2 Paths and Lorentz curves on J

Vladimir Balan

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

Within the framework of jet spaces endowed with non-linear connection, are characterized the special curves of these spaces (h-paths, v-paths and geodesics, Lorentz-type paths and electromagnetic Lagrangian-action minimizers) which extend the Riemannian classical electromagnetic field model. Remarkable special cases outline the extension and computer-drawn graphic Maple-V plots for paths are provided.

Mathematics Subject Classification 2000:58A20, 58B50, 53C80, 53C22, 78A35, 53-04, 78-04.

Key words: Lagrangian, jet space, nonlinear connection, Cartan connection, deflec- tion tensor, electromagnetic tensor, stationary curve, path, Lorentz equations.

1 Geometric objects on J

(T, M )

The geometrized framework on osculating first and higher-order osculating spaces was introduced and widely studied by Acad. R.Miron and collaborators ([4], [5]). As a complementary extension of the tangent (first-order osculating) framework in the last decade was developed with significant results the geometric approach on first- order jet spaces ([11], [9], [1], [3]).

In the sequel let ξ = (E=J¹(T, M), π, T ×M) be the first order jet bundle of mappingsϕ:T →M, whereT andM areC^∞real differentiable manifolds (dimT = m, dimM =n). The local jet coordinates on Ewill be denoted by

(t^α, xⁱ, y^A)_{(α,i,A)∈I∗}≡(y^µ)µ∈I, where the set of indicesI splits as follows

I=I_h∪I_v, I_h=I_h1∪I_h2, I_v =m+n+ 1, m+n+mn Ih1 = 1, m, Ih2=m+ 1, m+n, I∗=Ih1×Ih2×Iv. and the indices implicitly take values as described below:

α, β, . . .∈Ih1; i, j, . . .∈Ih2; A, B, . . .∈Iv; λ, µ, . . .∈I.

Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 1-10.∗

c

°Balkan Society of Geometers, Geometry Balkan Press 2003.

As well, when appropriate, we identifyA=m+n+n(i−m−1) +αasA≡¡_i

α

¢and denotey^A≡x(ⁱα) = ^∂x_∂tαⁱ.

We endowE with a the extended Lagrangian of electrodynamics ([9]) of the form L(t, x, y) = ˜gAB(t, x, y)y^Ay^B+UA(t, x)y^A+ Φ(t, x),

(1.1)

whereUA(t, x) is a 1-form onE, Φ∈F(E) and assume the Kronecker decomposition

˜

gAB≡g˜₍i

α)(^jβ) =h^αβ(t, x)gij(t, x, y), (1.2)

withhαβ andgij non-degenerate tensor fields. The derived Euler-Lagrange equations evidentiate a spray, which under certain restrictive conditions provides a non-linear connectionN ={N_µ^A}µ∈Ih,A∈Iv onE which leads to the splittingT E =HE⊕V E, whereV E=Ker π∗ [11, 5]. As well,N determines the local adapted basis ofX(E)

B={δα, δi, δA}(α,i,A)∈I∗≡ {δµ}µ∈I, (1.3)

with∂α= _∂t^∂α, ∂i =_∂x^∂i and

δα=∂α−N_α^AδA, δi=∂i−N_i^AδA, δA= ˙∂A= ∂

∂y^A. (1.4)

The dual basis ofB in (1.3) writes thenB^∗={δ^α, δⁱ, δ^A}(α,i,A)∈I∗≡ {δ^µ}µ∈I,where δ^α=dt^α, δⁱ=dxⁱ, δ^A≡δy^A=dy^A+N_α^Adt^α+N_i^Adxⁱ.

(1.5)

The existence of Lagrangian-derived non-linear connections in the general Kronecker case represents still an open problem ([9]). However, in the following cases where ˜g admits a particular Kronecker splitting, the problem is tractable.

We note as particular case the ARL (almost Riemann Lagrange) jet case, where the tensor fieldhαβ(t) is a metric tensor field onT; then the Lagrangian (1.1) produces the canonical nonlinear connectionN ={N(ⁱα)

β , N(ⁱα)

j } of coefficients N(ⁱα)

β =−

¯¯

¯_αβ^γ

¯¯

¯y(ⁱγ), N(ⁱα)

j =¯

¯_jkⁱ ¯

¯y(^kα) +¹₄g^ik(2∂αgjk+hαβU(^kβ)^j), (1.6)

whereU(^kβ)^j=δ[jU¡_k]

β

¢means the h2−curl ofU; generally, we denote τ[i...j]=τi...j− τj...i and τ_{i...j}=τi...j+τj...i. Also we have

˜

gAB= ¹₂∂˙_AB² L.

