Polar reciprocals of trajectories are an elegant alternative to hodographs for motion in a central force field. Their principal advantage is that the transformation from a trajectory to its polar reciprocal is its own inverse. The form of the polar reciprocals of Kepler orbits is established, and a geometrical construction is presented for the orbits of the Kepler problem starting from their polar reciprocals. No obscure knowledge of conics is required to demonstrate the validity of the method. Unlike a graphical procedure suggested by Feynman and extended by Derbes, the method based on polar reciprocals works without changes for elliptical, parabolic, and hyperbolic trajectories.
REFERENCES
1.
H.
Abelson
, A.
diSessa
, and L.
Rudolph
, “Velocity space and the geometry of planetary orbits
,” Am. J. Phys.
43
, 579
–589
(1975
).2.
A.
González-Villanueva
, H. N.
Núñez-Yépez
, and A. L.
Salas-Brito
, “In veolcity space the Kepler orbits are circular
,” Eur. J. Phys.
17
, 168
–171
(1996
).3.
T. A.
Apostolatos
, “Hodograph: A useful geometrical tool for solving some difficult problems in dynamics
,” Am. J. Phys.
71
, 261
–266
(2003
).4.
E. I.
Butikov
, “Comment on ‘Eccentricity as a vector’
,” Eur. J. Phys.
25
, L41
–L43
(2004
).5.
C. I.
Mungan
, “Another comment on ‘Eccentricity as a vector’
,” Eur. J. Phys.
26
, L7
–L9
(2005
).6.
D. L.
Goodstein
and J. R.
Goodstein
, Feynman’s Lost Lecture: The Motion of Planets Around the Sun
(W. W. Norton
, New York
, 1996
).7.
8.
R.
Weinstock
, Math. Intell.
21
, 71
–73
(1999
), review of Ref. 6.9.
D.
Derbes
, “Reinventing the wheel: Hodographic solutions to the Kepler problems
,” Am. J. Phys.
69
, 481
–489
(2001
).10.
E.
Guillaumin–España
, A. L.
Salas-Brito
, and H. N.
Núñez-Yépez
, “Tracing a planet’s orbit with a straight edge and a compass with the help of the hodograph and the Hamilton vector
,” Am. J. Phys.
71
, 585
–589
(2003
).11.
D. W.
Tiberiis
, “Comment on ‘Reinventing the wheel: Hodographic solutions to Kepler problems,’ by David Derbes [Am. J. Phys. 69(4), 481–489 (2001)]
,” Am. J. Phys.
70
, 79
(2002
).12.
I.
Newton
, Philosophiae Naturalis Principia Mathematica
(University of California Press
, Berkeley
, 1934
). See Proposition I, Corollary I, p. 41.13.
D.
Chakerian
, “Central force laws, hodographs, and polar reciprocals
,” Math. Mag.
74
, 3
–18
(2001
).14.
Trajectories for which L = 0 have to be excluded, but these are one-dimensional and easily drawn once the initial conditions are known. See
D. V.
Anosov
, “A note on the Kepler problem
,” J. Dyn. Control Syst.
8
, 413
–442
(2002
).15.
Equation (4) is implicit in the literature on hodographs. For example, Eq. (4) can be inferred from Eq. (3) of Ref. 4, which is a concise and clear treatment of hodographs for the Kepler problem. In
M. D.
Vivarelli
, “A configuration counterpart of the Kepler problem hodograph
,” Celest. Mech. Dyn. Astron.
68
, 305
–311
(1998
), algebraic considerations specific to the Kepler problem are used to motivate the introduction of a sum vector , which is the combination of vectors in Eq. (4).16.
H. S. M.
Coxeter
, Introduction to Geometry
, 2nd ed. (John Wiley & Sons
, New York
, 1969
).17.
This result could also have been established directly by using the fact that to recast Eq. (4) as the relation , where is the Laplace-Runge-Lenz vector as defined in
H.
Goldstein
, Classical Mechanics
, 2nd ed. (Addison-Wesley
, Reading, MA
, 1980
).18.
Problem 2(d) addresses the origin of the angular restriction on k* for e ≥ 1.
19.
See 〈www.susqu.edu/brakke/constructions/constructions.htm〉.
20.
E. C.
Wallace
and S. F.
West
, Roads to Geometry
, 2nd ed. (Prentice-Hall
, Upper Saddle River, NJ
, 1998
). This text discusses the inversion of a point in the unit circle on p. 278, and the other two constructions on pp. 191
–192
.21.
P. T.
Tam
, A Physicist’s Guide to Mathematica
, 2nd ed. (Academic
, Burlington, MA
, 2008
).22.
Other applications of the complex variable formalism to central force problems appear in
M. M.
D’Eliseo
, “The first-order orbital equation
,” Am. J. Phys.
75
, 352
–355
(2007
).23.
Alternatively, Euclid’s proposition III.36 on secants of a circle (see Ref. 16, p. 8).
24.
A brief but careful introduction to the Hohmann transfer of Fig. 5 is given in
R. D.
Gregory
, Classical Mechanics
(Cambridge U.P.
, Cambridge
, 2006
), Sec. 7.6.© 2011 American Association of Physics Teachers.
2011
American Association of Physics Teachers
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