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On the equation of motion and form factors of charged particles in classical electrodynamics
Physica 30 1453-1458
Cloetens, W. 1964
ON THE EQUATION OF MOTION AND FORM FACTORS OF CHARGED PARTICLES IN CLASSICAL ELECTRODYNAMICS by W. CLOETENS Facult6
des Sciences,
Universit6
Libre de Bruxelles,
Belgique
Synopsis 11 est bien connu que l’equation de Lorentz-Dirac valable pour une particle ponctuelle admet des solutions non physiques, c’est un systeme instable. Dans ce travail nous Btudions a la limite non relativiste la stabilite d’une particule “&endue” caracterisee par un facteur de forme (cut-off). Pour les forces temporelles nous demontrons que si nous nous fixons des conditions aux limites (ou la causalite) la particule chargee caracterisee par un facteur de forme toujours positif et une masse mecanique positive forme un systltme hereditaire, causal et stable. Aux solutions non physiques de la theorie de Lorentz-Dirac correspondent alors des solutions microscopiques et transitoires amorties par “radiation damping” apres un temps (cks)-1 qui est de l’ordre de grandeur du temps que prend la lumiere pour traverser le rayon classique de l’electron. La solution transitoire exprime l’adaptation de la particule chargee au champ exterieur qu’on branche. Dans les memes conditions la “bonne” solution de l’equation de Lorentz-Dirac peut etre interpretee comme une solution macroscopique et permanente. Pour les forces extkieures variant lentement pendant le temps (cks)-i la solution macroscopique “coincide” avec la bonne solution de l’equation de Lorentz-Dirac pour t > (c&)-i. Ceci est Bgalement vrai pour des forces positionnelles (par exemple le champ magnetique constant, l’oscillateur harmonique) . Pour des particules chargees caracterisees par certaines fonctions de cut-off ayant des zeros nous constatons que nous pouvons exciter indefiniment des self-oscihations dont la frequence est de l’ordre de &a. Si la frequence du champ exterieur est la m&me que la frequence des self-oscillations nous constatons une “resonnance”; pour ces frequences, la solution de l’equation du mouvement depend done tres sensiblement du facteur de structure choisi.
It is well known that the linearised Loren tz-Dirac 1) 2) equation admits unphysical solutions (run-aways). If one eliminates those unphysical solutions (e.g. refs. 1, 2, 3) by imposing a particular final acceleration one violates causality (pre-acceleration) for a time interval of the order of ~
e2
mcs
w IO-23 seconds
(1)
Sommerfeld, Markoff, Bohm and Weinstein and more recently Prigogine and Henin 5) have suggested that the existence of these run-away -
1453 -
w.
1454
CLOETENS
solutions comes from the assumption of point particles and that such unphysical solutions would cease to exist if the “structure” of the particle would be correctly taken into account. This suggestion is specially interesting at present because Prig o g i n e and H en i n 6) succeeded in writing covariant equations of motion for “extended particles” in arbitrary motion. In the present paper we shall, however, consider the special case of the non-relativistic approximation of these equations. This case corresponds to a linearization of the Prigogine-Henin electrodynamics and leads to expressions very similar to the Sommerfeld-BohmWeinstein equations for an extended charge. We obtain:
a(t) =
$
f(t) +
$
ro jdl.
jdh
--co
cos ck(t -
t’) g(k,-,, k) ii
(2)
0
with 10 = -
es
(3)
W&C2
where g&o,
k) =
d% e-* s spaoe
$
p(r).
The function p represents the effective “charge density” of the “point ticle” with a cut-off for the frequencies (for details see refs. 4 and 7). Following Prigogine and Henin we now define the mass as follows m=
m0
+ffu
(4) par-
(51
03
2e2 w&,1=
-
7cc2f
dk g&o, 4,
0
ma is the mechanical
mass,
m,l is the electromagnetic
mass.
To solve the integro-differential equation (2), one has to impose boundary conditions. For the cutoff functions we have used, we have proven that if one assumes that lim a(t) = 0, t-s-co
(7)
the equation of motion of the free particle admits only the solution
a(t) = 0.
(8)
For this reason, if one applies external forces for finite times only, which for
FORM
FACTORS
AND
1455
STABILITY
reasons of mathematical convenience we shall always take in the region of positive times, the equation of motion may be written a(t) = $-
f(t) + &
ye jdi. bh 0
cos ck(t - t’) g(ko, k) d(t).
(9)
0
We develop a method to solve easily this equation by introducing Laplacetransforms ; viz. S(p) ‘= / e--pt a(t) dt
(10)
0
E(9)
=
/
e--pt
f(t)
dt.
