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of higher order

Marcela Popescu and Paul Popescu

Abstract.A higher order Lagrangian or an affine Hamiltonian is totally singular if its vertical Hessian vanishes. A natural duality relation between totally singular Lagrangians and affine Hamiltonians is studied in the pa-per. We prove that the energy of a totally singular affine Hamiltonian has as a dual a suitable first order totally singular Lagrangian. Relations between the solutions of Euler and Hamilton equations of dual objects are studied by mean of semi-sprays. In order to generate examples fork >1, some natural lift procedures are constructed.

M.S.C. 2010: 53C80, 70H03, 70H05, 70H50.

Key words: totally singular Lagrangian and affine Hamiltonian, semi-spray.

1

Introduction

Some results and constructions from [14] are extended in this paper from the case k = 2 to the general case, k ≥ 2. Some physical and mathematical aspects that motivate the study of totally singular Lagrangians in the second order case can be found also in [1, 8, 7] and the references therein.

For hyperregular Lagrangians of higher order, the Legendre duality between La-grangians and Hamiltonians was studied in various papers (see [16] for recent results and references). But the class of hyperregular Lagrangians and Hamiltonians is too restrictive. We study Lagrangians and Hamiltonians of higher order that have null vertical Hessians, called in the paper astotally singular; they are the ,,most singular” Lagrangians and Hamiltonians. We consider in the paper that a totally singular La-grangian of orderkis allowed if it is in duality with a totally singular Hamiltonian of orderk. An allowed totally singular Lagrangian has a dual allowed totally singular Hamiltonian; for the converse situation, Theorem 2.1 asserts that, assuming some conditions, an allowed totally singular Hamiltonian of orderk has a dual allowed to-tally singular Lagrangian of orderkand both can be related to ordinary dual (allowed totally singular) Lagrangians and Hamiltonians of first order onTk−1_M.

∗

Balkan Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 122-132. c

In order to have consistent examples of totally singular Lagrangians and Hamil-tonians of higher order, lifting procedures are given in the last section. In this way, certain examples considered in [14, 9, 4] can be lifted to totally singular Lagrangians and Hamiltonians of higher order. Following [17], in an analogous manner one can study the time-dependent case. Further investigations on general jet spaces, complex spaces or using linear frames can be made following approaches in [6], [13] and [3] respectively.

2

Higher order Hamiltonians and Lagrangians

LetM be a differentiable manifold. We use a coordinate construction ofTk_M,k≥2,

as in [11], [12] or [16]. The fibered manifold (Tk_{M, π}

k, Tk−1M) is an affine bundle,

fork≥2. A section S :Tk_{M → T}k+1_{M of the affine bundle (T}k+1_{M, π}

k+1, TkM)

is called asemi-spray of orderkonM;S can be seen as well as a vector field on the manifoldTk_{M. ALagrangian of orderkonM is a differentiable functionL:T}k_{M →}

IRorL:W →IR, whereW ⊂Tk_{M is an open fibered submanifold. For example, in}

[11] and [12]W =T]k_{M =T}k_{M\{0} (where{0} is the image of the ,,null” section,}

i.e. the section of null velocities) andL:Tk_{M →IRis continuous.}

The totally singular Hamiltonians of orderk≥2, studied in our paper, are affine Hamiltonians as in [16]. Let us consider the affine bundle Tk_M πk_{→ T}k−1_{M and}

u∈ Tk−1_{M. The fiber T}k

uM = πk−1(u) ⊂TkM is a real affine space, modelled on

the real vector spaceTπ(u)M. Thevectorial dualof the affine spaceTukM isTuk†M =

Af f(Tk

uM, IR), whereAf f denotes affine morphisms. Denoting byTk†M = ∪ u∈Tk−1_M Tk†

u M andπ† :Tk†M →Tk−1M the canonical projection, then (Tk†M, π†, Tk−1M)

is a vector bundle. There is a canonical vector bundle morphism Π :Tk†M →Tk∗M, over the base Tk−1_{M. This projection is also a canonical projection of an affine}

bundle with type fiber IR. An affine Hamiltonian of order k onM is a section h: Tk∗_{M →T}k†_{M of this affine bundle (or of an open fibered submanifoldW ⊂T}k∗_M), i.e. Π◦h= 1Tk∗_M (or Π◦h= 1W respectively). Thus an affine Hamiltonian is not a

real function, but a section in an affine bundle with a one dimensional fiber.

