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’sOsckM 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
(xa =y0a, y1a, y2a, . . . , yka, 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 (xa(t) =y0a(t), y1a(t), . . . , yka(t), p1a(t), . . . , pka(t)), where
yAa(t) =dAty0a(t) A= 1, k, dAt = dA dtA, (1.1)
pαa(t) =dα−1t p1a(t), α= 1, k, dα−1t = dα−1 dtα−1.
1Faculty of Technical Sciences, 21000 Novi Sad, Serbia, e-mail: comirena@uns.ac.rs, http://imft.ftn.uns.ac.rs/~irena/
The allowable coordinate transformations are given by xa0 =xa0(xa)⇔xa=xa(xa0)
(1.2)
y1a0 =Baa0y1a, Baa0 =∂0axa0=∂axa0, ∂Aa= ∂
∂yAa A= 0, k,
y2a0 = Ã
1 0
!
(d1tBaa0)y1a+ Ã
1 1
!
Baa0y2a=d1t(Baa0y1a),
y3a0 = Ã
2 0
!
(d2tBaa0)y1a+ Ã
2 1
!
(d1tBaa0)y2a+ Ã
2 2
!
Baa0y3a=d2t(Baa0y1a), . . . ,
yAa0 = Ã
A−1 0
!
(dA−1t Baa0)y1a+ Ã
A−1 1
!
(dA−2t Baa0)y2a+· · ·
· · ·+ Ã
A−1 A−1
!
Baa0yAa=dA−1t (Baa0y1a), . . . ,
yka0 = Ã
k−1 0
!
(dk−1t Baa0)y1a+ Ã
k−1 1
!
(dk−2t Baa0)y2a+· · ·
· · ·+ Ã
k−1 k−1
!
Baa0yka=dk−1t (Baa0y1a),
p1a0 =Baa0p1a Baa0 =∂0a0xa= ∂xa
∂xa0 =Baa0(t),
p2a0 = Ã
1 0
!
(d1tBaa0)p1a+ Ã
1 1
!
Baa0p2a=d1t(Baa0p1a),
p3a0 = Ã
2 0
!
(d2tBaa0)p1a+ Ã
2 1
!
(d1tBaa0)p2a+ Ã
2 2
!
Baa0p3a=d2t(Baa0p1a),
p4a0 = Ã
3 0
!
(d3tBac0)p1c+ Ã
3 1
!
(d2tBca0)p2c+ Ã
3 2
!
(d1tBac0)p3c+ Ã
3 3
!
Bca0p4c, . . . ,
pαa0 = Ã
α−1 0
!
(dα−1t Baa0)p1a+ Ã
α−1 1
!
(dα−2t Baa0)p2a+· · ·
· · ·+ Ã
α−1 α−1
!
Baa0pαa, . . . ,
pka0 = Ã
k−1 0
!
(dk−1t Baa0)p1a+ Ã
k−1 1
!
(dk−2t Baa0)p2a+· · ·
· · ·+ Ã
k−1 k−1
! Baa0pka.
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 [y1a] = 1, rank [p1a] = 1) and continuous in those points of U, where y1a 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= ∂
∂xa = ∂
∂y0a, (1.3)
∂Aa = ∂
∂yAa A= 1, k, ∂αa= ∂
∂pαa, α= 1, k.
Theorem 1.2. The elements of B¯LH transform in the following way:
(1.4)
∂0a=(∂0ay0a0)∂0a0+(∂0ay1a0)∂1a0+(∂0ay2a0)∂2a0+(∂0ay3a0)∂3a0+· · ·+(∂0ayka0)∂ka0+ (∂0ap1a0)∂1a0+(∂0ap2a0)∂2a0+(∂0ap3a0)∂3a0+· · ·+(∂0apka0)∂ka0,
∂1a= (∂1ay1a0)∂1a0+(∂1ay2a0)∂2a0+(∂1ay3a0)∂3a0+· · ·+(∂1ayka0)∂ka0+ (∂1ap2a0)∂2a0+(∂1ap3a0)∂3a0+· · ·+(∂1apka0)∂ka0,
∂2a= (∂2ay2a0)∂2a0+(∂2ay3a0)∂3a0+· · ·+(∂2ayka0)∂ka0+ (∂2ap3a0)∂3a0+· · ·+(∂2apka0)∂ka0, ...
