its Poincar´e - Cartan form), and an auxiliary differential form corresponding to the at least 2-contact part of the Lepagean equivalent of the Lagrangian

on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary

ON HAMILTON p2–EQUATIONS IN SECOND-ORDER FIELD THEORY

DANA SMETANOV ´A

Abstract. In the present paper recent results on regularizations of first order variational problems are generalized to Lagrangians affine in the second deriva- tives. 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 varia- tional theory on fibered manifolds.

It is known that in field theory to a variational problem represented by a La- grangian one can associate different Hamilton equations corresponding to different Lepagean equivalents of the Lagrangian (Dedecker [1], Krupka [7]). Accord- ingly, these Hamilton equations depend upon a Lagrangian (resp. its Poincar´e - 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],Krupkov´a[11], [12],Krup- kov´aandSmetanov´a [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 inter- esting traditionally singular physical fields, the Dirac field, and the electromagnetic field (cf. Dedecker[1],Krupkov´aandSmetanov´a[13], [14]).

Throughout this paper, π : Y → X is s fibered manifold, and dimX = n, dimY = m+n. The r-jet prolongation of π is a fibered manifold denoted by

1991Mathematics Subject Classification. 35A15, 49L10, 49N60, 58E15.

Key words and phrases. Hamilton extremals, Hamiltonp2-equations, Lagrangian, Legendre transformation, Lepage form, Poincar´e-Cartan form, regularity.

Research supported by Grants MSM:J10/98:192400002, VS 96003 and FRVˇS 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 Krupkov´a for her valuable stimulation and discussions about principal ideas of the present work.

329

πr : J^rY → X and πr,s : J^rY → J^sY, 0 ≤ s ≤ r, we denote the natural jet projections. A fibered chart on Y (resp. an associated fibered chart on J^rY) is denoted by (V, ψ), ψ = (xⁱ, y^σ) (resp. (Vr, ψr), ψr = (xⁱ, y^σ, y_i^σ, . . . , y_i^σ_1...ir)). In what follows, we considerr= 1 orr= 2.

Recall that everyq-form η onJ^rY admits a unique (canonical) decomposition into a sum of q-forms onJ^r+1Y as follows:

π_r+1,r^∗ η=h(η) +

q

X

k=1

pk(η),

whereh(η) is a horizontal form, called thehorizontal part of η, andp_k(η), 1≤k≤ q, is ak-contact form, called thek-contact part ofη (see e.g. [5], [6] for review).

We use the following notations:

ω₀=dx¹∧dx²∧ · · · ∧dxⁿ, ω_i=i_∂/∂xiω₀, ω_ij =i_∂/∂xjω_i, and

ω^σ=dy^σ−y^σ_jdx^j, ω_i^σ=dy^σ_i −y^σ_ijdx^j.

By an r-th order Lagrangian we shall mean a horizontal n-form λon J^rY. A form ρ is called a Lepagean equivalent of a Lagrangian λif (up to a projection) h(ρ) =λ, andp₁(dρ) is aπ_r+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 Poincar´e–Cartan form associated toλ, and ν is an arbitrary n-form of order of contactness ≥2, i.e., such thath(ν) =p1(ν) = 0 (cf. Krupka [5], [6]). Recall that for a Lagrangian of order 1, Θ =θλ where θλ is the classical Poincar´e–Cartan form ofλ,

θλ=Lω0+ ∂L

∂y_i^σ ω^σ∧ωi.

Ifr= 2, Θ is no more unique, however, there is an invariant decomposition

(1.2) Θ =θ_λ+dφ,

where

θ_λ=Lω₀+ ∂L

∂y^σ_j −d_k ∂L

∂y_jk^σ

!

ω^σ∧ω_j+ ∂L

∂y^σ_ijω_i^σ∧ω_j andφdoes not depend uponλ(Krupka[6]).

With the help of Lepagean equivalents of a Lagrangian one obtains the following intrinsic formulation of theEuler–Lagrangeand Hamilton equations.

Theorem(Krupka[5]). Letλbe a Lagrangian onJ^rY,ρits Lepagean equivalent.

A section γ ofπis an extremal of λif and only if (1.3) J^2r−1γ^∗i_J2r−1ξdρ= 0 for every π-vertical vector fieldξon Y.