Considering some local coordinates (xi_{) onM, (x}i_{, y}(1)i_{, . . . , y}(k−1)i_{) on T}k−1_M,

and (xi_{, y}(1)i_{, . . . , y}(k−1)i_{, p}

i, T) on Tk†M, then the coordinates pi andT change

ac-cording to the rules

pi′ =

∂xi

∂xi′pi; T

′ _=T+ 1 kΓ

(k−1)

U (y

(k−1)i′

)∂x

i

∂xi′pi,

where

Γ(_Uk−1)=y(1)i ∂

∂xi +· · ·+ (k−1)y

(k−1)i ∂

∂y(k−2)i.

The the affine Hamiltonianh:T^k∗_{M →T}^k†_{M has the local form}

(2.1) h(xi, y(1)i, . . . , y(k−1)i, pi) = (xi, . . . , y(k−1)i, pi, H0(xi, . . . , y(k−1)i, pi))

and the local functionH0changes according to the rule

H₀′(xi′, y(1)i′, . . . , y(k−1)i′, pi′) = H0(xi, y(1)i, . . . , y(k−1)i, pi) +

1 kΓ

(k−1)

U (y

(k−1)i′

)∂x

i

There is aco-Legendre mapH:Tk∗_{M →T}k_{M, locally given by}

H(xi_{, y}(1)i_{, . . . , y}(k−1)i_{, p}

i) = (xi, y(1)i, . . . , y(k−1)i,

Hi(xi, y(1)i, . . . , y(k−1)i, pi) =

∂H0

∂pi

(xi, y(1)i, . . . , y(k−1)i, pi)).

Since ∂2H0′

∂pi′∂pj′ =

∂xi′

∂xi ∂x j′

∂xj ∂

2

H0

∂pi∂pj, it follows that hij = ∂2H0

∂pi∂pj defines a symmetric

bilinear d-form onTk−1_{M, called the vertical Hessianofh.}

For an affine Hamiltonianhof orderk(k≥2) and the local domain of coordinates U, one can consider the local functions on T∗_Tk−1_M:

(2.2) EU =p(0)iy(1)i+· · ·+ (k−1)p(k−1)iy(k)i+kH0(xi, y(1)i, . . . , y(k−1)i, p(1)i).

It can be easily proved (as in [16]) that the local functionsEU glue together to a global functionE0:T∗T^k−1M →IR, called theenergyofh.

We say that an affine Hamiltonian is totally singular if its vertical Hessian is null. Notice that the difference of two affine Hamiltonians of order k is a vectorial Hamiltonian of order k (i.e. a real function on Tk∗_{M, see [16]) and every affine} Hamiltonian of orderkis a sum of an affine totally singular Hamiltonian of order k and a vectorial Hamiltonian of orderk. If a totally singular affine Hamiltonianhof orderkonM has a local form (2.1), then the local functionH0 has the form

H0(xj, y(1)j, . . . , y(k−1)j, pi) = piSi(xj, y(1)j, . . . , y(k−1)j) +

f(xj, y(1)j, . . . , y(k−1)j), (2.3)

where (Si_{) defines an affine sectionS:T}k−1_{M →T}k_{M given locally by}

(xi, y(1)i, . . . , y(k−1)i)→S (xi, y(1)i, . . . , y(k−1)i, Si) andf ∈ F(Tk−1M).

It defines a semi-spray Γ0∈ X(Tk−1M), called theassociated semi-sprayofh:

(2.4) Γ0=y(1)i

∂

∂xi +· · ·+ (k−1)y

(k−1)i ∂

∂y(k−2)i +kS i ∂

∂y(k−1)i.