∂ka= (∂kayka0)∂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∗ ={dy0a, dy1a, . . . , dyka, dp1a, dp2a, . . . , dpka}.
Theorem 1.3. The elements of B¯LH∗ transform in the following way:
(1.5)
dy0a0 = (∂0ay0a0)dy0a
dy1a0 = (∂0ay1a0)dy0a+ (∂1ay1a0)dy1a, . . . ,
dyka0 = (∂0ayka0)dy0a+ (∂1ayka0)dy1a+· · ·+ (∂kayka0)dyka, dp1a0 = (∂0ap1a0)dy0a+ (∂1ap1a0)dp1a,
dp2a0 = (∂0ap2a0)dy0a+ (∂1ap2a0)dy1a+ (∂1ap2a0)dp1a+ (∂2ap2a0)dp2a, . . . ,
dpka0 = (∂0apka0)dy0a+ (∂1apka0)dy1a+· · ·+ (∂(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, δ1a, δ2a, . . . , δka}
is defined by (1.7)
δ0a=¡0
0
¢∂0a− ¡1
0
¢N0a1b∂1b − ¡2
0
¢N0a2b∂2b − ¡3
0
¢N0a3b∂3b − · · · − ¡k
0
¢N0akb∂kb
−¡0
0
¢N0a1b∂1b−¡1
0
¢N0a2b∂2b−¡2
0
¢N0a3b∂3b− · · · − ¡k−1
0
¢N0akb∂kb
δ1a= ¡1
1
¢∂1a − ¡2
1
¢N0a1b∂2b − ¡3
1
¢N0a2b∂3b − · · · − ¡k
1
¢N0a(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
¢N0a1b∂3b − · · · − ¡k
2
¢N0a(k−2)b∂kb
−¡2
2
¢N0a1b∂3b− · · · − ¡k−1
2
¢N0a(k−2)b∂kb
δ3a= ¡3
3
¢∂3a − · · · − ¡k
3
¢N0a(k−3)b∂kb
− · · · −¡k−1
3
¢N0a(k−3)b∂kb, . . . ,
δka= ¡k
k
¢∂ka
δ1a= ¡0
0
¢∂1a −¡1
0
¢N2b0a∂2b −¡2
0
¢N3b0a∂3b − · · · − ¡k
0
¢Nkb0a∂kb
δ2a= ¡1
1
¢∂2a −¡2
1
¢N2b0a∂3b − · · · − ¡k−1
1
¢N(k−1)b0a ∂kb
δ3b= ¡2
2
¢∂3b − · · · − ¡k−2
2
¢N(k−2)b0a ∂kb, . . . ,
δkb= ¡k−1
k−1
¢∂kb.