A section δ of the fibered manifold π_2r−1 is called a Hamilton extremal of ρ (Krupka[7]) if

(1.4) δ^∗i_ξdρ= 0,

for everyπ2r−1-vertical vector fieldξonJ^2r−1Y.

(1.3) are called theEuler–Lagrange equations and (1.4) theHamilton 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,i6= 2).

The aim of this paper is to consider Hamiltonp2-equations for a class of second order Lagrangians. Namely, we study Lagragians affine in second derivativesy^σ_kl, such that their Lepagean equivalent is of the form ρ=θλ+ν, where ν =p2(β), for ann-formβ defined onJ¹Y.

Recall that a section δ of the fibered manifold π_r is said to be holonomic if δ=J^rγfor a sectionγofπ. Clearly, ifγis an extremal thenJ^2r−1γis 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 byKrupkaandStˇˇ ep´ankov´a[9] in consequence with a study of second order Lagrangians with projectable Poincar´e–Cartan forms: Throughout this paper a Lepagean form is called regular if every its Hamilton extremal is holonomic.

Taking a Lepagean equivalent ofλin the formρ=θλ+p2(β), where β is defined on J¹Y, we can see that regularity conditions involve λ and β, and one can ask abouta proper choice β, such thatρis 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 J¹Y. 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) whereφ= 0, ν is 2-contact, and

ν =p2(β),

whereβ is defined onJ¹Y and such thatp_i(β) = 0 for alli≥3.

In a fiber chart, a Lagrangianλaffine in the second derivatives is expressed by (2.1) λ=Lω0, L=Le+Le^ij_σy_ij^σ,

where functions L,e Le^ij_σ do not depend on the variables y_kl^κ and the functions Le^ij_σ satisfy the conditionLe^ij_σ =Le^ji_σ. In view of the above considerations we obtain

(2.2)

ρ=

Le+Le^kl_ν y^ν_kl

ω₀+ ∂Le

∂y^σ_j +∂Le^kl_ν

∂y^σ_j y_kl^ν −d_kLe^jk_σ

!

ω^σ∧ω_j +Le^il_σω_i^σ∧ωj+f_σν^ijω^σ∧ω^ν∧ωij+g_σν^kijω^σ∧ω^ν_k∧ωij

+h^klij_σν ω_k^σ∧ω^ν_l ∧ωij,

where the functions f_σν^ij, g_σν^kij, h^klij_σν do not depend on the y_pq^κ’s and satisfy the conditions

f_σν^ij =−f_νσ^ij, f_σν^ij =−f_σν^ji, f_σν^ij =f_νσ^ji; g^kij_σν =−g_σν^kji;

(2.3)

h^klij_σν =−h^lkij_νσ , h^klij_σν =−h^klji_σν .

In general case, the Poincar´e–Cartan forms of a second order Lagrangians is defined onJ³Y, but for Lagrangians of the forms (2.1)θλis projectable ontoJ²Y. Our choice of the 2-contact part ν of ρ conserves the Lepagean form (2.2), (2.3) defined onJ²Y.

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 ofλ=hρare inbijective correspondence.

Theorem A. Let dimX ≥3. Let λ be a second-order Lagrangian affine in the variablesy^σ_ij, the formula(2.1)be its expression in a fiber chart(V, ψ),ψ= (xⁱ, y^σ) on Y. Letρbe a Lepagean equivalent of λof the form(2.2), (2.3).

Assume that the matrix

(2.5) A^klj_νσ |B_νκ^klpq

,

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 A^klj_νσ = ∂Le^kl_ν

∂y^σ_j − 1 2

∂Le^jk_σ

∂y^ν_l + ∂Le^jl_σ

∂y^ν_k

!

− g_σν^kjl − g^ljk_σν

! , and

B_νκ^klpq = h^kpql_νκ +h^lpqk_νκ , has rank mn(n + 3)/ 2.

Then ρis regular on π₂⁻¹(V), i.e., every Hamilton extremal δ :π(V)⊃U → J²Y ofρ is of the formδ=J²γ, whereγ is an extremal ofλ.