Let us consider some examples. 1◦_{. Let Γ}

0, given by formula (2.4), be the local form of a semi-spray. Then the

for-mulaH0(xi, y(1)i, . . . , y(k−1)i, pi) =Sipi defines a totally singular affine Hamiltonian

of orderk. 2◦_{. Let L}

0 :Tk−1M →IR be a regular Lagrangian of orderk−1 and let Γ0 be

the semi-spray defined by L0 (see [11]). Then, using the above example, a totally

singular affine Hamiltonian of orderkis obtained.

Let L : T M → IR and H : T∗_{M → IR be a totally singular Lagrangian and} Hamiltonian respectively, of first order, having local forms

L(xi, yi) =αi(xj)yi+β(xj), H(xi, pi) =piϕi(xj) +γ(xj),

whereα=αidxi ∈ X∗(M),ϕ=ϕi ∂_∂xi ∈ X(M) andβ, γ∈ F(M) (see [14]).

Accord-ing also to [14],LandH are in duality ifiϕdα=dβ andL(x, ϕ) +H(x, α)−α(ϕ) =

The energy of a totally singular affine Hamiltonianhgiven by (2.3) is

E=p(0)iy(1)i+. . .+ (k−1)p(k−2)iy(k−1)i+kp(k−1)iSi+kf.

We can viewE as a totally singular HamiltonianE :T∗_Tk−1_{M →IR. Let us look in}

that follows for a totally singular Lagrangian onTk−1_{M, that is in duality withE.}

Proposition 2.1. Locally, there is a (first order) totally singular Lagrangian L : ¯

U ⊂T Tk−1_{M →IR, which is dual to the (first order) totally singular Hamiltonian}

E:T Tk−1M →IR.

Proof. The vector field on Tk−1_{M, that corresponds to E is ϕ= Γ}

0, the

semi-spray defined by h, given by formula (2.4). We denote ϕ(0)i _{= y}(1)i_{,. . .,ϕ}(k−2)i ₌

(k−1)y(k−1)i_{, ϕ}(k−1)i_=kSi_(xj_{, y}(1)j_{, . . . , y}(k−1)j_{) andγ =f. Let us take α:= ¯α,}

with

(2.5) α¯ =α(0)idxi+α(1)idy(1)i+· · ·+α(k−1)idy(k−1)i

and denote

H0(xi, y(1)j, . . . , , y(k−1)j, p(k−1)j) =Sip(k−1)i+f.

The equalityiϕdα=dβ gives the following system of partial differential equations:

(2.6) ( _k∂H0

∂xi(xi, y(1)j, . . . , , y(k−1)j, α(k−1)j) + Γ0(α(0)j) = 0,

α(0)i+k_∂y∂H(1)0i+ Γ0(α(1)j) = 0, . . . , α(k−2)i+k_∂y∂H(k−01)i + Γ0(α(k−1)j) = 0.

Eliminating successively α(k−2)i,. . ., α(0)i in the last k−1 equations, the first

equation becomes:

Γk−1

0 (α(k−1)i) = (−1)kk

∂H0

∂xi (x

i_{, y}(1)i_{, . . . , y}(k−1)i_{, α}

(k−1)i) +

(−1)k−1_kΓ 0

µ ∂H0

∂y(1)i

¶

+· · · −kΓk−1 0

µ ∂H0

∂y(k−1)i

¶ . (2.7)

Let us denote byFi∈ F(Tk∗M) the right side of this equation. Let{zα}_α=1,(k+1)m

be a system of local coordinates on the manifold Tk∗_{M such that Γ}

0 = _∂z∂1. Then

the local form of the differential equation (2.7) is ∂k−1α(k−1)i

∂(z1₎k−1 =Fi(zα, α(k−1)i).Since

this differential equation has local solutions, the conclusion follows. ¤

Let us consider the canonical projectionsTk†_{M →}Π _Tk∗_{M =T}k−1_M×MT∗_M p1

→

Tk−1_{M. For a d-formα= (α}

i(xj, y(1)j, . . . , y(k−1)j)) onTk−1M, we denote byα′ :