(11)
0
If we assume that f(0) = 0 and that f(t) is continuous in the neighbourhood of t = 0, then we have
w =$
H(P) E(P)
where 00
H(P) =
1
[l - &ro! +p2c2k2 g(ko? 4 dk]
(13)
p2
Taking first the analytical continuation H+(p) the inverse Laplace transforms gives
(definition (18), (19), (20)),
lJ+i=
s$
H+(P) E(P) ept dp
(14)
y--ion
If we put fJj=-- P
(15)
C iw
=
u
(16)
-4 d--k)
g(k) = g(ko, 4 r(k) + d--h
(17)
where 7 is the Heaviside function, we get H(--ica)
=
1 +m
C1+$/osam -00
2(k -
a)
(18)
dk
1
1456
W. CLOETENS
It may be shown that H(--icu)
is analytical
above the straight
line d. In
the complex plane the real axis forms a cut. (see fig. 1).
Fig. 1. Contour of inverse Laplace transform
Fig. 2. r Contour in the complex k plane
We now observe that [H(--ice)]-l is a regular function in the upperhalfplane S+. If Im CY> 0 on the real axis we obtain an integral representation by using the Plemelj formula. +oO [H+(-Lx)]-1
The analytical given by
=
1+ $
continuation
H+(-icu)
YOB 12;
of H+(--ice)
dk + i;
in the
lower
half-plane
(19) S-
is
= [I +i;yoiEdk]’
where r Using we can function
r. @(a).
is a contour in the complex plane (see fig. 2). the principle of the variation of the argument easily prove 7) that if g(ko, k) is a continuous of k and if the inequality ma > --km
(20)
and the hodograph positive decreasing
(21)
is satisfied, then H(9) has no pole with positive real parts: for this reason the charged particle and its self-field constitutes a stable system in the sense of G. Doetsch and Lyapounoff s) (comparea)ra)). Moreover, it can now be verified that the theory is strictly causal and that what previously was the unphysical solution of the Lorentz-Dirac equation becomes now a transient microscopic solution damped by radiation-damping after a time of the order of (cko)-1 w lo-23 s. On the other hand, the physical solution of the Lorentz-Dirac equation
FORM
FACTORS
AND
1457
STABILITY
(the “good” solution) is indeed the correct asymptotic solution for t > (eke)-1 and slowly varying forces. It may then be shown that our solutions have a behaviour corresponding to the usual physical (“good”) Lorentz-Dirac solutions. This physical solution may be interpreted as a permanent macroscopic solution. Let us illustrate these arguments with a simple example: Let
(22) then if
f(t) =
I
g
if t 0,
&
(23)
(13-l is a unit time) our general methods yield the solutions 0 a(t) =
if t
eE m
f3t + $0(&)-l
(1 -
e-
if t > 0.
(24)
/ The transient solution 2 --- 3
eE m B(ck~)-1 e-3cko’
expresses the “adaptation” of the particle to the external field. Bohm and Weinstein, Th. Erber, and Prigogine and Henin have suggested that interesting phenomena should arise for form factors which have zeros. It is indeed very simple to prove that if g&a, k) has zeros andif the transfer function H(9) has pure imaginary poles one can excite undamped selfoscillations with an external pulse. For instance, if
(25) (26) if t 0, our general methods
yield easily the solution
0 a(t) =
fi i
(27)
if t
e(cko)--r sin 43 dot
if t > 0.
(28)
1458
FORM FACTORS
AND STABILITY
If we take an external oscillating force which has the same frequency as the self-oscillations, we get a “resonance” for the acceleration. We shall give a more detailed discussion in another paper. 7). Acknowledgements. We wish to thank Professor I. Prigogine, Dr. F. Henin, Dr. A. Kuszell and Professor Th. Erber for many interesting and helpful discussions. This research was supported by a scholarship of the IRSIA. Received 3-2-64
REFERENCES
1) Dirac, P. A. M., Proc. Roy. Sot. A167 (1938) 148. 2) Erber, Th., Fortschritte der Physik 9 (1961) 343. 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
Plass, G. N., Rev. mod. Phys. 33 (1961) 37. Haag, R., Z. Naturf. A10 (1955) 752. Prigogine, I. and Henin, F., Physica 28 (1962) 667. Prigogine, I. and Henin, F., (to appear). Cloetens, W., MBmoire de Licence - Brussels’ University (1963). Bull. AC. Roy. Belg. Cl. SC. (to appear). Doetsch, G., Laplace Transformation, Dover, New York, (1943). Herglotz, G., Giitt. Nachr. 1903, p. 357. Wildermuth, K., Z. Naturf. A10 (1955)450. Balescu, R., Statistical Mechanics of Charged Particles, Wiley-Interscience, New York, 1963. Bohm, D. and Weinstein, M., Phys. Rev. 74 (1948) 1789. Titchmarsh, I%., The Theory of Functions, Oxford, (1958).