(1.7)

More particular, in the ARLS (almost Riemann Lagrange separated) jet case,gij

is a metric tensor field onM, and both the nondegenerate metric tensorsh, gand the potentialsUA determine the nonlinear connectionN of coefficients

N(ⁱα)

β =−

¯¯

¯_αβ^γ

¯¯

¯y(ⁱγ), N(ⁱα)

j =¯

¯_jkⁱ ¯

¯y(^kα) +¹₄g^ik·hαβU(^k_β)^j. (1.8)

IfE is endowed with a non-linear connectionN ={N_α^A, N_i^A}, any linear connec- tion∇ ={L^λ_µν}λ,µ,ν∈I on E has its components relative to the adapted basis (1.3) provided by the relationsδ^λ(∇δνδµ) =L^λ_µν, ∀λ, µ, ν∈I. According to the three sets of indicesIh1, Ih2, Iv, these components group in 3³= 27 distinct subsets.

The subsets of nontrivial coefficients of∇can be strongly reduced for the connec- tions Γ(N) (called ”N-connections”), whose covariant derivative preserves the sections S(HE) andS(V E); these obey the conditions

L^λ_µν = 0, ∀(λ, µ)∈(I_h×I_v)∪(I_v×I_h).

(1.9)

Further, one may consider the special N-connectionsΓ∗(N), whose covariant deriv- atives preserve the distributions Span(δ_α)_α∈Ih

1 and Span(δ_i)_i∈Ih

2; they satisfy the supplementary relations

L^λ_µν = 0, ∀(λ, µ)∈(Ih₁×Ih₂)∪(Ih₂×Ih₁).

(1.10)

More particular, the so-called ”Γ-linear h-normal connections” Γn(N) [9] have just four essential sets of components

{L^α_βγ, Lⁱ_jγ, Lⁱ_jk, Lⁱ_jA} ≡ ∇, (1.11)

which provide the other 5 derived sets by means of L^A_Bγ≡L(ⁱα)

(^j_β)^γ=δ^β_αLⁱ_jγ−δⁱ_j¯

¯^β

αγ

¯¯, L^A_Bk≡L(ⁱα) (^j_β)^k =δ_α^β

¯¯

¯_jkⁱ

¯¯

¯,

L^A_BC≡L(ⁱα)

(^jβ)^C=δ^β_αLⁱ_jC, L^α_βj = 0, L^α_βC= 0.

We shall further consider the case whenhαβ(t) andgij(t, x, y) in the LagrangianLin (1.1) are non-degenerate, and we endowEwith a semi-Riemannian metric

G=hαβ(t)dt^α⊗dt^β

| {z }

h

+gij(t, x, y)dxⁱ⊗dx^j

| {z }

g

+ ˜gAB(t, x, y)δy^A⊗δy^B

| {z }

˜ g

, (1.12)

where ˜gAB ≡g˜₍i

α)(^jβ) =h^αβ(t)gij(t, x, y). In this case the so-calledthe Cartan linear connection, which is anh−normal connection, is metrical and satisfies the conditions ([11], [9])

Lⁱ_jγ =gik

2 ∂γgjk, Lⁱ_[jk]= 0, Lⁱ

[j¡_k]

α

¢= 0.

Its four essential sets of coefficients (1.11) are given by L^α_βγ=

¯¯

¯_βγ^α

¯¯

¯, Lⁱ_jγ= ¹₂g^ikδγgkj, Lⁱ_jk =

¯¯

¯_jkⁱ

¯¯

¯, Lⁱ_jA≡Lⁱ_j(^kγ)= ¹₂g^il(δ¡_{k

γ

¢g_jl}−δ(^lγ)gjk).

(1.13)

The adapted components of the torsionT and of the curvatureR of∇ are defined by the relations

δ^λ(T(δν, δµ)) =T_µν^λ , δ^λ(R(δν, δµ)δρ) =R_{ρ µν}^λ , ∀λ, µ, ν, ρ∈I.

Then the Cartan essential torsion coefficients are ([9]; for ARL case [11, Theorem 4.4])

{T(ⁱα)

γ (^jβ), T (ⁱα)

k (^jβ), T (ⁱα)

(^jβ) (^kγ), T_{β j}ⁱ , T_jAⁱ , T_{β γ}^A, T_{β j}^A, T_{i j}^A}.

