay^3a0)∂3a⁰+· · ·+(∂2ay^ka0)∂ka⁰+ (∂2ap3a⁰)∂^3a0+· · ·+(∂2apka⁰)∂^ka0, ...

∂ka= (∂kay^ka0)∂ka⁰

∂^1a= (∂^1ap1a⁰)∂^1a0+(∂^1ap2a⁰)∂^2a0+(∂^1ap3a⁰)∂^3a0+· · ·+(∂^1apka⁰)∂^ka0,

∂^2a= (∂^2ap2a⁰)∂^2a0+(∂^2ap3a⁰)∂^3a0+· · ·+(∂^2apka⁰)∂^ka0,

∂^3a= (∂^3ap3a⁰)∂^3a0+· · ·+(∂^3apka⁰)∂^ka0,

...

∂^ka= (∂^kapka⁰)∂^ka0.

The natural basis ofT^∗(GLH)^(nk)is

B¯_LH^∗ ={dy^0a, dy^1a, . . . , dy^ka, dp1a, dp2a, . . . , dpka}.

Theorem 1.3. The elements of B¯_LH^∗ transform in the following way:

(1.5)

dy^0a0 = (∂0ay^0a0)dy^0a

dy^1a0 = (∂0ay^1a0)dy^0a+ (∂1ay^1a0)dy^1a, . . . ,

dy^ka0 = (∂_0ay^ka0)dy^0a+ (∂_1ay^ka0)dy^1a+· · ·+ (∂_kay^ka0)dy^ka, dp1a⁰ = (∂0ap1a⁰)dy^0a+ (∂^1ap1a⁰)dp1a,

dp2a⁰ = (∂0ap2a⁰)dy^0a+ (∂1ap2a⁰)dy^1a+ (∂^1ap2a⁰)dp1a+ (∂^2ap2a⁰)dp2a, . . . ,

dpka⁰ = (∂0apka⁰)dy^0a+ (∂1apka⁰)dy^1a+· · ·+ (∂_(k−1)apka⁰)dy^(k−1)a+ (∂^1apka⁰)dp1a+· · ·+ (∂^kapka⁰)dpka.

Definition 1.2. The special adapted basis BLH of T(GLH)^(nk)

(1.6) BLH ={δ0a, δ1a, . . . , δka, δ^1a, δ^2a, . . . , δ^ka}

is defined by (1.7)

δ0a=¡₀

0

¢∂0a− ¡₁

0

¢N_0a^1b∂1b − ¡₂

0

¢N_0a^2b∂2b − ¡₃

0

¢N_0a^3b∂3b − · · · − ¡_k

0

¢N_0a^kb∂kb

−¡₀

0

¢N0a1b∂^1b−¡₁

0

¢N0a2b∂^2b−¡₂

0

¢N0a3b∂^3b− · · · − ¡_k−1

0

¢N0akb∂^kb

δ1a= ¡₁

1

¢∂1a − ¡₂

1

¢N0a^1b∂2b − ¡₃

1

¢N0a^2b∂3b − · · · − ¡_k

1

¢N_0a^(k−1)b∂kb

−¡₁

1

¢N0a1b∂^2b−¡₂

1

¢N0a2b∂^3b− · · · − ¡_k−1

1

¢N0a(k−1)b∂^kb

δ2a= ¡₂

2

¢∂2a − ¡₃

2

¢N0a^1b∂3b − · · · − ¡_k

2

¢N_0a^(k−2)b∂kb

−¡₂

2

¢N0a1b∂^3b− · · · − ¡_k−1

2

¢N0a(k−2)b∂^kb

δ3a= ¡₃

3

¢∂3a − · · · − ¡_k

3

¢N_0a^(k−3)b∂kb

− · · · −¡_k−1

3

¢N0a(k−3)b∂^kb, . . . ,

δka= ¡_k

k

¢∂ka

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−1)b^0a ∂^kb

δ^3b= ¡₂

2

¢∂^3b − · · · − ¡_k−2

2

¢N_(k−2)b^0a ∂^kb, . . . ,

δ^kb= ¡_k−1

k−1

¢∂^kb.