Tk−1_{M →T}k∗_{M =T}k−1_{M ×M T}∗_{M the map defined byα}′_{(z) = (z, α}

z). We say

that a maphα : Tk−1M →Tk†M is anα-Hamiltonian if Π◦hα =α′. Using local

coordinates, the local form ofhα is

(xj_{, y}(1)j_{, . . . , y}(k−1)j₎ h

→(xj_{, y}(1)j_{, . . . , y}(k−1)j_{, α}

and the local functionsh0change on the intersection of two coordinate charts

accord-are solutions of the Lagrange equation

(2.9) ∂L

The integral action of the affine Hamiltonian h along a curve γ: [0,1]→T∗M, t→γ (xi_{(t), p}

i(t)), is defined in [16] by the formula:

(2.10) I(γ) =

The critical condition (or Fermat condition in the case of an extremum) forγ, gives the Hamilton equation forhin the condensed form:

(2.11)

LethandLbe totally singular of orderk, having the local forms (2.3) and (2.8) respectively. Then a d-formα= (αi(xj, y(1)j, . . . , y(k−1)j)) onTk−1M corresponds to

Land a semi-spray Γ0 of order k−1, given by (2.4), corresponds toh. We say that

Lis in dualitywith hif the formula (2.7) holds, withα(k−1)i =αi and hα =h◦α′

Lemma 2.1. IfL is in duality withh, then fora= 1, k−1 one have ∂H0

∂xi (x

i_{, y}(1)i_{, . . . , y}(k−1)i_{, α}

i) = −

∂L ∂xi(x

i_{, y}(1)i_{, . . . , y}(k−1)i_{, S}i_),

∂H0

∂y(a)i(x

i_{, y}(1)i_{, . . . , y}(k−1)i_{, α}

i) = −

∂L ∂y(a)i(x

i_{, y}(1)i_{, . . . , y}(k−1)i_{, S}i_).

Proof. It suffices to prove only the first relation, since the proof of each of the other relations follows the same idea. Using relationsh0=h◦α′and (2.8), we obtain

L(xj_{, y}(1)j_{, . . . , y}(k)j_{) =kα}

j(y(k)j−Si)−f. Thus the first relation holds. ¤

We say also that a totally singular Lagrangian of orderk is allowed if there is a semi-spray Γ0, of orderk−1, and a d-formα= (αi) such that the following formula

holds:

Γk0−1(αi) = (−1)k−1k

∂L ∂xi(x

i_{, y}(1)i_{, . . . , y}(k−1)i_{, S}i_{) + (−1)}k−2_kΓ 0

µ ∂L ∂y(1)i

¶ +

· · · −kΓk0−1 µ

∂L ∂y(k−1)i

¶ .

It is easy to see that a totally singular Lagrangian of orderkis allowed if it is in duality with a totally singular Hamiltonian of orderk. Thus a local dual of a totally singular Hamiltonian of orderk is allowed. The following result can be proved by a straightforward verification, using local coordinates.

Proposition 2.3. Letα= (αi(xj, y(1)j, . . . , y(k−1)j))be a d-form onTk−1M andh:

Tk−1_{M →T}k†_{M be anα-Hamiltonian such that there is a1-formα¯ ∈ X}∗_(Tk−1_M)

whereαis the top component ofα, i.e.¯ α¯=α(0)idxi+α(1)idy(1)i+· · ·+α(k−1)idy(k−1)i,

withα(k−1)i=αi. Then the formula

(2.12) L(x

j_{, y}(1)j_{, . . . , y}(k−1)j_{, Y}(0)i_{, Y}(1)i_{, . . . , Y}(k−1)i_{) = (Y}(0)i_−y(1)i_)α

(0)i+ · · ·+ (Y(k−2)i_−y(k−1)i_)α

(k−2)i+Y(k−1)iα(k−1)i−h

defines a totally singular Lagrangian onTk−1_M.