The nontrivial non-holonomy coefficientsω^λ_µν are described by the relations [δµ, δν] =ω_µν^AδA≡T_µν^AδA, ∀µ, ν ∈Ih,

[δµ, δB] =ω_µB^A δA≡∂BN_µ^AδA, ∀µ∈Ih,

and are explicitly provided for the ARL case in [9, Theorem 2.3]. Ultimately, the five essential and three derived nontrivial sets of curvatureN-tensor fields are respectively

{R_{β γδ}^α , R_{j km}ⁱ , R_{j γλ}ⁱ , R_{j λA}ⁱ , R_{j CD}ⁱ }, {R(ⁱα)

(^j_β) ^γδ, R(ⁱα)

(^j_β) ^λk, R(ⁱα) (^j_β) ^µA}, forλ∈Ih, µ∈I.

In this framework, the Liouville fieldC=y^Aδ_A on (E, N,∇) produces thedeflec- tion tensor fields

d^A_µ =δ^A∇δµC, µ∈I, A∈Iv,

which lead further to the associated toNand∇electromagnetic 2-formF =FAµδy^A∧ δy^µ, of nontrivial components















FAβ≡F₍ⁱ_α)β=¹₂³

h^αγgiky(^k[γ)´

|β]

FAB ≡F₍i

α)(^jβ) = ¹²˜g¡_[i

α

¢Cy^C

|¡_j]

β

¢

FAj ≡F₍ⁱ_α)j= ¹₂ d¡_[i

α

¢j]=¹₂ y¡_[i

α

¢|j]= ¹₂

³

y(^kγ)h^αγg_k[i

´

|j], (1.14)

where|α,|iand|A are the covariant derivations given by∇δµ, forµ∈Ih1, Ih2 andIv

respectively. Considering the raising/lowering of the indices performed by the metric tensor fieldG,F providesthe electromagnetic force

F˜ =F_A^µδµ⊗δ^A (1.15)

of nontrivial essential components,

F_A^α=h^αβFAβ, F_Aⁱ =g^ijFAj, F_A^C=g^CDFAD.

We note that in the particular ARLS case, the Cartan connection has just two basic nontrivial coefficients

{L^α_βγ=¯¯^α

βγ

¯¯, Lⁱ_jk=¯¯ⁱ

jk

¯¯},

and its non-trivial torsionN−fields are {T(^mγ)

αβ , T(^mγ)

ij , T(^mγ)

αj }([9]).

Moreover, form= 1, n= 4 andh11= 1, one finds as particular case, the pseudo- Riemannian weak gravitational model endowed with the metric gij(x) =ηij+εij(x), where the weakness of the gravitational fieldg_ijis expressed by its decomposition into the flat Minkowski metricnij =diag(−1,1,1,1) and a small perturbationεij(x), a symmetric tensor field with|εij(x)|<<1.

2 Paths and Lorentz curves on J

(T, M )

We consider in the following on (E, N,∇) smooth curves c : J = [a, b] ⊂ R →E, having their images inside a chart ˜U ⊂E, locally given by

c(s) = (t^α(s), xⁱ(s), y^A(s))≡(y^µ(s)),∀t∈J.

Definitions.a) The fieldV^µ= ^δyds^µ defined oncis calledN-velocity fieldof the curve c. Its components are explicitely given by

{V^µ}µ∈I ≡ µ

t˙^α,x˙ⁱ,δy^a

ds = ˙y^A+N_β^At˙^β+N_j^Ax˙^j

¶

(α,i,A)∈I∗

where we denote by dot thes-derivation. We denote by F =F^µδµ theN-force field onc, which provides the motion of the test-body alongc and whose components are explicitely given by

F^µ=∇V^µ ds

not= δV^µ

ds +L^µ_νρV^νV^ρ.

b) We callc stationary curvewith respect to∇ iffF = 0 along the curve.

c) The curvec is called

• h−curve, ifπv(V) = 0, and

• v−curve, ifπh(V) = 0,

where byπhandπvwe denoted respectively theh−andv−projectors of the canonic splitting induced byN.

d) An h−/v−curve which satisfies also the extra condition F = 0, is called h−/v−path, respectively.