Definition 1.3. The special adapted basisB^∗_LH of T^∗(GLH)^(nk) is (1.8) B_LH^∗ ={δy^0a, δy^1a, . . . , δy^ka, δp1a, δp2a, . . . , δpka}, where

δy^0a=dy^0a=dx^a (1.9)

δy^1a= µ1

1

¶ dy^1a+

µ1 0

¶

M_0b^1ady^0b

δy^2a= µ2

2

¶ dy^2a+

µ2 1

¶

M_0b^1ady^1b+ µ2

0

¶

M_0b^2ady^0b,

δy^3a= µ3

3

¶ dy^3a+

µ3 2

¶

M_0b^1ady^2b+ µ3

1

¶

M_0b^2ady^1b+ µ3

0

¶

M_0b^3ady^0b, . . . ,

δy^ka= µk

k

¶ dy^ka+

µ k k−1

¶

M_0b^1ady^(k−1)b+ µ k

k−2

¶

M_0b^2ady^(k−2)b+· · ·

· · ·+ µk

0

¶

M_0b^kady^0b,

δp1a= µ0

0

¶

M0a1bdy^0b+ µ0

0

¶ dp1a,

δp2a= µ1

0

¶

M0a2bdy^0b+ µ1

1

¶

M0a1bdy^1b+ µ1

0

¶

M_0a^1bdp1b+ µ1

1

¶ dp2a,

δp3a= µ2

0

¶

M0a3bdy^0b+ µ2

1

¶

M0a2bdy^1b+ µ2

2

¶

M0a1bdy^2b+ µ2

0

¶

M_0a^2bdp1b+ µ2

1

¶

M_0a^1bdp2b+ µ2

2

¶

dp3a, . . . ,

δpka= µk−1

0

¶

M0akbdy^0b+ µk−1

1

¶

M0a(k−1)bdy^1b+· · ·

· · ·+ µk−1

k−1

¶

M0a1bdy^(k−1)b+ µk−1

0

¶

M_0a^(k−1)bdp1b

+ µk−1

1

¶

M_0a^(k−2)bdp2b+· · ·+ µk−1

k−1

¶ dpka.

In [9], there are given the conditions for M’s and N’s such that the elements ofBLH andB_LH^∗ are tensors and when these bases are dual to each other.

The J¯structure in (GLH)^(nk)

Definition 1.4. The k-tangent structure J¯is aF linear mapping J¯:T^∗(GLH)^(nk)→T^∗(GLH)^(nk)

defined by

Jdy¯ ^0a= 0,Jdy¯ ^1a=dy^0a,Jdy¯ ^2a= 2dy^1a, . . . ,Jdy¯ ^ka=kdy^(k−1)a (1.10)

Jdp¯ 1a = 0,Jdp¯ 2a =dp1a,Jdp¯ 3a= 2dp2a, . . . ,Jdp¯ ka= (k−1)dp_(k−1)a, from which it follows

J¯ = dy^0a⊗∂_1a+ 2dy^1a⊗∂_2a+· · ·+kdy^(k−1)a⊗∂_ka+ (1.11)

dp1a⊗∂^2a+ 2dp2a⊗∂^3a+· · ·+ (k−1)dp(k−1)a⊗∂^ka. In [9], it is proved that the ¯J structure in the special adapted bases BLH

andB_LH^∗ is given by

J¯ = δy^0a⊗δ1a+2δy^1a⊗δ2a+3δy^2a⊗δ3a+· · ·+kδy^(k−1)a⊗δka+ (1.12)

δp1a⊗δ^2a+2δp2a⊗δ^3a+3δp3a⊗δ^4a+· · ·+(k−1)δp(k−1)a⊗δ^ka. Remark 1.1. From (1.11) and (1.12) it follows that thek-tangent structure J¯in the natural and special adapted bases has the same coordinates.