Cloetens, W. 1964
ON THE EQUATION OF MOTION AND FORM FACTORS OF CHARGED PARTICLES IN CLASSICAL ELECTRODYNAMICS by W. CLOETENS Facult6
des Sciences,
Universit6
Libre de Bruxelles,
Belgique
Synopsis 11 est bien connu que l’equation de Lorentz-Dirac valable pour une particle ponctuelle admet des solutions non physiques, c’est un systeme instable. Dans ce travail nous Btudions a la limite non relativiste la stabilite d’une particule “&endue” caracterisee par un facteur de forme (cut-off). Pour les forces temporelles nous demontrons que si nous nous fixons des conditions aux limites (ou la causalite) la particule chargee caracterisee par un facteur de forme toujours positif et une masse mecanique positive forme un systltme hereditaire, causal et stable. Aux solutions non physiques de la theorie de Lorentz-Dirac correspondent alors des solutions microscopiques et transitoires amorties par “radiation damping” apres un temps (cks)-1 qui est de l’ordre de grandeur du temps que prend la lumiere pour traverser le rayon classique de l’electron. La solution transitoire exprime l’adaptation de la particule chargee au champ exterieur qu’on branche. Dans les memes conditions la “bonne” solution de l’equation de Lorentz-Dirac peut etre interpretee comme une solution macroscopique et permanente. Pour les forces extkieures variant lentement pendant le temps (cks)-i la solution macroscopique “coincide” avec la bonne solution de l’equation de Lorentz-Dirac pour t > (c&)-i. Ceci est Bgalement vrai pour des forces positionnelles (par exemple le champ magnetique constant, l’oscillateur harmonique) . Pour des particules chargees caracterisees par certaines fonctions de cut-off ayant des zeros nous constatons que nous pouvons exciter indefiniment des self-oscihations dont la frequence est de l’ordre de &a. Si la frequence du champ exterieur est la m&me que la frequence des self-oscillations nous constatons une “resonnance”; pour ces frequences, la solution de l’equation du mouvement depend done tres sensiblement du facteur de structure choisi.
It is well known that the linearised Loren tz-Dirac 1) 2) equation admits unphysical solutions (run-aways). If one eliminates those unphysical solutions (e.g. refs. 1, 2, 3) by imposing a particular final acceleration one violates causality (pre-acceleration) for a time interval of the order of ~
e2
mcs
w IO-23 seconds
(1)
Sommerfeld, Markoff, Bohm and Weinstein and more recently Prigogine and Henin 5) have suggested that the existence of these run-away -
1453 -
w.
1454
CLOETENS
solutions comes from the assumption of point particles and that such unphysical solutions would cease to exist if the “structure” of the particle would be correctly taken into account. This suggestion is specially interesting at present because Prig o g i n e and H en i n 6) succeeded in writing covariant equations of motion for “extended particles” in arbitrary motion. In the present paper we shall, however, consider the special case of the non-relativistic approximation of these equations. This case corresponds to a linearization of the Prigogine-Henin electrodynamics and leads to expressions very similar to the Sommerfeld-BohmWeinstein equations for an extended charge. We obtain:
a(t) =
$
f(t) +
$
ro jdl.
jdh
--co
cos ck(t -
t’) g(k,-,, k) ii
(2)
0
with 10 = -
es
(3)
W&C2
where g&o,
k) =
d% e-* s spaoe
$
p(r).
The function p represents the effective “charge density” of the “point ticle” with a cut-off for the frequencies (for details see refs. 4 and 7). Following Prigogine and Henin we now define the mass as follows m=
m0
+ffu
(4) par-
(51
03
2e2 w&,1=
-
7cc2f
dk g&o, 4,
0
ma is the mechanical
mass,
m,l is the electromagnetic
mass.
To solve the integro-differential equation (2), one has to impose boundary conditions. For the cutoff functions we have used, we have proven that if one assumes that lim a(t) = 0, t-s-co
(7)
the equation of motion of the free particle admits only the solution
a(t) = 0.
(8)
For this reason, if one applies external forces for finite times only, which for
FORM
FACTORS
AND
1455
STABILITY
reasons of mathematical convenience we shall always take in the region of positive times, the equation of motion may be written a(t) = $-
f(t) + &
ye jdi. bh 0
cos ck(t - t’) g(ko, k) d(t).
(9)
0
We develop a method to solve easily this equation by introducing Laplacetransforms ; viz. S(p) ‘= / e--pt a(t) dt
(10)
0
E(9)
=
/
e--pt
f(t)
dt.