Definition 1.3. The special adapted basisB∗LH of T∗(GLH)(nk) is (1.8) BLH∗ ={δy0a, δy1a, . . . , δyka, δp1a, δp2a, . . . , δpka}, where
δy0a=dy0a=dxa (1.9)
δy1a= µ1
1
¶ dy1a+
µ1 0
¶
M0b1ady0b
δy2a= µ2
2
¶ dy2a+
µ2 1
¶
M0b1ady1b+ µ2
0
¶
M0b2ady0b,
δy3a= µ3
3
¶ dy3a+
µ3 2
¶
M0b1ady2b+ µ3
1
¶
M0b2ady1b+ µ3
0
¶
M0b3ady0b, . . . ,
δyka= µk
k
¶ dyka+
µ k k−1
¶
M0b1ady(k−1)b+ µ k
k−2
¶
M0b2ady(k−2)b+· · ·
· · ·+ µk
0
¶
M0bkady0b,
δp1a= µ0
0
¶
M0a1bdy0b+ µ0
0
¶ dp1a,
δp2a= µ1
0
¶
M0a2bdy0b+ µ1
1
¶
M0a1bdy1b+ µ1
0
¶
M0a1bdp1b+ µ1
1
¶ dp2a,
δp3a= µ2
0
¶
M0a3bdy0b+ µ2
1
¶
M0a2bdy1b+ µ2
2
¶
M0a1bdy2b+ µ2
0
¶
M0a2bdp1b+ µ2
1
¶
M0a1bdp2b+ µ2
2
¶
dp3a, . . . ,
δpka= µk−1
0
¶
M0akbdy0b+ µk−1
1
¶
M0a(k−1)bdy1b+· · ·
· · ·+ µk−1
k−1
¶
M0a1bdy(k−1)b+ µk−1
0
¶
M0a(k−1)bdp1b
+ µk−1
1
¶
M0a(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 andBLH∗ 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=dy0a,Jdy¯ 2a= 2dy1a, . . . ,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¯ = dy0a⊗∂1a+ 2dy1a⊗∂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
andBLH∗ is given by
J¯ = δy0a⊗δ1a+2δy1a⊗δ2a+3δy2a⊗δ3a+· · ·+kδy(k−1)a⊗δka+ (1.12)
δp1a⊗δ2a+2δp2a⊗δ3a+3δp3a⊗δ4a+· · ·+(k−1)δp(k−1)a⊗δka. 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 =δy0a,Jδy¯ 2a= 2δy1a, . . . ,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(y0a, y1a, . . . , yka, p1a, p2a, . . . , pka) and M0(y0a+dy0a, y1a+dy1a, . . . , yka+dyka, 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 = dy0a∂0a+dy1a∂1a+· · ·+dyka∂ka+ (1.14)
dp1a∂1a+dp2a∂2a+· · ·+dpka∂ka. It is proved ([9]) thatdris coordinate invariant, i.e.
dr = δy0aδ0a+δy1aδ1a+· · ·+δykaδka+ (1.15)
δp1aδ1a+δp2aδ2a+· · ·+δpkaδka.
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
¶
δy0aδka,
Γ¯1 = µk
1
¶
δy1aδka+ µk−1
0
¶
δy0aδ(k−1)a+ µk−1
0
¶
δp1aδka,
Γ¯2 = µk
2
¶
δy2aδka+ µk−1
1
¶
δy1aδ(k−1)a+ µk−2
0
¶
δy0aδ(k−2)a+ µk−1
1
¶
δp2aδka+ µk−2
0
¶
δp1aδ(k−1)a,
Γ¯3 = µk
3
¶
δy3aδka+ µk−1
2
¶
δy2aδ(k−1)a+ µk−2
1
¶
δy1aδ(k−2)a+ µk−3
0
¶
δy0aδ(k−3)a+ µk−1
2
¶
δp3aδka+ 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
¶
δy1aδ2a+ µ1
0
¶
δy0aδ1a+ µk−1
k−2
¶
δp(k−1)aδka+ µk−2
k−3
¶
δp(k−2)aδ(k−1)a+· · ·+ µ1
0
¶
δp1aδ2a,
Γ¯k = µk
k
¶
δykaδka+ µk−1
k−1
¶
δy(k−1)aδ(k−1)A+· · ·+ µ1
1
¶
δy1aδ1a+ µ0
0
¶
δy0aδ0a+ µk−1
k−1
¶
δpkaδka+ µk−2
k−2
¶
δp(k−1)aδ(k−1)a+· · ·+ µ0
0
¶
δp1aδ1a.