It is also proved that the following relations are valid

Jδy¯ ^0a = 0,Jδy¯ ^1a =δy^0a,Jδy¯ ^2a= 2δy^1a, . . . ,Jδy¯ ^ka=kδy^(k−1)a (1.13)

Jδp¯ 1a= 0,Jδp¯ 2a=δp1a,Jδp¯ 3a= 2δp2a, . . . ,Jδp¯ ka= (k−1)δp_(k−1)a, The Liouville vector field. If M(y^0a, y^1a, . . . , y^ka, p1a, p2a, . . . , pka) and M⁰(y^0a+dy^0a, y^1a+dy^1a, . . . , y^ka+dy^ka, p1a+dp1a, p2a+dp2a, . . . , pka+dpka) are two points in (GLH)^(nk), then the vector M M⁰ expressed in the natural basisT(GLH)^(nk)has the form [9]

M M⁰=dr = dy^0a∂_0a+dy^1a∂_1a+· · ·+dy^ka∂_ka+ (1.14)

dp1a∂^1a+dp2a∂^2a+· · ·+dpka∂^ka. It is proved ([9]) thatdris coordinate invariant, i.e.

dr = δy^0aδ0a+δy^1aδ1a+· · ·+δy^kaδka+ (1.15)

δp1aδ^1a+δp2aδ^2a+· · ·+δpkaδ^ka.

Definition 1.5. The Liouville vector fieldsΓ¯0,Γ¯1,Γ¯2, . . . ,Γ¯k are defined by (1.16)

Γ¯k=dr, J¯Γ¯A= ¯ΓAJ¯= (k−(A−1))¯ΓA−1, A¯= 1, k, J¯¯Γ0= ¯Γ0J¯= 0.

Remark 1.2. From (1.14)-(1.15) it is obvious thatdr has the same com- ponents in the natural and special adapted bases. The same property has the structure ¯J (see Remark 1.1). This fact allows that the action ¯J on drcan be written by the equations of the same form in both coordinate systems.

In (GLH)^(nk) it is difficult to construct vector fields, but using dr, the structure ¯J, one family of the Liouville vector field can, be constructed.

From (1.16) it follows

J¯Γ¯0= ¯Γ0J¯= 0, J¯¯Γ1= ¯Γ1J¯=kΓ¯0, J¯Γ¯2= ¯Γ2J¯= (k−1)¯Γ1, (1.17)

. . . ,J¯Γ¯k−1= ¯Γk−1J¯= 2¯Γk−2, J¯Γ¯k = ¯ΓkJ¯= ¯Γk−1.

Theorem 1.4. The Liouville vector fields Γ¯0,¯Γ1, . . . ,Γ¯k from (GLH)^(nk) ex- pressed in the special adapted basis B of T(GLH)^(nk), have the form [9]

(1.18) Γ¯0 =

µk 0

¶

δy^0aδka,

Γ¯1 = µk

1

¶

δy^1aδka+ µk−1

0

¶

δy^0aδ_(k−1)a+ µk−1

0

¶

δp1aδ^ka,

Γ¯2 = µk

2

¶

δy^2aδka+ µk−1

1

¶

δy^1aδ(k−1)a+ µk−2

0

¶

δy^0aδ(k−2)a+ µk−1

1

¶

δp2aδ^ka+ µk−2

0

¶

δp1aδ^(k−1)a,

Γ¯3 = µk

3

¶

δy^3aδka+ µk−1

2

¶

δy^2aδ(k−1)a+ µk−2

1

¶

δy^1aδ(k−2)a+ µk−3

0

¶

δy^0aδ_(k−3)a+ µk−1

2

¶

δp3aδ^ka+ µk−2

1

¶

δp2aδ^(k−1)a+ µk−3

0

¶

δp1aδ^(k−2)a, . . . ,

Γ¯k−1 = µ k

k−1

¶

δy^(k−1)aδka+ µk−1

k−2

¶

δy^(k−2)aδ(k−1)a+· · ·+ µ2

1

¶

δy^1aδ2a+ µ1

0

¶

δy^0aδ1a+ µk−1

k−2

¶

δp_(k−1)aδ^ka+ µk−2

k−3

¶

δp(k−2)aδ^(k−1)a+· · ·+ µ1

0

¶

δp1aδ^2a,

Γ¯k = µk

k

¶

δy^kaδka+ µk−1

k−1

¶

δy^(k−1)aδ_(k−1)A+· · ·+ µ1

1

¶

δy^1aδ1a+ µ0

0

¶

δy^0aδ0a+ µk−1

k−1

¶

δpkaδ^ka+ µk−2

k−2

¶

δp(k−1)aδ^(k−1)a+· · ·+ µ0

0

¶

δp1aδ^1a.