Analytically, these curves are described by

Theorem 1. Let c : J ⊂ R → E be a curve. Then the curves defined above are characterized as follows:

a)c is an h−curve iff

V^A= 0 ⇔ ^δy_ds^A = 0 ⇔ y˙^A+N_α^At˙^α+N_j^Ax˙^j = 0, ∀A∈Iv. (2.16)

b) cis av−curve iff

V^µ= 0, ∀µ∈I_h ⇔ ^δy_ds^µ = 0,∀µ∈I_h ⇔ c(s) = (t₀, x₀, y(s)), s∈J.

(2.17)

c)cis anh−path (”stationaryh−curve or ”horizontal geodesic”) iff besides (2.16) it satisfies

dV^µ

ds +L^µ_νρV^νV^ρ= 0, ∀µ∈Ih. (2.18)

d)c is av−path (”stationaryv−curve or ”vertical geodesic”) iff besides (2.17) it satisfies

δV^A

ds +L^A_BCV^BV^C= 0, ∀A∈Iv. (2.19)

We note that the implicit sum in the right term of (2.18)/(2.19) involves just hori- zontal/vertical index types. The proof is computational.

Consider the triple (E, N, G), where the metric G in the one in (1.12), N is a fixed nonlinear connection, and ∇ is the Cartan connection attached to G of basic coefficients (1.13). Then qne can derive the electromagnetic tensor fields in (1.14) and (1.15) and we have

Definition.A curvec is calledLorentz curveon (E, N, G) iff Gνρ

∇V^ρ

ds =FAνV^A ⇔ ∇V^µ

ds =F_A^µV^A. (2.20)

Theorem 2.([1, 3])The Lorentz equations (2.20) have the equivalent form t¨^α+L^α_βCt˙^βV^C+L^α_jCx˙^jV^C+L^α_βγt˙^βt˙^γ+L^α_jγx˙^jt˙^γ+L^α_βkt˙^βx˙^k+L^α_jkx˙^jx˙^k=F_B^αV^B (2.21)

¨

xⁱ+Lⁱ_βCt˙^βV^C+Lⁱ_jCx˙^jV^C+Lⁱ_βγt˙^βt˙^γ+Lⁱ_jγx˙^jt˙^γ+Lⁱ_βkt˙^βx˙^k+Lⁱ_jkx˙^jx˙^k=F_BⁱV^B (2.22)

V˙^A+N_α^At˙^α+N_i^Ax˙ⁱ+L^A_CβV^Ct˙^β+L^A_CjV^Cx˙^j+L^A_BCV^BV^C =F_B^AV^B, (2.23)

whereV^A= ˙y^A+N_β^At˙^β+N_i^Ax˙ⁱ, A∈Iv.

Remarks.a) The Lorentz h-pathssatisfy the extra conditionsV^A = 0, A∈Iv and since the right side of (2.21)-(2.23) is identically vanishing, they coincide with the usualh-paths of (E, N, G).

b) TheLorentzv-pathshave fixed base-point, i.e.,

V^µ= 0, µ∈Ih ⇔ (t, x) = (t0, x0)∈T×M, and hence the associated Lorentz equations rewrite

F_B^αV^B = 0, F_BⁱV^B= 0, F_B^AV^B= ˙V^A+L^A_BCV^BV^C.

c) In the ARLS casewith the nonlinear connection (1.6) induced by the Lagrangian, the electromagnetic tensors simplify to

F_A^α≡F(^αi_β)^γ = 0, F_Aⁱ =g^ijF˜Aj=−1

4g^ijUAj, F_A^B = 0, (2.24)

and the nonvanishing Cartan connection essential coefficients reduce to L^α_βγ=¯

¯_βγ^α¯

¯, Lⁱ_jk=¯

¯_jkⁱ ¯

¯, L^A_Bγ ≡L(ⁱα)

(^jβ)^γ =−δ_jⁱ¯

¯_αγ^β ¯

¯, L^A_Bk≡L(ⁱα)

(^jβ)^k =−δ_α^β¯

¯_jkⁱ ¯

¯.

Then the Lorentz equations (2.21)-(2.23) get the typical shape

¨t^α+¯

¯^α

βγ

¯¯t˙^βt˙^γ = 0, x¨ⁱ+¯

¯ⁱ

jk

¯¯x˙^jx˙^k =−1

4g^ijUAjV^A, V˙^A= 0.

