We formulate a brachistochrone problem in Lorentzian geometry and we prove a variational principle valid for brachistochrones in stationary manifolds. This variational principle is stated in terms of geodesics in a suitable sub-Riemannian structure on ℳ. Moreover, we prove the regularity of the solutions of our variational problem and we determine a differential equation satisfied by the brachistochrones. Some explicit examples are computed.
REFERENCES
1.
F.
Goldstein
and C. M.
Bender
, “Relativistic brachistochrone
,” J. Math. Phys.
27
, 507
–511
(1985
).2.
G.
Kamath
, “The brachistochrone in almost flat space
,” J. Math. Phys.
29
, 2268
–2272
(1988
).3.
V.
Perlick
, “The brachistochrone problem in a stationary space-time
,” J. Math. Phys.
32
, 3148
–3157
(1991
).4.
W. Liu and H. J. Sussman, “Shortest paths for sub-Riemannian metrics on rank-2 distribution,” Memoirs AMS 564, 1995, Vol. 118.
5.
R.
Montgomery
, “Singular extremals on Lie groups
,” Math. Control Signals
7
, 217
–234
(1994
).6.
J. K. Beem, P. E. Ehrlich, and K. L. Easly, Global Lorentzian Geometry (Marcel Dekker, New York, 1996).
7.
S. W. Hawking and G. F. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, London, 1973).
8.
B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity (Academic, New York, 1983).
9.
I.
Kovner
, “Fermat principle in arbitrary gravitational fields
,” Astrophys. J.
351
, 114
–120
(1990
).10.
F. Giannoni, A. Masiello, and P. Piccione, “A timelike extension of Fermat’s principle in general relativity and applications,” to appear in Calculus of Variations and PDE.
11.
V.
Perlick
, “On Fermat’s principle in ceneral relativity: I. The general case
,” Class. Quantum Grav.
7
, 1319
–1331
(1990
).12.
R. Palais, Foundations of Global Nonlinear Analysis (Benjamin, New York, 1968).
13.
S. Lang, Differential Manifolds (Springer-Verlag, Berlin, 1985).
14.
R.
Montgomery
, “A survey of singular curves in sub-Riemannian geometry
,” J. Dyn. Controlled Syst.
1
, 49
–90
(1995
).15.
M. Spivak, A Comprehensive Introduction to Differential Geometry, 2nd ed. (Publish or Perish, Inc., Wilmington, DE), Vol. 2.
16.
W. Rindler, Essential Relativity (Springer, New York, 1977).
This content is only available via PDF.
© 1997 American Institute of Physics.
1997
American Institute of Physics
You do not currently have access to this content.
