A second-order variational principle for the Lorentz force equation: conjugacy and bifurcation

Giambò, Roberto; Javaloyes, Miguel Ángel

doi:10.1017/s0308210506000497

Cited by 2 publications

(1 citation statement)

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“…A basic idea that I shall use is that of light lift [60,71,61]. It has been introduced in [60] for the case of spacelike dimensional reduction (which leads to the relativistic Lorentz force equation) and since then it has been used to solve problems of existence and multiplicity for stationary points of the charged particle action [71,36]. It has been introduced in the lightlike dimensional reduction context of this work in [61] and then taken up again in [34].…”

Section: The Light Lift and The Action Functionalmentioning

confidence: 99%

Causality of spacetimes admitting a parallel null vector and weak KAM theory

Minguzzi

2012

Preprint

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The causal spacetimes admitting a covariantly constant null vector provide a connection between relativistic and non-relativistic physics. We explore this relationship in several directions. We start proving a formula which relates the Lorentzian distance in the full spacetime with the least action of a mechanical system living in a quotient classical space time. The timelike eikonal equation satisfied by the Lorentzian distance is proved to be equivalent to the Hamilton-Jacobi equation for the least action. We also prove that the Legendre transform on the classical base corresponds to the musical isomorphism on the light cone, and the Young-Fenchel inequality is nothing but a well known geometric inequality in Lorentzian geometry. A strategy to simplify the dynamics passing to a reference frame moving with the E.-L. flow is explained. It is then proved that the causality properties can be conveniently expressed in terms of the least action. In particular, strong causality coincides with stable causality and is equivalent to the lower semi-continuity of the classical least action on the diagonal, while global hyperbolicity is equivalent to the coercivity condition on the action functional. The classical Tonelli's theorem in the calculus of variations corresponds, then, to the well known result that global hyperbolicity implies causal simplicity. The well known problem of recasting the metric in a global Rosen form is shown to be equivalent to that of finding global solutions to the Hamilton-Jacobi equation having complete characteristics. * Work presented at the conferences "Non-commutative structures and non-relativistic (super)symmetries", Tours (2010) [59], and "Weak KAM Theory in Italy", Cortona (2011). This first version includes results which are mostly on causality theory; the next version will include a chapter with more results related to weak KAM theory. Some relevant references can be missing in this version.

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