Notable Curves in Geometrized J^{1}(T, M) Framework

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, deflection tensor, electromagnetic tensor, stationary curve, path, Lorentz equations.

1 Geometric objects on J^{1}(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 firstorder jet spaces ([11], [9], [1], [3]).

In the sequel let ξ = (E = J^{1}(T, M), π, T × M) be the first order jet bundle of mappings φ : T → M, where T and M are C^{∞} real differentiable manifolds (dim T = m, dim M = n). The local jet coordinates on E will be denoted by

(t^{α}, x^{i}, y^{A})_{(α,i,A)∈I_{∗}} ≡ (y^{μ})_{μ∈I},

where the set of indices I splits as follows I = I_{h} ∪ I_{v}, I_{h} = I_{h_{1}} ∪ I_{h_{2}}, I_{v} = \overline{m + n + 1, m + n + mn}

I_{h_{1}} = \overline{1, m}, I_{h_{2}} = \overline{m + 1, m + n}, I_{∗} = I_{h_{1}} × I_{h_{2}} × I_{v}.

and the indices implicitly take values as described below:

α, β, … ∈ I_{h_{1}}; i, j, … ∈ I_{h_{2}}; A, B, … ∈ I_{v}; λ, μ, … ∈ I.

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

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As well, when appropriate, we identify A = m + n + n(i − m − 1) + α as A ≡ \binom{i}{α} and denote y^{A} ≡ x^{\binom{i}{α}} = \frac{∂x^{i}}{∂t^{α}}.

We endow E with a the extended Lagrangian of electrodynamics ([9]) of the form

L(t, x, y) = g̃_{AB}(t, x, y)y^{A}y^{B} + U_{A}(t, x)y^{A} + Φ(t, x), (1.1)

where U_{A}(t, x) is a 1-form on E, Φ ∈ F (E) and assume the Kronecker decomposition

g̃_{AB} ≡ g̃_{\binom{i}{α}}\binom{j}{β} = h^{αβ}(t, x)g_{ij}(t, x, y), (1.2)

with h_{αβ} and g_{ij} non-degenerate tensor fields. The derived Euler-Lagrange equations evidentiate a spray, which under certain restrictive conditions provides a non-linear connection N = {N_{μ}^{A}}_{μ∈I_{h},A∈I_{v}} on E which leads to the splitting T E = HE ⊕ V E, where V E = Ker π_{∗} [11, 5]. As well, N determines the local adapted basis of X (E)

B = {δ_{α}, δ_{i}, δ_{A}}_{(α,i,A)∈I_{∗}} ≡ {δ_{μ}}_{μ∈I}, (1.3)

with ∂_{α} = \frac{∂}{∂t^{α}} , ∂_{i} = \frac{∂}{∂x^{i}} and

δ_{α} = ∂_{α} − N_{α}^{A}δ_{A}, δ_{i} = ∂_{i} − N_{i}^{A}δ_{A}, δ_{A} = ˙∂_{A} = \frac{∂}{∂y^{A}} . (1.4)

The dual basis of B in (1.3) writes then B^{∗} = {δ^{α}, δ^{i}, δ^{A}}_{(α,i,A)∈I_{∗}} ≡ {δ^{μ}}_{μ∈I}, where

δ^{α} = dt^{α}, δ^{i} = dx^{i}, δ^{A} ≡ δy^{A} = dy^{A} + N_{α}^{A}dt^{α} + N_{i}^{A}dx^{i}. (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 g̃ admits a particular Kronecker splitting, the problem is tractable.

We note as particular case the ARL (almost Riemann Lagrange) jet case, where the tensor field h_{αβ}(t) is a metric tensor field on T; then the Lagrangian (1.1) produces

the canonical nonlinear connection N = \{N\binom{i}{α}_{β}, N\binom{i}{α}_{j}\} of coefficients

N\binom{i}{α}_{β} = − |_{αβ}^{γ}| y^{\binom{i}{γ}}, N\binom{i}{α}_{j} = |_{jk}^{i}| y^{\binom{k}{α}} + \frac{1}{4}g^{ik}(2∂_{α}g_{jk} + h_{αβ}U_{\binom{k}{β}j}), (1.6)

where U_{\binom{k}{β}j} = δ_{[j}U_{\binom{k]}{β}} means the h_{2}−curl of U; generally, we denote τ_{[i...j]} = τ_{i...j} −

