The Jacobi metric derived from the line element by one of the authors is shown to reduce to the standard formulation in the non-relativistic approximation. We obtain the Jacobi metric for various stationary metrics. Finally, the Jacobi-Maupertuis metric is formulated for time-dependent metrics by including the Eisenhart-Duval lift, known as the Jacobi-Eisenhart metric.
REFERENCES
1.
B.
Rink
, Lecture notes on “Geometric Mechanics and Dynamics,” http://www.few.vu.nl/˜brink/Preview.pdf.2.
A. A.
Izquierdo
, M. A. G.
Leon
, J. M.
Guilarte
, and M. T.
Mayado
, “Jacobi metric and Morse theory of dynamical systems
,” e-print arXiv:math-ph/0212017.3.
P. L.
Maupertuis
, Accord de Différentes Loix de la Nature qui Avoient Jusqu’ici paru Incompatibles
(Académie Internationale d’Histoire des Sciences
, Paris
, 1744
).4.
S.
Nair
, S.
Ober-Blöbaum
, and J. E.
Marsden
, “The Jacobi-Maupertuis principle in variational integrators
,” AIP Conf. Proc.
1168
, 464
(2009
).5.
O. C.
Pin
, “Curvature and mechanics
,” Adv. Math.
15
, 269
–311
(1975
).6.
V. G.
Gurzadyan
and G. K.
Savvidy
, “Collective relaxation of stellar systems
,” Astron. Astrophys.
160
, 203
–210
(1986
).7.
A. V.
Tsiganov
, “The Maupertuis principle and canonical transformations of the extended phase space
,” J. Nonlinear Math. Phys.
8
, 157
–182
(2001
).8.
G. W.
Gibbons
, “The Jacobi-metric for timelike geodesics in static spacetimes
,” Classical Quantum Gravity
33
, 025004
(2015
).9.
G. W.
Gibbons
, C. A. R.
Herdeiro
, C. M.
Warnick
, and M. C.
Werner
, “Stationary metrics and optical Zermelo-Randers-Finsler geometry
,” Phys. Rev. D
79
, 044022
(2009
).10.
G.
Randers
, “On an asymmetrical metric in the four-space of general relativity
,” Phys. Rev.
59
, 195
(1941
).11.
C.
Robles
, “Geodesics in Randers spaces of constant curvature
,” Trans. Am. Math. Soc.
359
, 1633
(2007
).12.
D. C.
Brody
, G. W.
Gibbons
, and D. M.
Meier
, “A Riemannian approach to Randers geodesics
,” J. Geom. Phys.
106
, 98
(2016
).13.
D. C.
Brody
, G. W.
Gibbons
, and D. M.
Meier
, “Time-optimal navigation through quantum wind
,” New J. Phys.
17
, 033048
(2015
).14.
E.
Zermelo
, “Uber das navigationsproblem bei ruhender oder ver anderlicher windverteilung
,” Z. Angew. Math. Mech.
11
, 114
(1931
).15.
L. P.
Eisenhart
, “Dynamical trajectories and geodesics
,” Ann. Math.
30
, 591
–606
(1928
).16.
C.
Duval
, G.
Burdet
, H. P.
Künzle
, and M.
Perrin
, “Bargmann structures and Newton-Cartan theory
,” Phys. Rev. D
31
, 1841
(1985
).17.
T.
Houri
, “Liouville integrability of Hamiltonian systems and spacetime symmetry
,” www.geocities.jp/football_physicien/publication.html.18.
T.
Iwai
and N.
Katayama
, “On extended Taub-NUT metrics
,” J. Geom. Phys.
12
, 55
–75
(1993
).19.
Y.
Grandati
, A.
Bérard
, and H.
Mohrbach
, “Bohlin-Arnold-Vassiliev’s duality and conserved quantities
,” e-print arXiv:0803.2610v2 [math-ph], 1–8
.20.
S.
Chanda
, P.
Guha
, and R.
Roychowdhury
, “Taub-NUT as Bertrand spactime with magnetic fields
,” J. Geom. Symmetry Phys.
41
, 33
–67
(2016
); e-print arXiv:1503.08183v4.21.
V.
Perlick
, “Bertrand spacetimes
,” Classical Quantum Gravity
9
, 1009
–1021
(1992
).22.
P.
Das
, R.
Sk
, and S.
Ghosh
, “Motion of charged particle in Reissner—Nordström spacetime: A Jacobi metric approach
,” e-print arXiv:1609.04577 [gr-qc].23.
M.
Cariglia
and F. K.
Alves
, “The Eisenhart lift: A didactical introduction of modern geometrical concepts from Hamiltonian dynamics
,” Eur. J. Phys.
36
, 025018
(2015
).24.
M.
Cariglia
, G. W.
Gibbons
, J. W.
van Holten
, P. A.
Horvathy
, and P. M.
Zhang
, “Conformal killing tensors and covariant Hamiltonian dynamics
,” J. Math. Phys.
55
, 122702
(2014
).25.
M.
Cariglia
, “Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
,” Phys. Rev. D
86
, 084050
(2012
).26.
M.
Cariglia
, C.
Duval
, G. W.
Gibbons
, and P. A.
Horvathy
, “Eisenhart lifts and symmetries of time-dependent systems
,” Ann. Phys.
373
, 631
–654
(2016
).27.
M.
Cariglia
, “Null lifts and projective dynamics
,” Ann. Phys.
362
, 642
–658
(2015
).© 2017 Author(s).
2017
Author(s)
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