Proof. Expressing the Hamiltonp2-equations (1.4) in fiber coordinates we get along δthe following system of first-order equations for sectionδ:

mn² equations

∂Le^kl_ν

∂y^σ_j −1 2

∂Le^jk_σ

∂y_l^ν +∂Le^jl_σ

∂y^ν_k

!

−g_σν^kjl−g^ljk_σν

!∂y^σ

∂x^j −y_j^σ (2.6)

+2 h^kijl_νσ +h^lijk_νσ ∂y^σ_i

∂x^j −y_ij^σ

= 0, mnequations

∂²Le

∂y^σ_j∂y_k^ν + ∂²Le^pq_κ

∂y_j^σ∂y_k^νy_pq^κ − ∂

∂y_k^νd_pLe^jp_σ −∂Le^kj_ν

∂y^σ −4f_σν^jk+ 2d_ig_σν^kij

! (2.7)

× ∂y^σ

∂x^j −y^σ_j

+ ∂Le^ij_σ

∂y^ν_k −∂Le^kj_ν

∂y_i^σ + 2g^kij_σν −2g_νσ^ikj−4dlh^kilj_νσ

!

× ∂y^σ_i

∂x^j −y^σ_ij

+ 2 h^ikjl_σν +h^lkji_σν ∂y^σ_il

∂x^j −y_ilj^σ

+

2∂f_σκ^ij

∂y^ν_k +∂g^kij_κν

∂y^σ −∂g_σν^kij

∂y^κ

∂y^σ

∂x^j −y_j^σ ∂y^κ

∂xⁱ −y^κ_i

+ 2

2∂h^lkij_κν

∂y^σ +∂g_σκ^lij

∂y^ν_k −∂g^kij_σν

∂y^κ_l

∂y^σ

∂xⁱ −y^σ_i ∂y^κ_l

∂x^j −y_lj^κ

+ 2

∂h^lkij_σν

∂y^κ_p +∂h^kpij_νκ

∂y^σ_l +∂h^plij_κσ

∂y_k^ν

∂y_p^κ

∂xⁱ −y^κ_pi ∂y_l^σ

∂x^j −y_lj^σ

= 0, andmequations

∂Le

∂y^ν +∂Le^pq_κ

∂y^ν y_pq^κ −d_j ∂Le

∂y^ν_j +∂Le^pq_κ

∂y^ν_j y_pq^κ

!

+d_jd_kLe^jk_ν

! (2.8)

+ ∂²Le

∂y^σ_j∂y^ν + ∂²Le^pq_κ

∂y_j^σ∂y^νy^κ_pq− ∂²Le

∂y^σ∂y_j^ν − ∂⁸Le^pq_κ

∂y^σ∂y_j^νy_pq^κ − ∂

∂y^νdkLe^jk_σ + ∂

∂y^σd_kLe^jk_ν + 2d_if_σν^ij ∂y^σ

∂x^j −y^σ_j

+ ∂Le^kj_σ

∂y^ν − ∂²Le

∂y_j^σ∂y^ν_k − ∂²Le^pq_κ

∂y^σ_j∂y_k^νy^κ_pq+ ∂

∂y_k^σd_pLe^jp_ν + 4f_νσ^jk−2d_ig^kij_νσ

!

× ∂y^σ_k

∂x^j −y^σ_kj

+ ∂Le^kl_σ

∂y^ν_j −1 2

∂Le^jk_ν

∂y_l^σ +∂Le^jl_ν

∂y^σ_k

!

−g^kjl_νσ −g_νσ^ljk

!∂y^σ_kl

∂x^j −y_klj^σ

+ 2 ∂f_σν^ij

∂y^κ +∂f_κσ^ij

∂y^ν +∂f_νκ^ij

∂y^σ

∂y^κ

∂xⁱ −y^κ_i ∂y^σ

∂x^j −y_j^σ

+ 2

2∂f_σν^ij

∂y_k^κ +∂g^kij_νκ

∂y^σ −∂g_σκ^kij

∂y^ν

∂y^κ_k

∂xⁱ −y^κ_ki ∂y^σ

∂x^j −y_j^σ

+

2∂h^lkij_κσ

∂y^ν +∂g_νκ^lij

∂y_k^σ −∂g^kij_νσ

∂y^κ_l

∂y^κ_l

∂xⁱ −x^κ_li ∂y^σ_k

∂x^j −y^σ_kj

= 0.