The restriction ofLtoTkM has the formL0(xj, . . . , y(k)j) =y(k)iαi−h. Thus if

a totally singular LagrangianL0 on TkM has the property that α= (αi) is the top

component of a 1-formα′ _onTk−1_{M, then L}

0 is the restriction toTkM of a totally

singular LagrangianL onTk−1_{M (sinceT}k_{M ⊂T T}k−1_M).

Lethbe a totally singular Hamiltonian of orderk onM, having the correspond-ing local functionH0(xj, y(1)j, . . . , y(k−1)j, pi) =piSi(xj, y(1)j, . . . , y(k−1)j) +f(xj,

y(1)j_{, . . . , y}(k−1)j_{). We can consider the local 1-form}

¯

α = α(0)idxi +α(1)idy(1)i+ · · · +α(k−1)idy(k−1)i that is a solution of the system

(2.6). Considering the d-form αon Tk−1_{M, defined by its top component, we can}

construct a totally singular Lagrangian of orderk onM.

Theorem 2.1. Let h be a totally singular affine Hamiltonian of order k. If the system (2.7) has a d-formα= (α(k−1)i)onTk−1M as a global solution, then there is

an allowed totally singular Lagrangian,L:T Tk−1_{M →IR (onT}k−1_{M), such that:}

2. The restriction of L to Tk_{M ⊂ T T}k−1_{M is an allowed totally singular}

La-grangianL1:TkM →IR (of orderkon M).

3. The pairs(h, L1)and(E, L)are each dual pairs.

Proof. Usingα(k−1)iin (2.6), one obtain a 1-form ¯α∈ X∗(Tk−1M) given by (2.5).

Using Proposition 2.3, one obtains L. The definitions of E in 2.2 and L from 2.12, prove the first statement. One hasL1(xj, y(1)j, . . . , y(k−1)j, y(k)i) =y(k)iα(k−1)i−hα¯,

where α is the d-form on Tk−1_{M defined by (α}

(k−1)i) and the α-Hamiltonian is

hα=h◦α′, Using Proposition 2.2 one obtain the second statement. The construction

of ¯αshows thatL1is in duality withh; the last statement follows using 1. ¤

We notice that Theorem 2.1 can be adapted in the case when the d-form α = (α(k−1)i) is a solution of the system (2.7) on an open fibered submanifold of

Tk∗_{M →T}k−1_M.

Proposition 2.4. Let t→γ1 (γi

1(t), γ (1)i

1 (t), . . . , γ (k−1)i

1 (t))be an integral curve of the

semi-sprayΓ0. Then:

1. the curve γ1 is the(k−1)-tangent lift of a curvet

γ

→(γi_{(t)), i.e. γ}

1=γ(k−1);

2. the curve t→γ2 (γi_{, ω}

i)inT∗M, where

ωi(t) =αi(γi(t),

dγ dt(t), . . . ,

1 (k−1)!

dk−1γ dtk−1(t))

is a solution of the Hamilton equation of h;

3. the curve γ is a solution of the Euler equation ofL.

Proof. The first assertion follows using that Γ0 is a semi-spray. Along the curve

γ(k−1)_{one have} d

dt = Γ0. The conclusion of the second statement follows using relation

(2.7). In order to prove the third statement one use Lemma 2.1 and 2. ¤

According to [14], not all the solutions of the Lagrange equation of a totally singular Lagrangian of order 2 come from the integral curves of a semi-spray of order 1. More precisely, letL:T2M →IR,

L(xi, y(1)i, y(2)i) = 2y(2)iαi(xj, y(1)j)−2β(xi, y(1)i)

be a totally singular Lagrangian of second order. In [14] it is proved that the solutions of its Lagrange equations are the integral curves of a second order semi-spray onM, provided that the skew symmetric d-tensor ¯α given by ¯αij = _∂y∂αj(1)i −

∂αi

∂y(1)j is

non-degenerate.

