On geodesics of Finsler metrics via navigation problem

Huang, Libing; Mo, Xiaohuan

doi:10.1090/s0002-9939-2011-10726-3

Cited by 30 publications

(34 citation statements)

References 19 publications

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“…This work extends the results of [21,38] to so called 'conic'-Finsler metrics and obtains a similar form for the geodesics under broadly analogous conditions.…”

Section: Physical Constraints Encompassed and Not Encompassed By Shensupporting

confidence: 76%

“…Future work will include potentially applying the results in [21,42] to a generalisation of the setup described here. The desired generalisation would be to relax the condition that the function representing the constraint on the control Hamiltonian is an inner product, and to allow more general Minkowski norms to take this role instead.…”

Section: Physical Constraints Encompassed and Not Encompassed By Shenmentioning

confidence: 99%

“…In the absence of a solution to the navigation problem analogous to the role of Randers metrics in the Riemannian case, alternative methods must been sought. The central result of [21] allows one to determin the geodesics of the Finsler metric solving the navigation problem on SU (n) for whichĤ c (t) is constrained such that F (p) (Ĥ c (t)) = 1. The geodesics of the (yet unknown) metric solving the navigation problem in such a generalised case are:…”

Section: Physical Constraints Encompassed and Not Encompassed By Shenmentioning

confidence: 99%

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Zermelo navigation in the quantum brachistochrone

Russell¹,

Stepney²

2015

J. Phys. A: Math. Theor.

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We analyse the optimal times for implementing unitary quantum gates in a constrained finite dimensional controlled quantum system. The family of constraints studied is that the permitted set of (time dependent) Hamiltonians is the unit ball of a norm induced by an inner product on . We also consider a generalization of this to arbitrary norms. We construct a Randers metric, by applying a theorem of Shen on Zermelo navigation, the geodesics of which are the time optimal trajectories compatible with the prescribed constraint. We determine all geodesics and the corresponding time optimal Hamiltonian for a specific constraint on the control i.e. for any given value of . Some of the results of Carlini et al are re-derived using alternative methods. A first order system of differential equations for the optimal Hamiltonian is obtained and shown to be of the form of the Euler–Poincaré equations. We illustrate that this method can form a methodology for determining which physical substrates are effective at supporting the implementation of fast quantum computation.

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“…This work extends the results of [21,38] to so called 'conic'-Finsler metrics and obtains a similar form for the geodesics under broadly analogous conditions.…”

Section: Physical Constraints Encompassed and Not Encompassed By Shensupporting

confidence: 76%

Section: Physical Constraints Encompassed and Not Encompassed By Shenmentioning

confidence: 99%

Section: Physical Constraints Encompassed and Not Encompassed By Shenmentioning

confidence: 99%

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Zermelo navigation in the quantum brachistochrone

Russell¹,

Stepney²

2015

J. Phys. A: Math. Theor.

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“…We refer to [48,49,50,47,51,4,52,53,54,8,55,56,57,5,58,59,60] for details on Randers spaces and on the Zermelo navigation problem.…”

Section: The Randers-zermelo Elliptic Wildfiresmentioning

confidence: 99%

A Finsler geodesic spray paradigm for wildfire spread modelling

Markvorsen

2016

Nonlinear Analysis: Real World Applications

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One of the finest and most powerful assets of Finsler geometry is its ability to model, describe, and analyze in precise geometric terms an abundance of physical phenomena that are genuinely asymmetric, see e.g. [1,2,3,4,5,6,7,8,9]. In this paper we show how wildfires can be naturally included into this family. Specifically we show how the celebrated and much applied Richards' equations for the large scale elliptic wildfire spreads have a rather simple Finsler-geometric formulation. The general Finsler framework can be explicitly 'integrated' to provide detailed -and curvature sensitive -geodesic solutions to the wildfire spread problem. The methods presented here stem directly from first principles of 2-dimensional Finsler geometry, and they can be readily extracted from the seminal monographs [10] and [11], but we will take special care to introduce and exemplify the necessary framework for the implementation of the geometric machinery into this new application -not least in order to facilitate and support the dialog between geometers and the wildfire modelling community. The 'integration' part alluded to above is obtained via the geodesics of the ensuing Finsler metric which represents the local fire templates. The 'paradigm' part of the present proposal is thus concerned with the corresponding shift of attention from the actual fire-lines to consider instead the geodesic spray -the 'fire-particles' -which together, side by side, mold the fire-lines at each instant of time and thence eventually constitute the local and global structure of the wildfire spread.

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“…Recently, Huang-Mo gave a geometric description of the geodesics of Finsler metrics produced from any Finsler metric F and any homothetic field W in terms of the navigation representation [5]. They showed that these curves are given by composing geodesics of F with the flow generated by −W (see Theorem 4.5 below).…”

Section: Momentioning

confidence: 99%

On Some Finsler Metrics of Non-positive Constant Flag Curvature

Mo¹

2011

Results. Math.

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α, β)-norms on R N induce Minkowski metrics, and the construction of related homothetic vector fields gives a family of new Finsler metrics of non-positive constant flag curvature for each non-trivial (α, β)-norm. The dimension of this family is at least 1 2 (N 2 − N + 4). In particular, we generalize the Funk metric on the unit ball via navigation representation of the standard Euclidean norm and the radial vector field. Finally, we describe the geodesics of these new Finsler metrics with constant flag curvature.Mathematics Subject Classification (2010). Primary 58E20; Secondary 53B40.

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On geodesics of Finsler metrics via navigation problem

Cited by 30 publications

References 19 publications

Zermelo navigation in the quantum brachistochrone

Zermelo navigation in the quantum brachistochrone

A Finsler geodesic spray paradigm for wildfire spread modelling

On Some Finsler Metrics of Non-positive Constant Flag Curvature

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