Theorem 1.5. The Liouville vector fields Γ¯0,Γ¯1, . . . ,Γ¯k in (GLH)^(nk) in the natural basis B¯ of T(GLH)^(nk) have the form obtained from (1.18) if δy^Aa, δpαa, δAa, δ^αa, are substituted by dy^Aa, dpαa, ∂Aa, ∂^αa respectively for ev- ery A= 0, k,α= 1, k.

Proof. The proof follows from Definition 1.5 and Remark 1.2.

2. The Zermello’s conditions

Letc^∗ be a curve in (GLH)^(nk)such that

c^∗:t∈[0,1]→x^a(t)∂a+d¹_tx^a(t)∂1a+· · ·+d^k_tx^a(t)∂ka+ (2.1)

p1a(t)∂^1a+· · ·+d^k−1_t p1a∂^ka=

=y^0a(t)∂0a+y^1a∂1a+· · ·+y^ka∂ka+ p1a∂^1a+· · ·+pka∂^ka andImc^∗⊂U.

The integral of actionIc^∗ is (2.2) I_c^∗=

Z1

0

F(x, y⁽¹⁾, y⁽²⁾, . . . , y^(k), p₍₁₎, p₍₂₎, . . . , p_(k))dt.

Ic^∗ does not depend on the parametrization of the curvec^∗: x^a=y^0a(t) =y^0a, y^Aa =d^A_tx^a =d^Ax^a

dt^A , A= 1, k, p1a =p1a(t), pαa =d^α−1_t p1a =d^α−1p1a

dt^α−1 α= 1, k if

Z1

0

F(x, y¹, y², . . . , y^k, p1, p2, . . . , pk)dt= (2.3)

= Z1

0

F(x, y¹⁰, y²⁰, . . ., y^k0, p1⁰, p2⁰, . . . , pk⁰)ds,

wheres=s(t) is at leastC^k function,s⁰(t)>0 fort∈[0,1],s(0) = 0,s(1) = 1 and

y^Aa0 =ds^Ax^a =d^Ax^a

ds^A , A= 1, k, (2.4)

pαa⁰ =ds^αp1a =d^α−1p1a

ds^α−1 , α= 1, k.

The equations, which give the conditions when (2.3) is satisfied are called Zer- mello’s conditions. The equality (2.3) will be satisfied if along the curvec^∗ we have

(2.5)

F(x, y¹, y², . . . , y^k, p1, p2, . . . , pk) =F(x, y¹⁰, y²⁰, . . . , y^k0, p1⁰, p2⁰, . . . , pk⁰)s⁰,

where s⁰ = ^ds_dt. To express (2.5) in invariant form we need some relations. We shall use the notation

s^(α)= d^αs

dt^α, α= 1, k.

Asx^a=x^a(s),s=s(t), we have y^1a = dx^a

ds s⁰=y^1a(s, s⁰), (2.6)

y^2a = ∂y^1a

∂s s⁰+∂y^1a

∂s⁰ s⁰⁰=y^2a(s, s⁰s⁰⁰), y^3a = ∂y^2a

∂s s⁰+∂y^2a

∂s⁰ s⁰⁰+∂y^2a

∂s⁰⁰s⁰⁰⁰ =y^3a(s, s⁰, s⁰⁰, s⁰⁰⁰), . . . , y^ka = ∂y^(k−1)a

∂s s⁰+∂y^(k−1)a

∂s⁰ s⁰⁰+· · ·+∂y^(k−1)a

∂s^(k−1) s^(k), p_1a = p_1a(s), s=s(t),

p_2a = ∂p1a

∂s s⁰=p_2a(s, s⁰), p3a = ∂p2a

∂s s⁰+∂p2a

∂s⁰ s⁰⁰=p3a(s, s⁰, s⁰⁰), . . . , pka = ∂p_(k−1)a

∂s s⁰+∂p_(k−1)a

∂s⁰ s⁰⁰+· · ·+∂p_(k−1)a

∂s^(k−2) s^(k−1)

= pka(s, s⁰. . . , s^(k−1)).

Using the notations (2.7) A^a₀ =dy^0a

ds =dx^a

ds =y^1a0, A^a_A= d^A

dt^AA^a₀=d^A_tA^a₀ A= 1, k−1,

(2.8) B_a¹=dp1a

ds =p2a⁰, B_a^α= d^α−1B_a¹

dt^α−1 =d^α−1_t B_a¹, α= 1, k−1 and the Leibniz rule for differentiation we can prove the following theorem.

