Newton's Theorem of Revolving Orbits derives the force that is necessary to explain a particular precession that leaves the shape of an orbit unchanged. Newton showed that for an orbiting body that is already subject to any central force, the inclusion of an additional inverse-cube central force will change that body's angular speed without affecting its radial motion; this leads to the orbit's precession. After reviewing the relevant concepts of Kepler orbits, we present a full description and explanation of Newton's theorem using modern physical and mathematical approaches aimed at a general audience. We specifically highlight the use of the transformation between a rotating reference frame and an inertial reference frame, making this a suitable topic for an upper-level undergraduate mechanics course.

1.
S. I.
Newton
,
I. B.
Cohen
,
A.
Whitman
, and
J.
Budenz
,
The Principia: Mathematical Principles of Natural Philosophy the Authoritative Translation and Guide
(
University of California Press
,
2016
).
Google Scholar
 
2.
S.
Chandrasekhar
,
Newton's Principia for the Common Reader
(
Oxford/Clarendon Press
,
Oxford, NY
,
1995
).
Google Scholar
Crossref
Search ADS
 
3.
E. T.
Whittaker
,
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
(
Cambridge U. P
.,
London, England
,
1904
).
Google Scholar
 
4.
S. H.
Lamb
,
Dynamics
(
Cambridge U. P
.,
London, England
,
1914
).
Google Scholar
 
5.
D.
Lynden-Bell
 and
R. M.
Lynden-Bell
, “
On the shapes of Newton's revolving orbits
,”
Notes Rec. R. Soc. London
51
(
2
),
195
–
198
(
1997
).
https://doi.org/10.1098/rsnr.1997.0016
Google Scholar
Crossref
Search ADS
 
6.
D.
Lynden-Bell
 and
S.
Jin
, “
Analytic central orbits and their transformation group
,”
Mon. Not. R. Astron. Soc.
386
(
1
),
245
–
260
(
2008
).
https://doi.org/10.1111/j.1365-2966.2008.13018.x
Google Scholar
Crossref
Search ADS
 
7.
D.
Lynden-Bell
, “
Hamilton's eccentricity vector generalized to Newton wonders
,”
Observatory
126
,
176
–
182
(
2006
).
Google Scholar
 
8.
H. D.
Nguyen
, “
Rearing its ugly head: The cosmological constant and Newton's greatest blunder
,”
Am. Math. Mon.
115
(
5
),
415
–
430
(
2008
).
https://doi.org/10.1080/00029890.2008.11920544
Google Scholar
Crossref
Search ADS
 
9.
P.
Christian
, “
Newton's theorem of revolving orbits in curved spacetime
,” arXiv:1602.00697 [gr-qc] (
2016
).
Google Scholar
 
10.
K.
Symon
,
Mechanics
, Addison-Wesley World Student Series (
Addison Wesley
,
Reading, MA
,
1971
).
Google Scholar
 
11.
Goldstein
,
Poole
, and
Safko
,
Classical Mechanics
, 3rd ed. (
Addison Wesley
,
San Francisco, CA
,
2002
).
Google Scholar
 
12.
S. T.
Thornton
 and
J. B.
Marion
,
Classical Dynamics of Particles and Systems
, 5th ed. (
Brooks/Cole-Thomson Learning
,
Belmont, CA
,
2004
).
Google Scholar
 
13.
See supplementary material online for the treatment of defining/viewing the static elliptical orbit from the revolving reference frame.
© 2024 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2024
Author(s)

Supplementary Material

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