Note that in this case (gdependent on xonly), the Berwald connection [11] has the same coefficients as the Cartan connection, and hence the associated Lorentz curves, h- and v-paths are described by the same equations. The Lorentzh-paths obey the extra equations

˙

y^A+N_β^At˙^β+N_j^Ax˙^j= 0, A∈Iv, which write explicitely

˙ y(ⁱα)−

¯¯

¯_αβ^γ

¯¯

¯y(ⁱγ) ˙t^β+ µ¯

¯_jkⁱ ¯

¯y(^kα) + 1

4g^ikhαβU(^kβ)^j

¶

˙ x^j= 0.

As well, the Lorentz v-paths for the Cartan connection satisfy the extra condition

−V^A= 2 ˙V^A, having as solutions flat rays within the fibers ofE - in accordance with the particular caseJ¹(R, M)≡T M studied in [6].

d) In the ARLSUcase (ARLS uniparametric case, form= 1 ands=t¹=t, [2]), for h11 = 1, we recapture the known results derived in [4, 6] for the tangent space case. For this, after shifting the indices left by one unit (Ih2 = 1, n, Iv=n+ 1,2n), y^A ≡y(ⁱ1) ^not= yⁱ, set locally h11 = 1 and we can usethe Finsler-Lagrange tangent space notationsfrom [5].

If we consider the Lagrangian (1.1) of the particular form L(x, y) =mc γij(x)yⁱy^j+2e

mUi(x)yⁱ+ Φ(x), (2.25)

with γij pseudo-Riemannian metric,U =Uidxⁱ 1-form on M and Φ∈F(M), then the fundamental tensor derived fromL via (1.7) is

˜

g(ⁱ1)(^j1)(t, x, y) =gij(x) =mc γij(x), the non-linear connection induced byLhas the components

N₁^A= 0, N(ⁱ1)

j =¯

¯_jkⁱ ¯

¯y^k+g^ikU(^k1)^j, i= 1, n, A=n+ 1,2n,

whereU(^k1) = ^meAk. In this case, the Cartan (1.13) and Berwald canonic connections have just null and Christoffel (re-indexed) components. For∇Cartan connection, the

Lorentz equations (2.22) confine to the known ones of Lagrange spaces ([5], [4, p.

171])

¨

xⁱ+ 2Gⁱ(x, y) = 0, yⁱ= dxⁱ

ds, i= 1, m (2.26)

of the Lagrangian spray derived from the LagrangianLin (2.25) for Φ constant, Gⁱ= 1

2γⁱ_jky^jy^k+ e

2m²cγ^ijA[j;k]y^k, where ”;k” expresses the canonic covariant derivative on (M, γij).

We note that in the absence of the electromagnetic forceFµA, the equations (2.20) rewritten in the form (2.26) become the equations of stationary curves of the con- nection ∇. Hence, in the absence of the covector potential U, the equations (2.20) becomethe equations of geodesics of the manifoldM and the equations ofh−paths become the Lorentz equations.

3 Electromagnetic Lagrangian extremals

In the ARLS case the extremals of the energy action E(L) =

Z

T

L(t, x, y) dvolT

(3.27)

of the LagrangianLin (1.1) are shown to satisfy the PDE system ([8]) h^αβ(∂βy(ⁱα) + 2G(ⁱα)

β ) = 0, i= 1, n.

(3.28)

In (3.28), an essential role plays the spray G(ⁱα)

β =¹G(ⁱα)

β +²G(ⁱα)

β associated to L, where







1G(ⁱα)

β =−¹₂

¯¯

¯_αβ^γ

¯¯

¯y(ⁱγ)

2G(ⁱα)

β = ¹₂

¯¯

¯_jkⁱ

¯¯

¯y(^jα)y(^kβ) +_4m¹ ˜g(ⁱα)(^lβ)(U₍l

ε)sy^(sε)+∂εU₍^l_ε)+U(^lγ)

¯¯_γε^ε ¯

¯−∂lΦ), which provides the canonicL-induced nonlinear connectionN in (1.8) via

N(ⁱα)

β = 2∂(¹G(ⁱα)

β )

∂y(^jγ) y(^jγ), N(ⁱα)

j = 2∂(²G(ⁱδ)

ε h^δε)

∂y(^jγ) hαγ.

We note that in the ARLSU case for m= 1 and h11= 1, using the conventions above, the extremals of the Lagrangian action are characterized by the equations

¨ xⁱ+¯

¯_jkⁱ ¯

¯x˙^jx˙^k= 1

4(F_jⁱy^j+g^ij∂jΦ),

and for constant Φ these coincide with the extended Lorentz paths produced by the Liouville tensor field.