The system (2.6) can be viewed as a system ofmn² (algebraic) linear homoge- neous equations for

mn+mn n+ 1

2

=mn n+ 3

2

unknowns

∂y^σ

∂xⁱ −y^σ_i

, and

∂y^σ_j

∂xⁱ −y^σ_ij

, j ≤i.

According to the (algebraic) Frobenius theorem, this system has a unique (zero) solution if and only if the rank of the matrix of system, i. e., A^klj_νσ |B_νκ^klpq

is equal to the number of unknowns, i. e.,mn((n+ 3)/2). Let dimX =n≥3, then

mn²=mnn 2 +n

2

≥mn n+ 3

2

,

as desired. Since rank of matrix (2.5) is maximal, by assumption, we obtain

∂y^σ

∂xⁱ ◦δ=y_i^σ◦δ, ∂y^σ_j

∂xⁱ ◦δ=y^σ_ij◦δ, j ≤i, proving that δ=J²γ. Substituting this into (2.8) we get

∂Le

∂y^ν +∂Le^pq_κ

∂y^ν y_pq^κ −d_j ∂Le

∂y^ν_j +∂Le^pq_κ

∂y_j^ν y_pq^κ

!

+d_jd_kLe^jk_ν

!

◦J³γ

= ∂L

∂y^ν −d_j ∂L

∂y_j^ν +d_jd_k ∂L

∂y_jk^ν

!

◦J³γ = 0, i. e.,γ is an extremal ofλ.

Theorem B. Let λ be a second-order Lagrangian affine in the variables y_ij^σ, 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 ofλof the form (2.2),(2.3). Suppose thatρsatisfies the conditions

(2.9) h^klij_σν = 0.

Assume that the matrix (2.10) A^klj_νσ = ∂Le^kl_ν

∂y_j^σ − 1 2

∂Le^jk_σ

∂y_l^ν + ∂Le^jl_σ

∂y_k^ν

!

− g^kjl_σν − g^ljk_σν

!

with mn² rows (resp. mn columns) labelled by(ν, k, l)(resp. (σ, j)), has the max- imal rank (i.e. rankA^klj_νσ =mn).

Then every Hamilton extremalδ:π(V)⊃U →J²Y ofρis of the formπ_2,1◦δ= J¹γ, whereγ is an extremal ofλ.

Proof. Substituting (2.9) into Hamiltonp2-equations (2.6), and using the condition rankA^klj_νσ =mnwe obtain

∂y^σ◦δ

∂x^j = y^σ_j ◦δ.

The previous condition means π2,1◦δ = J¹γ. However, the last equations (2.8) now mean thatγ is an extremal ofλ.

3. Legendre transformation onJ²Y for second order Lagrangians affine in second derivatives

Writing the Lepagean equivalent (2.2), (2.3) in the form of a noninvariant de- composition in the canonical basis (dxⁱ, dy^σ,dy^σ_i,dy^σ_ij) of 1-forms we get

ρ=−Hω0+pⁱ_σdy^σ∧ωi+p^ij_σdy_i^σ∧ωj

+f_σν^ij dy^σ∧dy^ν∧ωij+g_σν^kijdy^σ∧dy^ν_k∧ωij+h^klij_σν dy^σ_k ∧dy^ν_l ∧ωij, where

H =−L+

∂L

∂y^σ_i −d_jLe^ij_σ

y_i^σ+eL^ij_σy^σ_ij+2f_σν^ijy^σ_iy_j^ν− g_σν^kij+g^jik_σν y^σ_iy^ν_jk

−1

2 h^klij_σν +h^ilkj_σν +h^kjil_σν +h^ijkl_σν y^σ_iky^ν_jl, pⁱ_σ = ∂L

∂y_i^σ −d_jLe^ij_σ −4f_σν^ijy^ν_j − g_σν^kij+g_σν^jik y_jk^ν , p^ij_σ =Le^ij_σ + g_νσ^ikj+g_νσ^jki

y^ν_k−2 h^kilj_νσ +h^likj_νσ y^ν_kl. (3.1)