Theorem 2.1. y^Aa,pαa ands^(α),A,α= 1, k are connected by formulae:

y^1a =A^a₀s⁰, (2.9)

y^2a = µ1

0

¶ A^a₁s⁰+

µ1 1

¶ A^a₀s⁰⁰,

y^3a = µ2

0

¶ A^a₂s⁰+

µ2 1

¶ A^a₁s⁰⁰+

µ2 2

¶

A^a₀s⁰⁰⁰, . . . ,

y^Aa=

µA−1 0

¶

A^a_A−1s⁰+

µA−1 1

¶

A^a_A−2s⁰⁰+· · ·+

µA−1 A−1

¶

A^a₀s^(A), . . . ,

y^ka= µk−1

0

¶

A^a_k−1s⁰+ µk−1

1

¶

A^a_k−2s⁰⁰+· · ·+ µk−1

k−1

¶ A^a₀s^(k), p1a =p1a(s), s=s(t)

p2a =B¹_as⁰ p3a =

µ1 0

¶ B_a²s⁰+

µ1 1

¶

B_a¹s⁰⁰, . . . ,

pαa= µα−2

0

¶

B_a^α−1s⁰+ µα−2

1

¶

B^α−2_a s⁰⁰+· · ·+ µα−2

α−2

¶

B_a¹s^(α−1), . . . ,

pka= µk−2

0

¶

B_a^k−1s⁰+ µk−2

1

¶

B^k−2_a s⁰⁰+· · ·+ µk−2

k−2

¶

B_a¹s^(k−1). Theorem 2.2. The following relations are valid:

(2.10) A^a_A=

µA−1 0

¶∂A^a_A−1

∂s s⁰+ µA−1

1

¶∂AA−2

∂s s⁰⁰+· · ·+

µA−1 A−1

¶∂A^a₀

∂s s^(A),

(2.11) A^a_A= dy^Aa

ds , A= 1, k−1,

(2.12) B_a^α= µα−2

0

¶∂B^α−1_a

∂s s⁰+ µα−2

1

¶∂B_a^α−2

∂s s⁰⁰+· · ·+ µα−2

α−2

¶

B¹_as^(α−1),

(2.13) B_a^α=dpαa

ds , α= 2, k−1.

Proof.

A^a_A = d^A_tA^a₀=d^A−1_t (d¹_tA^a₀) =d^A−1_t µ∂A^a₀

∂s s⁰

¶

=

d^A−1_t ∂

∂s(A^a₀s⁰) = ∂

∂sd^A−1_t (A^a₀s⁰) =

∂

∂s

·µA−1 0

¶

A^a_A−1s⁰+ µA−1

1

¶

A^a_A−2s⁰⁰+· · ·+

µA−1 A−1

¶ A^a₀s^(A)

¸ . If we take∂/∂sfrom the sum in the middle bracket we obtain (2.10), and if we substitutey^Aa from (2.9) we obtain (2.11). In the similar way we have

B_a^α = d^α−1_t B¹_a=d^α−2_t (d¹_tB_a¹) =d^α−2_t µ

d¹_tdp1a

ds

¶

=

d^α−2_t

· ∂

∂s µ∂p1a

∂s s⁰

¶¸

= ∂

∂sd^α−2_t (B_a¹s⁰) =

∂

∂s

·µα−2 0

¶

B^α−1_a s⁰+ µα−2

1

¶

B_a^α−2s⁰⁰+· · ·+ µα−2

α−2

¶

B_a¹s^(α−1)

¸ . If we take _∂s^∂ from the sum in the middle bracket, we obtain (2.12) and if we substitutepαa from (2.9) we obtain (2.13).

