Lagrangian mechanics and the geometry of configuration spacetime

Trümper, Manfred

doi:10.1016/0003-4916(83)90305-6

Cited by 25 publications

(12 citation statements)

References 9 publications

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“…The Lagrangian (3.43) is therefore formally identical to the one describing the motion of a charged particle minimally coupled to an electromagnetic field through the vector potential A i and the scalar potential U and moving on a Riemannian manifold with metric γ ij . For a generic Augustinian structure, the Euler-Lagrange equations of motion derived from L take the form [9]:…”

Section: Proofmentioning

confidence: 99%

Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

Bekaert

Morand

2016

Journal of Mathematical Physics

156

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The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric.Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e. compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive.We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e. the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the role of the Levi-Civita connection is played by a whole class of privileged origins, the so-called torsional Newton-Cartan (TNC) geometries recently investigated in the literature. Finally, we discuss a generalisation of Newtonian connections to the torsional case.

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Section: Proofmentioning

confidence: 99%

Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

Bekaert

Morand

2016

Journal of Mathematical Physics

156

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“…can be derived via a variational principle from a Lagrangian built in terms of the Leibnizian structure and the potential 1-form. As shown in [9,5], such a Lagrangian defines a (possibly nondegenerate) Lagrangian metric onM which reduces on the absolute spaces to the absolute collections of rulersγ, and furthermore constitutes the necessary and sufficient structure needed to supplement a Leibnizian structure in order to define a unique Newtonian connection. Besides its purely mathematical interest, Newton-Cartan geometry has recently known a revival of interest triggered first by condensed-matter applications [10,11] following the seminal work by Son [12] (cf.…”

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confidence: 99%

Connections and dynamical trajectories in generalised Newton-Cartan gravity. II. An ambient perspective

Bekaert

Morand

2018

Journal of Mathematical Physics

109

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Connections compatible with degenerate metric structures are known to possess peculiar features: on the one hand, the compatibility conditions involve restrictions on the torsion; on the other hand, torsionfree compatible connections are not unique, the arbitrariness being encoded in a tensor field whose type depends on the metric structure. Nonrelativistic structures typically fall under this scheme, the paradigmatic example being a contravariant degenerate metric whose kernel is spanned by a one-form. Torsionfree compatible (i.e. Galilean) connections are characterised by the gift of a two-form (the force field). Whenever the two-form is closed, the connection is said Newtonian. Such a nonrelativistic spacetime is known to admit an ambient description as the orbit space of a gravitational wave with parallel rays. The leaves of the null foliation are endowed with a nonrelativistic structure dual to the Newtonian one, dubbed Carrollian spacetime. We propose a generalisation of this unifying framework by introducing a new non-Lorentzian ambient metric structure of which we study the geometry. We characterise the space of (torsional) connections preserving such a metric structure which is shown to project to (resp. embed) the most general class of (torsional) Galilean (resp. Carrollian) connections.

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“…The physically important condition is that equations containing one or another connection are invariant under all Newton-Cartan gauge symmetries. For further discussions on the ambiguity among the class of Newton-Cartan metric compatible connections we refer to [22,59,63] 8 An explicit expression for the boost invariant Newtonian potential appears in [64]; this can also be recovered directly from the expressions in [65], which first introduced the relevant and appropriate parametrization. In this paper, we follow more closely the notation of [32].…”

Section: )mentioning

confidence: 99%

Homogeneous nonrelativistic geometries as coset spaces

Hartong

Keeler

Obers

2018

Class. Quantum Grav.

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We generalize the coset procedure of homogeneous spacetimes in (pseudo-)Riemannian geometry to non-Lorentzian geometries. These are manifolds endowed with nowhere vanishing invertible vielbeins that transform under local non-Lorentzian tangent space transformations. In particular we focus on nonrelativistic symmetry algebras that give rise to (torsional) Newton-Cartan geometries, for which we demonstrate how the Newton-Cartan metric complex is determined by degenerate co-and contravariant symmetric bilinear forms on the coset. In specific cases we also show the connection of the resulting nonrelativistic coset spacetimes to pseudo-Riemannian cosets via Inönü-Wigner contraction of relativistic algebras as well as null reduction. Our construction is of use for example when considering limits of the AdS/CFT correspondence in which nonrelativistic spacetimes appear as gravitational backgrounds for nonrelativistic string or gravity theories.In view of their importance in nonrelativistic holography 2 , Lifshitz and Schrödinger spacetimes were already studied via coset constructions in [44] (and applied in e.g. [45-47]). Since nonrelativistic algebras are not semi-simple and have a degenerate Cartan-Killing metric, the standard (Riemannian) coset method of contracting the (inverse) vielbeins with the (inverse) Cartan-Killing metric, obtaining the metric and its inverse, does not apply. Instead of the Cartan-Killing metric, [44] uses the most general non-degenerate group invariant symmetric bilinear form, as [48] did for the Nappi-Witten WZW model on the non-semi simple group E c 2 , which is the centrally extended 2-dimensional Euclidean group. Thus [44] recovers the so-called Lifshitz and Schrödinger spacetimes, first proposed in [49, 50] and [51, 52], from a pseudo-Riemannian coset construction by performing cosets on the Lifshitz and Schrödinger groups respectively.However, aside from the somewhat counterintuitive concept of proposing locally relativistic spacetimes as duals to nonrelativistic field theories [34], difficulties with field profile reconstruction [53][54][55] as well as spacetime reconstruction [56] motivate the consideration of alternatives. As advocated in e.g. [25], such nonrelativistic spacetimes are more naturally viewed as Newton-Cartan spacetimes, described by a clock 1-form, degenerate spatial metric and an extra U (1) connection that contains the Newtonian potential. As we will see in this paper, the key observation is first of all that for certain choices of subgroup H ⊂ G the invariant bilinear form on the coset is necessarily degenerate. Consequently there are two (degenerate) distinct group-invariant bilinear forms on the coset: a covariant bilinear form (Ω ab ) and a corresponding dual bilinear form (Ω ab ). These objects are obviously not inverses of each other as they are both degenerate. With these two objects, we can construct the Newton-Cartan geometric data 3 for several interesting cases.Since some nonrelativistic algebras can be obtained using Inönü-Wigner contractions of relativi...

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Lagrangian mechanics and the geometry of configuration spacetime

Cited by 25 publications

References 9 publications

Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

Connections and dynamical trajectories in generalised Newton-Cartan gravity. II. An ambient perspective

Homogeneous nonrelativistic geometries as coset spaces

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