4 Numerical simulation

In the ARLS uniparametric case detailed above, consider n = 2, M endowed with the Lagrangian L in (1.1) with g = mcγij, Φ = 0 and the potential ¯U given by U¯ = ε(x¹dx²−x²dx¹), ε ∈ R. Then, denoting by a = εe(m²c)⁻¹ the control pa- rameter of electromagnetid field strength, the appropriately rescaled Lorentz-type equations (2.26) read

¨ xⁱ+¯

¯ⁱ

jk

¯¯x˙^jx˙^k= (−1)ⁱ⁺¹a(gⁱ¹x˙²+gⁱ²x˙¹), i= 1,2.

(4.29)

We exemplify further the influence of the electromagnetic forceF derived from ˜U via (2.24) onh-paths for three cases: R², H² and S². Using Maple V programming were obtained computer-drawn images representing the Lorentz-type sheaves of curves (the left-bended lines in the drawings) which are obtained for fixed non-zero values ofa(a=−512 for Euclidean case,a=−1024 for the Poincare half-plane, a= 2 for the sphere respectively).

R² H² S²

We note that, when the influence of the generalized electric potentials Ui(x) dis- appears (i.e., fora= 0 regarded as a limit case), one obtains the sheaves of geodesics of the manifold M (marked with thick lines). Hence the geodesics - the solutions for a= 0 of the system (4.29) deform to Lorentz curves, under the controlled by a influence of the generalized electromagnetic tensor field.

Acknowledgments. The author is grateful to Professor C.Udri¸ste for the interesting insights on the present subject. The present work was partially supported by Grant CNCSIS MEN 477 (75) / 2003.

References

[1] V. Balan,Lorentz-type equations in first-order jet spaces endowed with nonlinear connection, Proceedings of The First French-Romanian Colloquium of Numerical Physics, October 30-31, 2000, Bucharest, Romania, Geometry Balkan Press 2002, pp. 105-114.

[2] V.Balan, Synge-Beil and Riemann-Jacobi jet structures with applications to physics, Jour. of Math. and Math. Sci, Hindawi Publ. Corp., 27 (2003), 1693- 1702.

[3] V. Balan, K. Trencevski, Stationary Lorentz curves in geometrized jet spaces.

Numerical simulation and analytic solutions, Algebras, Groups and Geometries, I.B.R., Palm Harbor, to appear.

[4] R. Miron,The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publishers, 1994.

[5] R. Miron, M.Anastasiei, Vector Bundles. Lagrange Spaces. Applications to Relativity, Geometry Balkan Press, 1996.

[6] R. Miron, R.Ro¸sca, M.Anastasiei, K.Buchner, New aspects in Lagrangian rela- tivity, Found. of Phys. Lett., v. 5 (1992), n.2, 141-171.

[7] R. Miron, M. Tatoiu-Radivoiovici,A Lagrangian theory of electromagnetism, Rep.

Math. Phys., 27 (1989), 49-84.

[8] M.Neagu, The geometry of autonomous metrical multi-time Lagrange space of electrodynamics, International Journal of Mathematics and Mathematical Sci- ences, Hindawi Publishing Corporation, (2001),

http://xxx.lanl.gov/abs/math.DG/0010091, (2000).

[9] M. Neagu, Generalized metrical multi-time Lagrangian geometry of physical fields, http://xxx.lanl.gov/abs/math.DG/0011003, 2000.

[10] M. Neagu, C. Udri¸ste, From PDE Systems and Metrics to Geometric Multi-Time Field Theories, Sem. Mec. 79, West Univ. of Timisoara, 2001.

http://xxx.lanl.gov/abs/math.DG/0009071, 2000.

[11] M. Neagu, C. Udri¸ste, The geometry of metrical multi-time Lagrange spaces, http://xxx.lanl.gov/abs/math.DG/0009071, 2000.

[12] D.J. Saunders,The Geometry of Jet Bundles, Cambridge University Press, 1989.

[13] C.Udri¸ste,Geometric Dynamics, Kluwer Acad. Publishers, 2000.

Vladimir Balan

University Politehnica of Bucharest, Department Mathematics I Splaiul Independent¸ei 313, RO-77206 Bucharest, Romania Email address: [email protected]