τ_{j...i} and τ_{{i...j}} = τ_{i...j} + τ_{j...i}. Also we have

g̃_{AB} = \frac{1}{2} ˙∂_{AB}^{2}L. (1.7)

More particular, in the ARLS (almost Riemann Lagrange separated) jet case, g_{ij} is a metric tensor field on M, and both the nondegenerate metric tensors h, g and the potentials U_{A} determine the nonlinear connection N of coefficients

N\binom{i}{α}_{β} = − |_{αβ}^{γ}| y^{\binom{i}{γ}}, N\binom{i}{α}_{j} = |_{jk}^{i}| y^{\binom{k}{α}} + \frac{1}{4}g^{ik} · h_{αβ}U_{\binom{k}{β}j}. (1.8)

If E is endowed with a non-linear connection N = {N_{α}^{A}, N_{i}^{A}}, any linear connection ∇ = {L_{μν}^{λ}}_{λ,μ,ν∈I} on E has its components relative to the adapted basis (1.3) provided by the relations δ^{λ}(∇_{δ_{ν}} δ_{μ}) = L_{μν}^{λ}, ∀λ, μ, ν ∈ I. According to the three sets of indices I_{h_{1}}, I_{h_{2}}, I_{v}, these components group in 3^{3} = 27 distinct subsets.

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

L_{μν}^{λ} = 0, ∀ (λ, μ) ∈ (I_{h} × I_{v}) ∪ (I_{v} × I_{h}). (1.9)

Further, one may consider the special N-connections Γ_{∗}(N), whose covariant derivatives preserve the distributions Span(δ_{α})_{α∈I_{h_{1}}} and Span(δ_{i})_{i∈I_{h_{2}}} ; they satisfy the supplementary relations

L_{μν}^{λ} = 0, ∀ (λ, μ) ∈ (I_{h_{1}} × I_{h_{2}}) ∪ (I_{h_{2}} × I_{h_{1}}). (1.10)

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

{L_{βγ}^{α}, L_{jγ}^{i}, L_{jk}^{i}, L_{jA}^{i}} ≡ ∇, (1.11)

which provide the other 5 derived sets by means of

\begin{matrix}L_{Bγ}^{A} ≡ L\binom{i}{α}_{\binom{j}{β}γ} = δ_{α}^{β}L_{jγ}^{i} − δ_{j}^{i} |_{αγ}^{β} | , L_{Bk}^{A} ≡ L\binom{i}{α}_{\binom{j}{β}k} = δ_{α}^{β} |_{jk}^{i}| , \\ L_{BC}^{A} ≡ L\binom{i}{α}_{\binom{j}{β}C} = δ_{α}^{β}L_{jC}^{i}, L_{βj}^{α} = 0, L_{βC}^{α} = 0. \end{matrix}

We shall further consider the case when h_{αβ}(t) and g_{ij}(t, x, y) in the Lagrangian L in (1.1) are non-degenerate, and we endow E with a semi-Riemannian metric

G = \underbrace{h_{αβ}(t)dt^{α} ⊗ dt^{β}}_{h} + \underbrace{g_{ij}(t, x, y)dx^{i} ⊗ dx^{j}}_{g} + \underbrace{g̃_{AB}(t, x, y)δy^{A} ⊗ δy^{B}}_{g̃}, (1.12)

where g̃_{AB} ≡ g̃_{\binom{i}{α}}\binom{j}{β} = h^{αβ}(t)g_{ij}(t, x, y). In this case the so-called the Cartan linear connection, which is an h−normal connection, is metrical and satisfies the conditions ([11], [9])

L_{jγ}^{i} = \frac{g_{ik}}{2} ∂_{γ}g_{jk}, L_{[jk]}^{i} = 0, L_{[j\binom{k]}{α}}^{i} = 0.