The explicit form of (2.6) as follows

y^1a =y^1a0s⁰, y^Aa0 =d^A_sy^0a(s), A= 1, k, (2.14)

y^2a =y^2a0(s⁰)²+y^1a0s⁰⁰,

y^3a =y^3a0(s⁰)³+y^2a03s⁰s⁰⁰+y^1a0s⁰⁰⁰,

y^4a =y^4a0(s⁰)⁴+y^3a06(s⁰)²s⁰⁰+y^2a0(3(s⁰⁰)²+ 4s⁰s⁰⁰⁰) +y^1a0s^0v, . . . , p1a =p1a(s), pαa⁰ =d^α−1_s p1a, α= 2, k,

p2a =p2a⁰s⁰,

p3a =p3a⁰s⁰²+p2a⁰s⁰⁰,

p4a =p4a⁰(s⁰)³+ 3p3a⁰s⁰s⁰⁰+p2a⁰s⁰⁰⁰,

p5a =p5a⁰(s⁰)⁴+p4a⁰6(s⁰²)s⁰⁰+p3a⁰(3(s⁰⁰)²+ 4s⁰s⁰⁰⁰) +p2a⁰s^0v, . . . . From (2.14) it follows

Theorem 2.3. The following relations are valid:

(2.15) ∂y^1a

∂s⁰ =∂y^2a

∂s⁰⁰ =· · ·= ∂y^ka

∂s^(k) =y^1a0 = dy^0a ds

(2.16) ∂y^Aa

∂s^(B) = A B

∂y^(A−1)a

∂s^(B−1) =· · ·= µA

B

¶∂y^(A−B)a

∂s

(2.17) ∂p2a

∂s⁰ = ∂p3a

∂s⁰⁰ =· · ·= ∂pka

∂s^(k−1) =dp1a

ds =p2a⁰

∂pαa

∂s^(β) = (α−1) β

∂p_(α−1)a

∂s^(β−1) =· · ·= µα−1

β

¶∂p_(α−β)a

∂s , (2.18)

k≥A≥B≥0, k≥α≥β+ 1≥1.

Now we return to the purpose of this examination.

If we take the partial derivatives of F given by (2.5) with respect to s⁰, s⁰⁰, . . . , s^(k), taking into account (2.6), we get

(2.19) (∂1aF)∂y^1a

∂s⁰ + (∂2aF)∂y^2a

∂s⁰ + (∂3aF)∂y^3a

∂s⁰ +· · ·+ (∂kaF)∂y^ka

∂s⁰ + (∂^2aF)∂p2a

∂s⁰ + (∂^3aF)∂p3a

∂s⁰ +· · ·+ (∂^kaF)∂pka

∂s⁰ =F(x, y¹⁰, . . . , y^k0, p1, . . . , pk⁰) (∂2aF)∂y^2a

∂s⁰⁰ + (∂3aF)∂y^3a

∂s⁰⁰ +· · ·+ (∂kaF)∂y^ka

∂s⁰⁰ + (∂^3aF)∂p3a

∂s⁰⁰ + (∂^4aF)∂p4a

∂s⁰⁰ +· · ·+ (∂^kaF)∂pka

∂s⁰⁰ = 0, . . . , (∂_(k−1)aF)∂y^(k−1)a

∂s^(k−1) + (∂kaF) ∂y^ka

∂s^(k−1)+ (∂^kaF) ∂pka

∂s^(k−1) = 0, (∂kaF)∂y^ka

∂s^(k) = 0.

On the left-hand side of (2.19) in all equationsF =F(x, y¹, . . . , y^k, p1, . . . , pk).

If in (2.19) we substitute (2.16) and (2.18) in the form

∂y^Aa

∂s^(B) = µA

B

¶∂y^(A−B)a

∂s =

µA B

¶

y^(A−B+1)adt ds

∂pαa

∂s^(β) = µα−1

β

¶∂p_(α−β)a

∂s =

µα−1 β

¶

p_(α−β+1)adt ds we obtain

[ µ1

1

¶

y^1a∂1a+ µ2

1

¶

y^2a∂2a+· · ·+ µk

1

¶

y^ka∂ka](F) + (2.20)

[ µ1

1

¶

p2a∂^2a+ µ2

1

¶

p3a∂^3a+· · ·+ µk−1

1

¶

pka∂^ka](F) =

F(x, y¹⁰, . . . , y^k0, p⁰₁, . . . , p⁰_k)ds

dt =F(x, y¹, . . . , y^k, p1, . . . , pk) =F {[

µ2 2

¶

y^1a∂2a+ µ3

2

¶

y^2a∂3a+· · ·+ µk

2

¶

y^(k−1)a∂ka] +