© Hindawi Publishing Corp. TRAJECTORIES UNDER A VECTORIAL POTENTIAL ON STATIONARY MANIFOLDS

By using variational methods, we study the existence and 
multiplicity of trajectories under a vectorial potential on (standard) stationary Lorentzian manifolds possibly with boundary.


Introduction and statement of the results. The pair (ᏸ,g) is called
Lorentzian manifold if ᏸ is a connected finite-dimensional smooth manifold with dim ᏸ ≥ 2 and g is a Lorentzian metric on ᏸ, that is, g is a smooth symmetric two covariant tensor field such that for any z ∈ ᏸ, the bilinear form g(z)[•, •] induced on T z ᏸ is nondegenerate and of index ν(g) = 1.Its points are called events.A Lorentzian manifold (ᏸ,g) is called (standard) stationary if ᏸ is a product manifold ᏸ = ᏹ × R, ᏹ any connected manifold (1.1) and g can be written as for any z = (x, t) ∈ ᏸ, ζ = (ξ, τ), ζ = (ξ ,τ ) ∈ T z ᏸ = T x ᏹ × R, where •, • , δ, and β are, respectively, a Riemannian metric on ᏹ, a smooth vector field, and a smooth scalar field on ᏹ.We refer to [13,15,17] for all the background material assumed in this paper.Let A be a smooth stationary vector field on ᏸ, that is, and let Ᏺ be the (1, 1) tensor field associated to curl A. In this paper, we look for smooth curves γ : [0, 1] → (trajectories) which solve the problem where z and w are two fixed events of , is an open connected subset of ᏸ.
When γ is a trajectory, there exists E γ ∈ R such that γ(s), γ(s) L = E γ ∀s ∈ [0, 1] (1.5) (see Remark 2.2) thus its causal character is well defined.We point out that (1.4) represents the Lorentz world-force law which determinates the motion of relativistic particles submitted to an electromagnetic field, when we take into account timelike curves γ (see [17, page 88]).In this case, A 1 is called vectorial potential and A 2 is called scalar potential, see [12].It is clear that this problem generalizes the one of the geodesic connectedness (see, e.g., [6,11,13]).Our problem has a variational nature.Indeed trajectories connecting two events are the critical points of the functional on a suitable infinite-dimensional manifold, see [5] and Section 2. When the manifold is (standard) static, existence and multiplicity results for these trajectories have been found in [2] and very recently timelike trajectories on stationary complete manifolds have been studied in [9].For results on periodic trajectories, we refer to [4,9,14].
In the following, for any vector ξ ∈ T ᏹ, we set |ξ| = ξ, ξ .Now, we are ready to state our first result, where we assume the completeness of ᏸ.
Theorem 1.1.Let (ᏸ, •, • L ) be a stationary Lorentzian manifold with ᏹ complete and assume that (i) there exist η, b, d ∈ R such that Then, for each two given events in ᏸ a trajectory γ joining them exists.Moreover, if ᏹ is noncontractible in itself, then, for each two given events of ᏸ a sequence {γ m } of trajectories joining them exists.
Remark 1.2.A gauge transformation does not modify (1.4).Indeed adding to A any irrotational vector field B independent on t, say B(x, t) = (∇V (x),a 0 ) with V ∈ Ꮿ 2 (ᏹ, R) and a 0 ∈ R, the critical points of the corresponding functional satisfy the same Euler-Lagrange equation.Thus it is enough that A + B satisfies assumption (ii) of Theorem 1.1 for such B (in particular it suffices that A 2 is bounded from below).
We also deal with noncomplete stationary Lorentzian manifolds having boundaries satisfying some convexity assumptions.Let be an open domain of a Lorentzian manifold ᏸ, ∂ its differentiable topological boundary and = ∪ ∂.We recall the following definition.
Definition 1.3 (global convexity, variational point of view).We say that ∂ is convex if and only if for one, and then for all, nonnegative function Φ on such that it results that In [3], it has been proved that the previous definition is equivalent to the following one.Definition 1.4 (global convexity, geometrical point of view).We say that ∂ is convex if for any z, w ∈ the range of any geodesic γ : .11)We recall that also the definition of causal convexity can be given, see, for example, [7] (see also [3]).
We use the following definition.
Definition 1.5.A manifold (, •, • L ), with = Ᏸ × R, is said to be a stationary Lorentzian manifold with differentiable boundary ∂ = ∂Ᏸ × R if a stationary Lorentzian manifold (ᏸ,g), with ᏸ = ᏹ × R, exists such that Ᏸ is an open domain of ᏹ, g restricted to is •, • L , and Ᏸ is a complete manifold with differentiable boundary.