Its four essential sets of coefficients (1.11) are given by

\begin{matrix}L_{βγ}^{α} = |_{βγ}^{α} | , L_{jγ}^{i} = \frac{1}{2}g^{ik}δ_{γ}g_{kj}, L_{jk}^{i} = |_{jk}^{i}| , \\ L_{jA}^{i} ≡ L_{j\binom{k}{γ}}^{i} = \frac{1}{2}g^{il}(δ_{\binom{\{k}{γ}}g_{jl\}} − δ_{\binom{l}{γ}}g_{jk}). \end{matrix} (1.13)

The adapted components of the torsion T and of the curvature R of ∇ are defined by the relations

δ^{λ}(T (δ_{ν}, δ_{μ})) = T_{μν}^{λ}, δ^{λ}(R(δ_{ν}, δ_{μ})δ_{ρ}) = 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 \binom{i}{α}_{γ \binom{j}{β}}, T \binom{i}{α}_{k \binom{j}{β}}, T\begin{matrix}( & {i \atop α} & ) \\ \binom{j}{β} & \binom{k}{γ} \end{matrix}, T_{β j}^{i}, T_{jA}^{i}, T_{β γ}^{A}, T_{β j}^{A}, T_{i j}^{A}\}.

The nontrivial non-holonomy coefficients ω_{μν}^{λ} are described by the relations

\begin{matrix}[δ_{μ}, δ_{ν}] = ω_{μν}^{A}δ_{A} ≡ T_{μν}^{A}δ_{A}, & ∀μ, ν ∈ I_{h}, \\ [δ_{μ}, δ_{B}] = ω_{μB}^{A}δ_{A} ≡ ∂_{B}N_{μ}^{A}δ_{A}, & ∀μ ∈ I_{h}, \end{matrix}

and are explicitly provided for the ARL case in [9, Theorem 2.3]. Ultimately, the five essential and three derived nontrivial sets of curvature N-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 \binom{i}{α}_{\binom{j}{β} γδ}, R \binom{i}{α}_{\binom{j}{β} λk}, R \binom{i}{α}_{\binom{j}{β} μA}\},

for λ ∈ I_{h}, μ ∈ I.

In this framework, the Liouville field C = y^{A}δ_{A} on (E, N, ∇) produces the deflection tensor fields

d_{μ}^{A} = δ^{A}∇_{δ_{μ}}C, μ ∈ I, A ∈ I_{v},

which lead further to the associated to N and ∇ electromagnetic 2-form F = F_{Aμ}δy^{A}∧ δy^{μ}, of nontrivial components

\{ \begin{matrix}F_{Aβ} ≡ F_{\binom{i}{α}β} = \frac{1}{2} (h^{αγ}g_{ik}y^{\binom{k}{[γ}})_{|β]} \\ F_{AB} ≡ F_{\binom{i}{α}}\binom{j}{β} = \frac{1}{2}g̃_{\binom{[i}{α}C}y_{|\binom{j]}{β}}^{C} \\ F_{Aj} ≡ F_{\binom{i}{α}j} = \frac{1}{2} d_{\binom{[i}{α}j]} = \frac{1}{2} y_{\binom{[i}{α}|j]} = \frac{1}{2} (y^{\binom{k}{γ}}h^{αγ}g_{k[i})_{|j]} , \end{matrix} (1.14)

where |α, |i and |A are the covariant derivations given by ∇_{δ_{μ}}, for μ ∈ I_{h_{1}}, I_{h_{2}} and I_{v} respectively. Considering the raising/lowering of the indices performed by the metric tensor field G, F provides the electromagnetic force

F̃ = F_{A}^{μ}δ_{μ} ⊗ δ^{A} (1.15)

of nontrivial essential components,

F_{A}^{α} = h^{αβ}F_{Aβ}, F_{A}^{i} = g^{ij}F_{Aj}, F_{A}^{C} = g^{CD}F_{AD}.

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

{L_{βγ}^{α} = |_{βγ}^{α} | , L_{jk}^{i} = |_{jk}^{i}|},

and its non-trivial torsion N−fields are \{T \binom{m}{γ}_{αβ}, T \binom{m}{γ}_{ij}, T \binom{m}{γ}_{αj}\} ([9]).