Ifp^ij_σ =p^ji_σ (i. e.,h^kilj_νσ +h^likj_νσ =h^kjli_νσ +h^ljki_νσ ) and

det













∂pⁱ_σ

∂y^ν_k

∂pⁱ_σ

∂y_kl^ν

∂p^ij_σ

∂y_k^ν

∂p^ij_σ

∂y_kl^ν











 6= 0,

then

(3.2) ψ2= (xⁱ, y^σ, y_i^σ, y^σ_ij)→(xⁱ, y^σ, pⁱ_σ, p^ij_σ) =χ

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, pⁱ_σ, p^ij_σ are called aHamiltonianandmomenta, respectively. Since the functionsf_σν^ij,g_σν^kij, h^likj_νσ (2.3) may depend upon the momenta pⁱ_σ (not upon p^ij_σ), the Hamilton p2- equations (1.4) in these “Legendre coordinates” take a rather complicated form:

∂H

∂y^σ =−∂pⁱ_σ

∂xⁱ + 4∂f_σν^ij

∂x^j

∂y^ν

∂xⁱ + 2 ∂f_κν^ij

∂y^σ +∂f_κσ^ij

∂y^ν +∂f_νκ^ij

∂y^σ ∂y^κ

∂xⁱ

∂y^ν

∂x^j

−4∂f_σν^ij

∂p^k_κ

∂p^k_κ

∂xⁱ

∂y^ν

∂x^j +∂g^kij_σν

∂x^j

∂y_k^ν

∂xⁱ + 2 ∂g_κν^kij

∂y^σ −∂g^kij_σν

∂y^κ ∂y^κ

∂xⁱ

∂y^ν_k

∂x^j

−2∂g^kij_σν

∂p^l_κ

∂p^l_κ

∂xⁱ

∂y^ν_k

∂x^j + 2∂h^klij_κν

∂y^σ

∂y_k^κ

∂xⁱ

∂y^ν_l

∂x^j,

∂H

∂pⁱ_σ = ∂y^σ

∂xⁱ + 2∂f_κν^jk

∂pⁱ_σ

∂y^κ

∂x^j

∂y^ν

∂x^k + 2∂g_κν^kjl

∂pⁱ_σ

∂y^κ

∂x^j

∂y^ν_k

∂x^l + 2∂h^kljm_κν

∂pⁱ_σ

∂y_k^κ

∂x^j

∂y^ν_l

∂x^m,

∂H

∂p^ijσ

= 1 2

∂y^σ_i

∂x^j + ∂y_j^σ

∂xⁱ

. However, ifdη= 0, where

η=f_σν^ij dy^σ∧dy^ν∧ωij+g^kij_σν dy^σ∧dy_k^ν∧ωij+h^klij_σν dy_k^σ∧dy_l^ν∧ωij, then

∂H

∂y^σ =−∂pⁱ_σ

∂xⁱ, ∂H

∂pⁱ_σ = ∂y^σ

∂xⁱ, ∂H

∂p^ijσ

= 1 2

∂y^σ_i

∂x^j + ∂y_j^σ

∂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.

Theorem C. Let λ be a second-order Lagrangian affine in the variables y_ij^σ, 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 ofλof the form (2.2),(2.3). Suppose thatρsatisfies the conditions h^klij_σν = 0. Suppose thatρadmits the Legendre transformation

ψ2 = (xⁱ, y^σ, y_i^σ, y_ij^σ)→ (xⁱ, y^σ, pⁱ_σ, p^ij_σ) = χ defined by(3.1),(3.2).

Thenπ_2,1◦δ=J¹γ, whereγ is an extremal ofλ.

Proof. Since, the functions h^klij_σν vanish, the Jacobi matrix of the Legendre trans- formation takes the form





∂pⁱ_σ

∂y_k^ν

∂pⁱ_σ

∂y_kl^ν

∂p^ij_σ

∂y_k^ν 0





The above matrix is regular if and only if the matrices _∂pi σ

∂y_kl^ν

, and _∂pij σ

∂y^ν_k

have the maximal rank. Explicit computations lead to

∂pⁱ_σ

∂y^ν_kl = ∂Le^kl_ν

∂y_i^σ − 1 2

∂Le^ik_σ

∂y_l^ν + ∂Le^il_σ

∂y^ν_k

!