3

Lifting procedures

We recall that if ¯α ∈ X∗_(Tk−1_{M) has the local expression ¯α = α}

(0)idxi +

α(1)idy(1)i+ · · · +α(k−1)idy(k−1)i, then the d-form α defined by (α(k−1)i) is called

its top component. We say that the d-form α on Tk−1_{M is non-degenerated if the}

matrix _µ

αij =

∂αi

∂y(k−1)j

¶

i,j=1,m

is non-degenerate in every point ofTk−1_{M of coordinates (x}j_{, y}(1)j_{, . . . , y}(k−1)j_{). We}

denote (αij_{) = (α}

ij)−1. Notice that the condition does not depend on coordinates.

We say that the d-formαonTk−1_{M iss-non-degenerated(the initialscomes from}

skew-symmetric) if the matrix

(3.1)

µ ˜ αij=

∂αi

∂y(k−1)j −

∂αj

∂y(k−1)i

¶

i,j=1,m

is non-degenerate in every point ofTk−1_{M of coordinates (x}j_{, y}(1)j_{, . . . , y}(k−1)j_{). We}

denote (˜αij_{) = (˜α}

ij)−1. Notice also that this condition does not depend on

coordi-nates.

LetLbe a totally singular Lagrangian of orderk≥2 having the form (2.8), such thatαis non-degenerated. Let us consider the local functions

(3.2) ti= ˜αij

µ

Γ(_Un−1)(αj) +

∂h0

∂y(k−1)j

¶ .

The following result can be proved by a straightforward verification using local coordinates.

Proposition 3.1. There is a global affine sectiont:Tk−1_{M →T}k_M,

t= Γ(_Uk−1)+ti ∂ ∂y(k)i.

We can consider the affine Hamiltonianhof order kgiven by

kH0(xj, y(1)j, . . . , y(k−1)j, pj) = piti−L(xi, . . . , y(k−1)i, ti) =

piti−ktiαi+kh0 = tipi+k(h0−tiαi).

A d-form on Tk−1_{M can be viewed as a section ω : T}k_{M → π}∗

kT∗M of the

vector bundle π∗

kT∗M → TkM, where πk : TkM → M is the canonical

projec-tion of a fibered manifold. A secprojec-tion ˜ω : TkM → ˜π∗

kT∗T M of the vector bundle

˜

π_k∗T∗T M → TkM is called a second d-form onTk−1M, where ˜πk :TkM → T M is

the canonical projection of a fibered manifold. Notice thatπ∗

kT∗M =TkM×MT∗M

and ˜π∗

kT∗T M = TkM ×T M T∗T M, as fibered products. There is a canonical

epi-morphism (i.e. a surjection on fibers) of vector bundles f1 : T∗T M → T∗M (of

cotangent vector bundles T∗_{T M → T M and T}∗_{M → M, over the canonical base} map T M). (It can be also obtained as a composition T∗_{T M → T T}∗_{M → T}∗_M, where T∗_{T M → T T}∗_{M is the canonical flip and T T}∗_{M → T}∗_{M is the canonical} projection.) Using local coordinates, f1 is given by (xi, yj, pi, qj)

f1

→ (xi_{, p}

i). Then

d-form ˜ω:Tk_{M →π˜}∗

kT∗T M induces a d-formω= ˜ω◦f˜1; we say thatωis the d-form

associatedwith ˜ω. We say that a second d-form isnon-degenerated if its associated d-form is non-degenerated.

If ˜ω is a non-degenerate second d-form that has the local expression (xi_{, y}(1)i_{, . . . , y}(k)i_)→ω˜ _(xi_{, y}(1)i_{, . . . , y}(k)i_{, β}

The fact thatS is a semi-spray can be proved by a straightforward calculation, using that the change rule of local functions{αi, βj} is

thus it is associated with the corresponding second d-form; these associations are not unique. But there are situations when if a d-form is given, one can construct in a canonical way a second d-form associated with. For example, if the d-formsis exact, i.e. there is a global functionL∈ F(Tk_{M) such that, using coordinates,ω}

i=_∂y∂L(k)i,

then ω is the top form of the differential dL and ω is associated with the second d-form ˜ωgiven locally by ( ∂L

∂y(k−1)i, ∂L

∂y(k)i). Below we consider a less trivial situation.