Moreover, for m = 1, n = 4 and h_{11} = 1, one finds as particular case, the pseudo-Riemannian weak gravitational model endowed with the metric g_{ij}(x) = η_{ij} +ε_{ij}(x), where the weakness of the gravitational field g_{ij} is expressed by its decomposition into the flat Minkowski metric n_{ij} = diag(−1, 1, 1, 1) and a small perturbation ε_{ij}(x), a symmetric tensor field with |ε_{ij}(x)| << 1.

2 Paths and Lorentz curves on J^{1}(T, M)

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

c(s) = (t^{α}(s), x^{i}(s), y^{A}(s)) ≡ (y^{μ}(s)), ∀t ∈ J.

Definitions. a) The field V^{μ} = \frac{δy^{μ}}{ds} defined on c is called N-velocity field of the curve c. Its components are explicitely given by

\{V^{μ}\}_{μ∈I} ≡ (ṫ^{α}, ẋ^{i}, \frac{δy^{a}}{ds} = ẏ^{A} + N_{β}^{A}ṫ^{β} + N_{j}^{A}ẋ^{j})_{(α,i,A)∈I_{∗}}

where we denote by dot the s-derivation. We denote by F = F^{μ}δ_{μ} the N-force field on c, which provides the motion of the test-body along c and whose components are explicitely given by

F^{μ} = \frac{∇V^{μ}}{ds} =^{not} \frac{δV^{μ}}{ds} + L_{νρ}^{μ}V^{ν}V^{ρ}.

b) We call c stationary curve with respect to ∇ iff F = 0 along the curve.

c) The curve c is called

• h−curve, if π_{v}(V) = 0, and

• v−curve, if π_{h}(V) = 0, where by π_{h} and π_{v} we denoted respectively the h− and v−projectors of the canonic splitting induced by N.

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 ⇔ \frac{δy^{A}}{ds} = 0 ⇔ ẏ^{A} + N_{α}^{A}ṫ^{α} + N_{j}^{A}ẋ^{j} = 0, ∀A ∈ I_{v}. (2.16)

b) c is a v−curve iff

V^{μ} = 0, ∀μ ∈ I_{h} ⇔ \frac{δy^{μ}}{ds} = 0, ∀μ ∈ I_{h} ⇔ c(s) = (t_{0}, x_{0}, y(s)), s ∈ J. (2.17)

c) c is an h−path (”stationary h−curve or ”horizontal geodesic”) iff besides (2.16) it satisfies

\frac{dV^{μ}}{ds} + L_{νρ}^{μ}V^{ν}V^{ρ} = 0, ∀μ ∈ I_{h}. (2.18)

d) c is a v−path (”stationary v−curve or ”vertical geodesic”) iff besides (2.17) it satisfies

\frac{δV^{A}}{ds} + L_{BC}^{A}V^{B}V^{C} = 0, ∀A ∈ I_{v}. (2.19)

We note that the implicit sum in the right term of (2.18)/(2.19) involves just horizontal/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 curve c is called Lorentz curve on (E, N, G) iff

G_{νρ} \frac{∇V^{ρ}}{ds} = F_{Aν}V^{A} ⇔ \frac{∇V^{μ}}{ds} = F_{A}^{μ}V^{A}. (2.20)

Theorem 2. ([1, 3]) The Lorentz equations (2.20) have the equivalent form

ẗ^{α} + L_{βC}^{α} ṫ^{β}V^{C} + L_{jC}^{α}ẋ^{j}V^{C} + L_{βγ}^{α} ṫ^{β}ṫ^{γ} + L_{jγ}^{α}ẋ^{j}ṫ^{γ} + L_{βk}^{α}ṫ^{β}ẋ^{k} + L_{jk}^{α}ẋ^{j}ẋ^{k} = F_{B}^{α}V^{B} (2.21)

ẍ^{i} + L_{βC}^{i} ṫ^{β}V^{C} + L_{jC}^{i}ẋ^{j}V^{C} + L_{βγ}^{i} ṫ^{β}ṫ^{γ} + L_{jγ}^{i}ẋ^{j}ṫ^{γ} + L_{βk}^{i}ṫ^{β}ẋ^{k} + L_{jk}^{i}ẋ^{j}ẋ^{k} = F_{B}^{i}V^{B} (2.22)