− g^kil_σν − g_σν^lik, i.e. in the notation (2.10),_∂pi

σ

∂y^ν_kl

= A^klj_νσ^T .

Accordingly, from Theorem B we obtainπ_2,1◦δ = J¹γ, whereγis an extremal ofλ.

For a deeper discussion on Legendre transformations and their geometric mean- ing we refer to [11], [12].

4. Projectability ontoJ¹Y

Theorem D. Letλbe a second-order Lagrangian affine in the variablesy^σ_ij, i. e., in fibered coordinates expressed by(2.1). Letρbe a Lepagean equivalent ofλof the form(2.2), (2.3). The following conditions are equivalent:

I.ρis projectable ontoJ¹Y. II. ρsatisfies the conditions

(4.1)

h^kilj_νσ +h^likj_νσ = 0, g^kjl_σν +g_σν^ljk=∂Le^kl_ν

∂y_j^σ −1 2

∂Le^jk_σ

∂y^ν_l +∂Le^jl_σ

∂y_k^ν

! . Proof. Taking into account

ρ=−Hω0+pⁱ_σdy^σ∧ωi+p^ij_σdy_i^σ∧ωj

+f_σν^ij dy^σ∧dy^ν∧ω_ij+g_σν^kijdy^σ∧dy^ν_k∧ω_ij+h^klij_σν dy^σ_k ∧dy^ν_l ∧ω_ij, it is sufficient to find conditions of the independenceH,pⁱ_σ, andp^ij_σ ony^σ_ij’s. Explicit computations lead to

∂pⁱ_σ

∂y^ν_kl =∂Le^kl_ν

∂y_i^σ −1 2

∂Le^ik_σ

∂y_l^ν +∂Le^il_σ

∂y_k^ν

!

−g^kil_σν −g_σν^lik= 0,

∂p^ij_σ

∂y^ν_kl =−2 h^kilj_νσ +h^likj_νσ

= 0,

∂H

∂y^ν_kl = ∂Le^kl_ν

∂y_i^σ −1 2

∂Le^ik_σ

∂y^ν_l +∂Le^il_σ

∂y^ν_k

!

−g^kil_σν −g_σν^lik

! y^σ_i

− h^kjli_νσ +h^kilj_νσ +h^ljki_νσ +h^likj_νσ y_ij^σ.

Corollary. Every second-order Lagrangian affine in the variables y_ij^κ has a Lep- agean equivalent projectable onto J¹Y.

Remark. If the functions f_σν^ij, g^kij_σν, h^klij_σν (2.2), (2.3) vanish, i. e., ρ = θλ the projectability conditions (4.1) take the form (cf. [9])

∂Le^kl_ν

∂y_j^σ −1 2

∂Le^jk_σ

∂y^ν_l +∂Le^jl_σ

∂y^ν_k

!

= 0.

Theorem E. Let λbe a second-order Lagrangian affine in the variables y_ij^σ, the formula (2.1) be its expression in a fiber chart (V, ψ),ψ= (xⁱ, y^σ)onY. Let ρbe a Lepagean equivalent ofλof the form(2.2), (2.3)and suppose that it is projectable ontoJ¹Y. Ifρsatisfies the conditions

(4.2)

h^klij_σν = 0,

∂f_σν^ij

∂y_k^κ =1 2

∂g^kij_κσ

∂y^ν −∂g_κν^kij

∂y^σ

g^ikj_νσ −g^kij_σν = 1 2

∂Le^ij_σ

∂y_k^ν −∂Le^kj_ν

∂y_i^σ

!

∂g_σκ^lij

∂y^ν_k −∂g_σν^kij

∂y^κ_l = 0.

and the matrix C_νσ^kj = ∂²Le

∂y^σ_j∂y_k^ν + ∂²Le^pq_κ

∂y_j^σ∂y_k^νy_pq^κ − ∂

∂y_k^νdpLe^jp_σ −∂Le^kj_ν

∂y^σ −4f_σν^jk+ 2dig_σν^kij

! , 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 → J¹Y of ρ is of the form δ=J¹γ, whereγ is an extremal ofλ.