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 δyAa, δpαa, δAa, δαa, are substituted by dyAa, 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]→xa(t)∂a+d1txa(t)∂1a+· · ·+dktxa(t)∂ka+ (2.1)
p1a(t)∂1a+· · ·+dk−1t p1a∂ka=
=y0a(t)∂0a+y1a∂1a+· · ·+yka∂ka+ p1a∂1a+· · ·+pka∂ka andImc∗⊂U.
The integral of actionIc∗ is (2.2) Ic∗=
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∗: xa=y0a(t) =y0a, yAa =dAtxa =dAxa
dtA , A= 1, k, p1a =p1a(t), pαa =dα−1t p1a =dα−1p1a
dtα−1 α= 1, k if
Z1
0
F(x, y1, y2, . . . , yk, p1, p2, . . . , pk)dt= (2.3)
= Z1
0
F(x, y10, y20, . . ., yk0, p10, p20, . . . , pk0)ds,
wheres=s(t) is at leastCk function,s0(t)>0 fort∈[0,1],s(0) = 0,s(1) = 1 and
yAa0 =dsAxa =dAxa
dsA , A= 1, k, (2.4)
pαa0 =dsαp1a =dα−1p1a
dsα−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, . . . , yk, p1, p2, . . . , pk) =F(x, y10, y20, . . . , yk0, p10, p20, . . . , pk0)s0,
where s0 = dsdt. To express (2.5) in invariant form we need some relations. We shall use the notation
s(α)= dαs
dtα, α= 1, k.
Asxa=xa(s),s=s(t), we have y1a = dxa
ds s0=y1a(s, s0), (2.6)
y2a = ∂y1a
∂s s0+∂y1a
∂s0 s00=y2a(s, s0s00), y3a = ∂y2a
∂s s0+∂y2a
∂s0 s00+∂y2a
∂s00s000 =y3a(s, s0, s00, s000), . . . , yka = ∂y(k−1)a
∂s s0+∂y(k−1)a
∂s0 s00+· · ·+∂y(k−1)a
∂s(k−1) s(k), p1a = p1a(s), s=s(t),
p2a = ∂p1a
∂s s0=p2a(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) Aa0 =dy0a
ds =dxa
ds =y1a0, AaA= dA
dtAAa0=dAtAa0 A= 1, k−1,
(2.8) Ba1=dp1a
ds =p2a0, Baα= dα−1Ba1
dtα−1 =dα−1t Ba1, α= 1, k−1 and the Leibniz rule for differentiation we can prove the following theorem.
Theorem 2.1. yAa,pαa ands(α),A,α= 1, k are connected by formulae:
y1a =Aa0s0, (2.9)
y2a = µ1
0
¶ Aa1s0+
µ1 1
¶ Aa0s00,
y3a = µ2
0
¶ Aa2s0+
µ2 1
¶ Aa1s00+
µ2 2
¶
Aa0s000, . . . ,
yAa=
µA−1 0
¶
AaA−1s0+
µA−1 1
¶
AaA−2s00+· · ·+
µA−1 A−1
¶
Aa0s(A), . . . ,
yka= µk−1
0
¶
Aak−1s0+ µk−1
1
¶
Aak−2s00+· · ·+ µk−1
k−1
¶ Aa0s(k), p1a =p1a(s), s=s(t)
p2a =B1as0 p3a =
µ1 0
¶ Ba2s0+
µ1 1
¶
Ba1s00, . . . ,
pαa= µα−2
0
¶
Baα−1s0+ µα−2
1
¶
Bα−2a s00+· · ·+ µα−2
α−2
¶
Ba1s(α−1), . . . ,
pka= µk−2
0
¶
Bak−1s0+ µk−2
1
¶
Bk−2a s00+· · ·+ µk−2
k−2
¶
Ba1s(k−1). Theorem 2.2. The following relations are valid:
(2.10) AaA=
µA−1 0
¶∂AaA−1
∂s s0+ µA−1
1
¶∂AA−2
∂s s00+· · ·+
µA−1 A−1
¶∂Aa0
∂s s(A),
(2.11) AaA= dyAa
ds , A= 1, k−1,
(2.12) Baα= µα−2
0
¶∂Bα−1a
∂s s0+ µα−2
1
¶∂Baα−2
∂s s00+· · ·+ µα−2
α−2
¶
B1as(α−1),
(2.13) Baα=dpαa
ds , α= 2, k−1.