V̇^{A} + N_{α}^{A}ṫ^{α} + N_{i}^{A}ẋ^{i} + L_{Cβ}^{A}V^{C} ṫ^{β} + L_{Cj}^{A}V^{C}ẋ^{j} + L_{BC}^{A}V^{B}V^{C} = F_{B}^{A}V^{B}, (2.23)

where V^{A} = ẏ^{A} + N_{β}^{A}ṫ^{β} + N_{i}^{A}ẋ^{i}, A ∈ I_{v}. Remarks. a) The Lorentz h-paths satisfy the extra conditions V^{A} = 0, A ∈ I_{v} and since the right side of (2.21)-(2.23) is identically vanishing, they coincide with the usual h-paths of (E, N, G).

b) The Lorentz v-paths have fixed base-point, i.e.,

V^{μ} = 0, μ ∈ I_{h} ⇔ (t, x) = (t_{0}, x_{0}) ∈ T × M,

and hence the associated Lorentz equations rewrite

F_{B}^{α}V^{B} = 0, F_{B}^{i}V^{B} = 0, F_{B}^{A}V^{B} = V̇^{A} + L_{BC}^{A}V^{B}V^{C}.

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

F_{A}^{α} ≡ F_{\binom{i}{β}γ}^{α} = 0, F_{A}^{i} = g^{ij}F̃_{Aj} = −\frac{1}{4}g^{ij}U_{Aj}, F_{A}^{B} = 0, (2.24)

and the nonvanishing Cartan connection essential coefficients reduce to

L_{βγ}^{α} = |_{βγ}^{α} | , L_{jk}^{i} = |_{jk}^{i}| , L_{Bγ}^{A} ≡ L\binom{i}{α}_{\binom{j}{β}γ} = −δ_{j}^{i} |_{αγ}^{β} | , L_{Bk}^{A} ≡ L\binom{i}{α}_{\binom{j}{β}k} = −δ_{α}^{β} |_{jk}^{i}| .

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

ẗ^{α} + |_{βγ}^{α} | ṫ^{β}ṫ^{γ} = 0, ẍ^{i} + |_{jk}^{i}| ẋ^{j}ẋ^{k} = −\frac{1}{4}g^{ij}U_{Aj}V^{A}, V̇^{A} = 0.

Note that in this case (g dependent on x only), 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 Lorentz h-paths obey the extra equations

ẏ^{A} + N_{β}^{A}ṫ^{β} + N_{j}^{A}ẋ^{j} = 0, A ∈ I_{v},

which write explicitely

ẏ^{\binom{i}{α}} − |_{αβ}^{γ}| y^{\binom{i}{γ}}ṫ^{β} + (|_{jk}^{i}| y^{\binom{k}{α}} + \frac{1}{4}g^{ik}h_{αβ}U_{\binom{k}{β}j}) ẋ^{j} = 0.

As well, the Lorentz v-paths for the Cartan connection satisfy the extra condition −V^{A} = 2V̇^{A}, having as solutions flat rays within the fibers of E - in accordance with the particular case J^{1}( R, M) ≡ T M studied in [6].

d) In the ARLSU case (ARLS uniparametric case, for m = 1 and s = t^{1} = t, [2]), for h_{11} = 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 (I_{h_{2}} = \overline{1, n}, I_{v} = \overline{n + 1, 2n}), y^{A} ≡ y^{\binom{i}{1}} =^{not} y^{i}, set locally h_{11} = 1 and we can use the Finsler-Lagrange tangent space notations from [5].

If we consider the Lagrangian (1.1) of the particular form

L(x, y) = mc γ_{ij}(x)y^{i}y^{j} + \frac{2e}{m}U_{i}(x)y^{i} + Φ(x), (2.25)

with γ_{ij} pseudo-Riemannian metric, U = U_{i}dx^{i} 1-form on M and Φ ∈ F (M), then the fundamental tensor derived from L via (1.7) is

g̃\binom{i}{1}\binom{j}{1}(t, x, y) = g_{ij}(x) = mc γ_{ij}(x),

the non-linear connection induced by L has the components

N_{1}^{A} = 0, N\binom{i}{1}_{j} = |_{jk}^{i}| y^{k} + g^{ik}U_{\binom{k}{1}j}, i = \overline{1, n}, A = \overline{n + 1, 2n},

where U\binom{k}{1} = \frac{e}{m}A_{k}. 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])

ẍ^{i} + 2G^{i}(x, y) = 0, y^{i} = \frac{dx^{i}}{ds} , i = \overline{1, m} (2.26)

of the Lagrangian spray derived from the Lagrangian L in (2.25) for Φ constant,

G^{i} = \frac{1}{2}γ_{jk}^{i}y^{j}y^{k} + \frac{e}{2m^{2}c}γ^{ij}A_{[j;k]}y^{k},

where ”; k” expresses the canonic covariant derivative on (M, γ_{ij}).