Remark that if is a stationary Lorentzian manifold with convex boundary, since ∂Ᏸ is differentiable, there exists a smooth function φ : Ᏸ → R satisfying (1.12)Moreover, Φ can be chosen such that, for any z = (x, t) ∈ ᏸ, and then We prove the following theorem.
Theorem 1.6.Let (, •, • L ) be a stationary Lorentzian manifold with differentiable and convex boundary; assume that the assumptions of Theorem 1.1 hold and that Then, for each two given events in a trajectory γ joining them exists.Moreover, if Ᏸ is noncontractible in itself, then, for each two given events of a sequence {γ m } of trajectories joining them exists.
We consider an open subset of the Minkowski spacetime.This spacetime is the model of special relativity which describes situations in which the gravitational effects are negligible.Given the vector field A, the 2-form curl A can be written as where E i , B i , i = 1, 2, 3, are differentiable functions.This 2-form, or the associated endomorphism field Ᏺ, is called the electromagnetic field.Moreover, E = 3 i=1 E i ∂ i is the electric field and B = 3 i=1 B i ∂ i the magnetic field.These concepts can be extended to the tangent space of any Lorentzian manifold, whenever a timelike tangent vector (which plays the role of ∂ t ) is fixed (see, e.g., [17, page 75]).Thus, hypothesis (1.15)only involves the magnetic field naturally associated to the decomposition = Ᏸ × R.
The paper is organized as follows.In Section 2, we state a variational principle which allows us to overcome the problems arising in the study of F because of the indefiniteness of the metric.Then in Section 3 we prove Theorem 1.1 by using classical critical point theory.Finally, in Section 4, thanks to a penalization technique (necessary in order to find trajectories not touching the boundary), we demonstrate Theorem 1.6.

A variational principle for trajectories.
From now on, we assume that an ᏹ is a submanifold of (R N , •, • ), for N sufficiently large.Thus (see [16]) is a submanifold of the Sobolev space H 1 ([0, 1], R N ).Fix z = (p, t 1 ), w = (q, t 2 ) ∈ ᏸ and consider the product manifold where (2.3) We recall that for any y ∈ H 1 ([0, 1], ᏹ), the tangent space at H 1 ([0, 1], ᏹ) is given by (see [16]) and for any t We will consider on ᐆ the functional F in (1.6) given explicitly by where ∇ L A denotes the gradient of the vector field A and (∇ L A) T its transpose, has a prime integral, in fact We recall that in [5] a new variational principle for the fundamental equations of the classical physics has been introduced; such a principle allows one to obtain a sort of unification of the gravitational and the electromagnetic fields.
The basic point of this variational principle is that the world-line of a material point is parametrized by a parameter s which carries some physical information, namely it is related to the rest mass and to the charge.In particular, the inertial mass turns out to be a constant of the motion, which is determined by the initial conditions and also the equality between the inertial and gravitational mass can be deduced.
By Lemma 2.1, to find trajectories joining two events, we have to investigate the existence of the critical points of functional (1.6) on ᐆ.Classical minimization arguments cannot be applied to functional F since it is strongly indefinite (i.e., it is unbounded both from above and from below and the Morse index of its critical points is +∞).As for the geodesic problem (see, e.g., [11]), when we deal with stationary manifolds and stationary vector fields, a variational principle can be proved.This variational principle (see [1] for the details) reduces the study of the critical points of F to the search of the critical points of a functional which is bounded from below under our assumptions on the coefficients of the metric and on the vector field.Remark that for any x ∈ Ω 1 (ᏹ) the functional F(x,•) has on H 1 (t 1 ,t 2 ) one and only one critical point, say that t = Ψ (x) (where Ψ can be explicitly determinated).Consider on Ω 1 (ᏹ) the functional which is smooth by the implicit function theorem and whose first variation is given by (where F x denotes the partial derivative of F with respect to x).Thus we get the following result.
Theorem 2.3.Let γ = (x, t) ∈ ᐆ.The following propositions are equivalent: (2.11) The functional J can be explicitly evaluated and it results that ds, (2.12) where 3. Proof of Theorem 1.1.By Lemma 2.1 and Theorem 2.3, we have to study the critical points of the functional (2.12) on Ω 1 (ᏹ).
Remark 3.1.The assumptions of Theorem 1.1 imply, by using the Hölder inequality, that where hence the functional J is bounded from below (we assume b = 1).
Before proving Theorem 1.1, we recall some definitions.If (X, h) is a Riemannian manifold modelled on a Hilbert space and f ∈ Ꮿ 1 (X, R), f satisfies the Palais-Smale condition if every sequence {y m } such that contains a converging subsequence, where ∇f (y) denotes the gradient of f at the point y with respect to the metric h and • is the norm on the tangent bundle induced by h.By standard arguments, it can be proved that J satisfies the Palais-Smale condition, (see, e.g., [1, Proposition 3.5]).The category, denoted with cat X Y , of a subspace Y of a topological space X is the least number of closed and contractible subset of X covering Y .If Y is not covered by a finite number of such subsets of X, we set cat X Y = +∞.