Proof.
AaA = dAtAa0=dA−1t (d1tAa0) =dA−1t µ∂Aa0
∂s s0
¶
=
dA−1t ∂
∂s(Aa0s0) = ∂
∂sdA−1t (Aa0s0) =
∂
∂s
·µA−1 0
¶
AaA−1s0+ µA−1
1
¶
AaA−2s00+· · ·+
µA−1 A−1
¶ Aa0s(A)
¸ . If we take∂/∂sfrom the sum in the middle bracket we obtain (2.10), and if we substituteyAa from (2.9) we obtain (2.11). In the similar way we have
Baα = dα−1t B1a=dα−2t (d1tBa1) =dα−2t µ
d1tdp1a
ds
¶
=
dα−2t
· ∂
∂s µ∂p1a
∂s s0
¶¸
= ∂
∂sdα−2t (Ba1s0) =
∂
∂s
·µα−2 0
¶
Bα−1a s0+ µα−2
1
¶
Baα−2s00+· · ·+ µα−2
α−2
¶
Ba1s(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
y1a =y1a0s0, yAa0 =dAsy0a(s), A= 1, k, (2.14)
y2a =y2a0(s0)2+y1a0s00,
y3a =y3a0(s0)3+y2a03s0s00+y1a0s000,
y4a =y4a0(s0)4+y3a06(s0)2s00+y2a0(3(s00)2+ 4s0s000) +y1a0s0v, . . . , p1a =p1a(s), pαa0 =dα−1s 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) +p2a0s0v, . . . . From (2.14) it follows
Theorem 2.3. The following relations are valid:
(2.15) ∂y1a
∂s0 =∂y2a
∂s00 =· · ·= ∂yka
∂s(k) =y1a0 = dy0a ds
(2.16) ∂yAa
∂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)∂y1a
∂s0 + (∂2aF)∂y2a
∂s0 + (∂3aF)∂y3a
∂s0 +· · ·+ (∂kaF)∂yka
∂s0 + (∂2aF)∂p2a
∂s0 + (∂3aF)∂p3a
∂s0 +· · ·+ (∂kaF)∂pka
∂s0 =F(x, y10, . . . , yk0, p1, . . . , pk0) (∂2aF)∂y2a
∂s00 + (∂3aF)∂y3a
∂s00 +· · ·+ (∂kaF)∂yka
∂s00 + (∂3aF)∂p3a
∂s00 + (∂4aF)∂p4a
∂s00 +· · ·+ (∂kaF)∂pka
∂s00 = 0, . . . , (∂(k−1)aF)∂y(k−1)a
∂s(k−1) + (∂kaF) ∂yka
∂s(k−1)+ (∂kaF) ∂pka
∂s(k−1) = 0, (∂kaF)∂yka
∂s(k) = 0.
On the left-hand side of (2.19) in all equationsF =F(x, y1, . . . , yk, p1, . . . , pk).
If in (2.19) we substitute (2.16) and (2.18) in the form
∂yAa
∂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
¶
y1a∂1a+ µ2
1
¶
y2a∂2a+· · ·+ µk
1
¶
yka∂ka](F) + (2.20)
[ µ1
1
¶
p2a∂2a+ µ2
1
¶
p3a∂3a+· · ·+ µk−1
1
¶
pka∂ka](F) =
F(x, y10, . . . , yk0, p01, . . . , p0k)ds
dt =F(x, y1, . . . , yk, p1, . . . , pk) =F {[
µ2 2
¶
y1a∂2a+ µ3
2
¶
y2a∂3a+· · ·+ µk
2
¶
y(k−1)a∂ka] +