We note that in the absence of the electromagnetic force F_{μ_{A}}, the equations (2.20) rewritten in the form (2.26) become the equations of stationary curves of the connection ∇. Hence, in the absence of the covector potential U, the equations (2.20) become the equations of geodesics of the manifold M and the equations of h − paths become the Lorentz equations.

3 Electromagnetic Lagrangian extremals

In the ARLS case the extremals of the energy action

E(L) = ∫_{T} L(t, x, y) dvol_{T} (3.27)

of the Lagrangian L in (1.1) are shown to satisfy the PDE system ([8])

h^{αβ}(∂_{β}y^{\binom{i}{α}} + 2G\binom{i}{α}_{β}) = 0, i = \overline{1, n}. (3.28)

In (3.28), an essential role plays the spray G\binom{i}{α}_{β} = ^{1}G\binom{i}{α}_{β} + ^{2}G\binom{i}{α}_{β} associated to L, where

\{ \begin{matrix}^{1}G\binom{i}{α}_{β} = −\frac{1}{2} |_{αβ}^{γ}| y^{\binom{i}{γ}} \\ ^{2}G\binom{i}{α}_{β} = \frac{1}{2} |_{jk}^{i}| y^{\binom{j}{α}}y^{\binom{k}{β}} + \frac{1}{4m}g̃^{\binom{i}{α}}^{\binom{l}{β}}(U_{\binom{l}{ε}s}y^{\binom{s}{ε}} + ∂_{ε}U\binom{l}{ε} + U_{\binom{l}{γ}} |_{γε}^{ε}| − ∂_{l}Φ), \end{matrix}

which provides the canonic L-induced nonlinear connection N in (1.8) via

N\binom{i}{α}_{β} = 2\frac{∂(^{1}G\binom{i}{α}_{β})}{∂y^{\binom{j}{γ}}}y^{\binom{j}{γ}}, N\binom{i}{α}_{j} = 2\frac{∂(^{2}G\binom{i}{δ}_{ε}h^{δε})}{∂y^{\binom{j}{γ}}}h_{αγ}.

We note that in the ARLSU case for m = 1 and h_{11} = 1, using the conventions above, the extremals of the Lagrangian action are characterized by the equations

ẍ^{i} + |_{jk}^{i}| ẋ^{j}ẋ^{k} = \frac{1}{4}(F_{j}^{i}y^{j} + g^{ij}∂_{j}Φ),

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 = mcγ_{ij}, Φ = 0 and the potential Ū given by Ū = ε(x^{1}dx^{2} − x^{2}dx^{1}), ε ∈ R. Then, denoting by a = εe(m^{2}c)^{−1} the control parameter of electromagnetid field strength, the appropriately rescaled Lorentz-type equations (2.26) read

ẍ^{i} + |_{jk}^{i}| ẋ^{j}ẋ^{k} = (−1)^{i+1}a(g^{i1}ẋ^{2} + g^{i2}ẋ^{1}), i = \overline{1, 2}. (4.29)

We exemplify further the influence of the electromagnetic force F derived from Ũ via (2.24) on h-paths for three cases: R^{2}, H^{2} and S^{2}. 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 of a (a = −512 for Euclidean case, a = −1024 for the Poincare half-plane, a = 2 for the sphere respectively).

R^{2}  H^{2}  S^{2}

We note that, when the influence of the generalized electric potentials U_{i}(x) disappears (i.e., for a = 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.Udrişte 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.

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Vladimir Balan University Politehnica of Bucharest, Department Mathematics I Splaiul Independenţei 313, RO-77206 Bucharest, Romania Email address: vbalan@mathem.pub.ro