Relativistic particles in the Kepler and Coulomb potentials may have trajectories that are qualitatively different from the trajectories found in nonrelativistic mechanics. Spiral scattering trajectories were pointed out by C. G. Darwin in 1913 in connection with the relativistic Rutherford scattering of classical charged particles. Relativistic trajectories are of current interest in connection with Cole and Zou’s computer simulation of the hydrogen ground state in classical physics.
REFERENCES
1.
See for example, H. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison-Wesley, New York, 2002), 3rd ed., pp. 316–317;
J. V. Jose and E. J. Saletan, Classical Dynamics: A Contemporary Approach (Cambridge, New York, 1998), pp. 211–212;
E. Saletan and A. H. Cromer, Theoretical Mechanics (Wiley, New York, 1971);
H. C. Corben and P. Stehle, Classical Mechanics (Dover, New York, 1994) (a re-publication of the 1960 edition). Goldstein, Poole, and Safko discuss the relativistic one-dimensional harmonic oscillator in Sec. 7.9. Jose and Saletan treat the relativistic Kepler problem in Sec. 5.1, but do not mention Darwin’s spiral trajectories.
2.
Such orbits appear in the work of
C. G.
Darwin
, “On some orbits of an electron
,” Philos. Mag.
25
, 201
–210
(1913
) in connection with the scattering of β particles in Rutherford’s model for an atom. Darwin wrote just before the publication of Bohr’s model for hydrogen.3.
D. C.
Cole
and Y.
Zou
, “Quantum mechanical ground state of Hydrogen obtained from classical electrodynamics
,” Phys. Lett. A
317
, 14
–20
(2003
).4.
T. H.
Boyer
, “Comments on Cole and Zou’s calculation of the Hydrogen ground state in classical physics
,” Found. Phys. Lett.
16
, 607
–611
(2003
).5.
A. E. Ruark and H. C. Urey, Atoms, Molecules, and Quanta (Dover, New York, 1965), p. 134.
6.
The existence of a lower limit for the angular momentum appears in the work of
U.
Torkelsson
, “The special and general relativistic effects on orbits around point masses
,” Eur. J. Phys.
19
, 459
–464
(1998
). He remarks, “For relativistic motion in the Coulomb potential the centrifugal barrier disappears at a small, but still finite, specific angular momentum.” He also notes that “there are no stable orbits” for small angular momentum, 7.
J. M. Aguirregabiria (private communication) has pointed out that other potentials of the form (where and are constants) also involve interesting nonrelativistic and relativistic restrictions on angular momentum for circular orbits.
8.
See the review by L. de la Pena and A. M. Cetto, The Quantum Dice - An Introduction to Stochastic Electrodynamics (Kluwer Academic, Dordrecht, 1996).
A brief introduction is given by
T. H.
Boyer
, “Random electrodynamics: The theory of classical electrodynamics with classical electromagnetic zero-point radiation
,” Phys. Rev. D
11
, 790
–808
(1975
).9.
T.
Marshall
and P.
Claverie
, “Stochastic electrodynamics of nonlinear systems. I. Particle in a central field of force
,” J. Math. Phys.
21
, 1918
–1925
(1980
);P.
Claverie
, L.
Pesquera
, and F.
Soto
, “Existence of a constant stationary solution for the hydrogen atom problem in stochastic electrodynamics
,” Phys. Lett. A
80
, 113
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(1980
);A.
Denis
, L.
Pesquera
, and P.
Claverie
, “Linear response of stochastic multiperiodic systems in stationary states with application to stochastic electrodynamics
,” Physica A
109
, 178
–192
(1981
);P.
Claverie
and F.
Soto
, “Nonrecurrence of the stochastic process for the hydrogen atom problem in stochastic electrodynamics
,” J. Math. Phys.
23
, 753
–759
(1982
).
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