Let ω be a bilinear d-form on Tk−1_{M and t : T}k−1_{M → T}k_{M be an affine}

section (or a semi-spray of orderk−1). We consider the d-vector field of order k, z:Tk_{M →π}∗

if the coordinates change. The first relation is obviously fulfilled. Since ∂ ∂y(k−1)i = ∂xi′

∂xi _∂y(k∂−1)i′ +

∂y(1)i′

∂xi _∂y(∂k)i′, then, fork >1, we have ¯θi=−_∂y∂(kωj¯−1)iz j₌

−∂xi′

∂xi ∂ω¯j′

∂y(k−1)i′

∂xj′

∂xjzj− ∂y(1)i′

∂xi ∂¯ωj′

∂y(k)i′

∂xj′

∂xjzj =−∂x i′

∂xi ∂ω¯j′

∂y(k−1)i′zj ′

− ∂y_∂x(1)ii′ωi′j′zj ′

=

∂xi′

∂xiθ¯i′+∂y

(1)i′

∂xi ω¯i′, thus the second relation also holds. ¤

Notice that if ω is a symmetric bilinear d-form on Tk−1_{M, then denoting by}

¯

ωi = zjωji, ¯θi = _∂y∂(kωj¯−1)izj we obtain in a similar way a canonical non-degenerate

second d-form ˜ω, associated with ωand an affine sectiont.

Let L be a totally singular Lagrangian of order k ≥ 2 having the form (2.8). Considering the section t : Tk−1_{M → T}k_{M defined by the formula (3.2) and the}

skew symmetric and non-degenerate bilinear form ˜α on Tk_{M defined by formula}

(3.1), we can construct a non-degenerate second d-form of order k and a section S : Tk_{M →T}k+1_{M, as above. Taking ¯α}

i =αij·(y(k)j−tj(xj, y(1)j, . . . , y(k−1)j)),

then we define a new Lagrangian ¯Lof orderk+ 1, using the formula

(3.5) L(x¯ j, y(1)j, . . . , y(k+1)j) = (k+ 1)(y(k+1)i−Si)¯αi(xj, y(1)j, . . . , y(k)j).

Then ¯α= (¯αi) is an s-non-degenerate d-form onTkM, since _∂y∂αi(¯k)j− ∂αj¯

∂y(k)i = 2αij

is a non-degenerated bilinear form. Notice that the ¯α-Hamiltonian of ¯Lis defined by the local functions ¯h0 =Siα¯i. We call ¯Las the lift ofL; it is easy to see that ¯L is

also s-non-degenerated (i.e. ¯αis s-non-degenerated). In the casek= 1, letL:T M →IR,L(xi_{, y}(1)i_{) =α}

i(xj)y(1)i+β(xj) be a totally

singular Lagrangian, whereα∈ X∗_{(M),β ∈ F(M). Then the formula}

¯

L(xi_{, y}(1)i_{, y}(2)i_{) = 2(y}(2)i_−Si_(xj_{, y}(1)j_))¯ω

i+β(xj),

where ¯ωi =y(1)jαij, αij = (dα)ij = ∂αi_∂xj − ∂αj

∂xi, defines a Lagrangian of second order

onM that has a null Hessian. IfLis non-degenerated, then ¯Lis s-non-degenerated, since _∂y∂ωi(1)¯ j −

∂ωj¯

∂y(1)i = 2αij.

Acknowledgments. The second author was partially supported by a CNCSIS Grant, cod. 536/2008, contr. 695/2009.

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Authors’ address:

Marcela Popescu and Paul Popescu

University of Craiova, Department of Applied Mathematics, 13 Al.I.Cuza st., 200585 Craiova, Romania.