Proof of Theorem 1.1.As J is bounded from below and satisfies the Palais-Smale condition on Ω 1 (ᏹ), it admits a minimum point x which corresponds to a critical point γ = (x, Ψ (x)) of F by virtue of Theorem 2.3.By Lemma 2.1, the proof of (i) is complete.Moreover, by a result of Fadell and Husseini (see [10]), the Ljusternik-Schnirelman category of (where and Ω 1 (Ᏸ), where Φ is as in (1.9), φ is as in (1.12), and (ψ ) ∈]0,1] is a family of nonnegative increasing functions in for some a > 0, b ≥ 0. We point out that the variational principle stated in Theorem 2.3 still holds since the penalizating term does not depend on t.The following lemma (see, e.g., [13]) plays a basic role in our penalization technique.We denote by and let {s m } be a sequence in [0, 1] such that Then for a C ∈ R. Then Proof.By (4.6) and the form of the penalization, we get Then, by (2.12) and the assumptions of Theorem 1.1, it results that for a suitable c ∈ R. Thus { ẋm 2 } is bounded and the proof follows by Lemma 4.1.
We omit the proof of the following proposition since it is a combination of the proof of [11,Theorem 3.3] and [4,Lemma 4.3].Proposition 4.3.Let J be as in (4.1).Then (i) for any ∈ ]0, 1] and for any c ∈ R, the sublevels are complete metric subspaces of Ω 1 (Ᏸ); (ii) for any ∈ ]0, 1], J satisfies the Palais-Smale condition.
By the previous proposition and Remark 3.1, there exists a family {x } of critical points of J satisfying (4.6).Thus, by Theorem 2.3, set γ = (x , Ψ (x )), we find a family {γ } of critical points of F such that Remark 4.4.It is easy to prove that a critical point γ of F satisfies the following equation: Thus, multiplying by γ, we get the existence of H (γ) ∈ R such that We set for any The following estimate on the family {µ } holds.
Lemma 4.5.There exists 0 ∈ ]0, 1] such that the family of functions Proof.For any ∈ ]0, 1] and s ∈ [0, 1], we set u (s) = Φ(γ (s)), so u is a Ꮿ 2 function on [0, 1].Let s be a minimum point for u .Since ψ is convex, ψ is nondecreasing, thus it results that Hence, it is enough to prove that {µ (s )} is bounded and to study the case in which inf Differentiating twice, we get (4.17) We consider on ᏸ the Riemannian metric given by for any γ = (x, t) ∈ ᏸ and ζ = (ξ, τ) ∈ T γ ᏸ.As H Φ L is a bilinear form, for some c 1 > 0.Moreover, as 0 is a regular value for Φ, for sufficiently small, for some c 2 > 0. Thus, by (4.17  for some c 3 > 0, so that for some c 4 > 0. Since and the proof is complete.
The same arguments used in [11,Lemma 4.7] allow us to obtain the following proposition.
Proposition 4.6.Let {γ } be a family in ᐆ such that for any ∈ ]0, 1], γ is a critical point of F and (4.6) holds.Then there exist an infinitesimal and decreasing sequence { m } in ]0, 1] and a curve γ

6 .
by classical arguments of Ljusternik-Schnirelman critical point theory, we get the existence of a sequence {x m } of critical points of J such that lim m→+∞ J x m = +∞.(3.5)Hence, set γ m = (x m , Ψ (x m )) for any m ∈ N, by Theorem 2.3 and Lemma 2.1, we get the existence of infinitely many trajectories joining the two given events such When we deal with open subsets of ᏸ, we need to penalize functionals F and J because Palais-Smale sequences converging to a critical point touching the boundary ∂ could exist.We consider for any ∈ ]0, 1] the functionals ) and(1.15)  follows, for sufficiently small,c 2 µ s ≤ c 1 x s ,x s + β x s ṫ2 s .(4.21)By (4.13), it results that
Usingthe Gronwall lemma, we prove that there exists σ > 0 such that [s M ,s M + σ ] ⊂ C, getting a contradiction.Indeed, for a η 1 > 0, as γ(s M ) ∈ ∂, there exists σ > 0 such that 0 ) ∈ , there exists a neighborhood I of s 0 such that u(s) = 0 for every s ∈ I. Thus γ is a trajectory joining z and w.Now, it suffices to prove that the range of γ is contained in .Let C = {s ∈ [0, 1] | γ(s) ∈ ∂} and assume that C is nonempty.Clearly, C is compact; say s M ∈ ]0, 1[ its maximum.Consider u as before; then for any s ∈ [s M ,s M + σ ]