Proof. Expressing the Hamiltonp2-equations (1.4) of a Lepagean equivalentρpro- jectable ontoJ¹Y in fiber coordinates and using (4.2) we get alongδthe following system of first-order equations:

mnequations

∂²Le

∂y^σ_j∂y_k^ν + ∂²Le^pq_κ

∂y_j^σ∂y_k^νy_pq^κ − ∂

∂y_k^νd_pLe^jp_σ −∂Le^kj_ν

∂y^σ −4f_σν^jk+ 2d_ig_σν^kij

! (4.3)

× ∂y^σ

∂x^j −y^σ_j

= 0, mequations

∂Le

∂y^ν +∂Le^pq_κ

∂y^ν y_pq^κ −dj

∂Le

∂y_j^ν +∂Le^pq_κ

∂y_j^ν y^κ_pq

!

+djdkLe^jk_ν

! (4.4)

+ ∂²Le

∂y_j^σ∂y^ν + ∂²Le^pq_κ

∂y^σ_j∂y^νy_pq^κ − ∂²Le

∂y^σ∂y_j^ν − ∂²Le^pq_κ

∂y^σ∂y^ν_jy^κ_pq− ∂

∂y^νd_kLe^jk_σ + ∂

∂y^σdkLe^jk_ν + 2dif_σν^ij ∂y^σ

∂x^j −y_j^σ

+ ∂Le^kj_σ

∂y^ν − ∂²Le

∂y^σ_j∂y_k^ν − ∂²Le^pq_κ

∂y_j^σ∂y_k^νy_pq^κ + ∂

∂y_k^σdpLe^jp_ν + 4f_νσ^jk−2dig_νσ^kij

!

× ∂y_k^σ

∂x^j −y_kj^σ

+ 2 ∂f_σν^ij

∂y^κ +∂f_κσ^ij

∂y^ν +∂f_νκ^ij

∂y^σ

∂y^κ

∂xⁱ −y_i^κ ∂y^σ

∂x^j −y^σ_j

= 0.

The matrixC_νσ^kj is regular. Hence, from equations (4.3) we obtain the formula

(4.5) ∂y^σ◦δ

∂x^j =y^σ_j ◦δ.

Substituting this into (4.4) we get

∂Le

∂y^ν +∂Le^pq_κ

∂y^ν y^κ_pq−d_j ∂Le

∂y^ν_j +∂Le^pq_κ

∂y^ν_j y^κ_pq

!

+d_jd_kLe^jk_ν

!

◦J³γ

= ∂L

∂y^ν −dj

∂L

∂y_j^ν +djdk

∂L

∂y^ν_jk

!

◦J³γ = 0, proving our assertion.

Remark. a)Letλbe a second-order Lagrangian (2.1), suppose that the functions Le^ij_σ satisfy the conditions

∂Le^ki_σ

∂y_j^ν = ∂Le^ki_ν

∂y^σ_j This means thatLe^ij_σ take the form

Le^ij_σ =1 2

∂f^j

∂y^σ_i + ∂fⁱ

∂y_j^σ

!

and the Lagrangian equivalent with a first-order Lagrangian.

We can choose the functionsg^kij_σν in a regular Lepagean equivalent (in the sense of Theorem E) in the following form

g^kij_σν =1 2

∂Le^kj_ν

∂y_i^σ −∂Le^ki_ν

∂y_j^σ

! +t^kij_σν,

where the functionst^kij_σν do not depend on the variablesy^ν_kl and satisfy the condi- tions

t^kij_σν =−t^kji_σν, t^kij_σν =−t^jik_σν, t^kij_σν =−t^ikj_νσ.

b) Let λbe a second-order Lagrangian (2.1) and suppose that the functions Le^ij_σ satisfy the conditions

∂Le^kl_ν

∂y_j^σ − 1 2

∂Le^jk_σ

∂y^ν_l + ∂Le^jl_σ

∂y^ν_k

!

= 0.

Then we can choose the functionsg_σν^kij as follows:

g_σν^kij= 1 4

∂Le^kj_ν

∂y^σ_i −∂Le^ij_σ

∂y^ν_k

! +t^ijk_σν, where thet^kij_σν’s are as above.

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