(11)
0
If we assume that f(0) = 0 and that f(t) is continuous in the neighbourhood of t = 0, then we have
w =$
H(P) E(P)
where 00
H(P) =
1
[l - &ro! +p2c2k2 g(ko? 4 dk]
(13)
p2
Taking first the analytical continuation H+(p) the inverse Laplace transforms gives
(definition (18), (19), (20)),
lJ+i=
s$
H+(P) E(P) ept dp
(14)
y--ion
If we put fJj=-- P
(15)
C iw
=
u
(16)
-4 d--k)
g(k) = g(ko, 4 r(k) + d--h
(17)
where 7 is the Heaviside function, we get H(--ica)
=
1 +m
C1+$/osam -00
2(k -
a)
(18)
dk
1
1456
W. CLOETENS
It may be shown that H(--icu)
is analytical
above the straight
line d. In
the complex plane the real axis forms a cut. (see fig. 1).
Fig. 1. Contour of inverse Laplace transform
Fig. 2. r Contour in the complex k plane
We now observe that [H(--ice)]-l is a regular function in the upperhalfplane S+. If Im CY> 0 on the real axis we obtain an integral representation by using the Plemelj formula. +oO [H+(-Lx)]-1
The analytical given by
=
1+ $
continuation
H+(-icu)
YOB 12;
of H+(--ice)
dk + i;
in the
lower
half-plane
(19) S-
is
= [I +i;yoiEdk]’
where r Using we can function
r. @(a).
is a contour in the complex plane (see fig. 2). the principle of the variation of the argument easily prove 7) that if g(ko, k) is a continuous of k and if the inequality ma > --km
(20)
and the hodograph positive decreasing
(21)
is satisfied, then H(9) has no pole with positive real parts: for this reason the charged particle and its self-field constitutes a stable system in the sense of G. Doetsch and Lyapounoff s) (comparea)ra)). Moreover, it can now be verified that the theory is strictly causal and that what previously was the unphysical solution of the Lorentz-Dirac equation becomes now a transient microscopic solution damped by radiation-damping after a time of the order of (cko)-1 w lo-23 s. On the other hand, the physical solution of the Lorentz-Dirac equation
FORM
FACTORS
AND
1457
STABILITY
(the “good” solution) is indeed the correct asymptotic solution for t > (eke)-1 and slowly varying forces. It may then be shown that our solutions have a behaviour corresponding to the usual physical (“good”) Lorentz-Dirac solutions. This physical solution may be interpreted as a permanent macroscopic solution. Let us illustrate these arguments with a simple example: Let
(22) then if
f(t) =
I
g
if t
&
(23)
(13-l is a unit time) our general methods yield the solutions 0 a(t) =
if t
eE m
f3t + $0(&)-l
(1 -
e-
if t > 0.
(24)
/ The transient solution 2 --- 3
eE m B(ck~)-1 e-3cko’
expresses the “adaptation” of the particle to the external field. Bohm and Weinstein, Th. Erber, and Prigogine and Henin have suggested that interesting phenomena should arise for form factors which have zeros. It is indeed very simple to prove that if g&a, k) has zeros andif the transfer function H(9) has pure imaginary poles one can excite undamped selfoscillations with an external pulse. For instance, if
(25) (26) if t
yield easily the solution
0 a(t) =
fi i
(27)
if t
e(cko)--r sin 43 dot
if t > 0.
(28)
1458
FORM FACTORS
AND STABILITY
If we take an external oscillating force which has the same frequency as the self-oscillations, we get a “resonance” for the acceleration. We shall give a more detailed discussion in another paper. 7). Acknowledgements. We wish to thank Professor I. Prigogine, Dr. F. Henin, Dr. A. Kuszell and Professor Th. Erber for many interesting and helpful discussions. This research was supported by a scholarship of the IRSIA. Received 3-2-64
REFERENCES
1) Dirac, P. A. M., Proc. Roy. Sot. A167 (1938) 148. 2) Erber, Th., Fortschritte der Physik 9 (1961) 343. 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
Plass, G. N., Rev. mod. Phys. 33 (1961) 37. Haag, R., Z. Naturf. A10 (1955) 752. Prigogine, I. and Henin, F., Physica 28 (1962) 667. Prigogine, I. and Henin, F., (to appear). Cloetens, W., MBmoire de Licence - Brussels’ University (1963). Bull. AC. Roy. Belg. Cl. SC. (to appear). Doetsch, G., Laplace Transformation, Dover, New York, (1943). Herglotz, G., Giitt. Nachr. 1903, p. 357. Wildermuth, K., Z. Naturf. A10 (1955)450. Balescu, R., Statistical Mechanics of Charged Particles, Wiley-Interscience, New York, 1963. Bohm, D. and Weinstein, M., Phys. Rev. 74 (1948) 1789. Titchmarsh, I%., The Theory of Functions, Oxford, (1958).