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On the instantaneous form of the two-body equations of motion in fokker-wheeler-feynman electrodynamics
Volume 85A, number 3
PHYSICS LETTERS
21 September 1981
ON THE INSTANTANEOUS FORM OF THE TWO-BODY EQUATIONS OF MOTION IN FOKKER—WHEELER—FEYNMAN ELECTRODYNAMICS V.1. ZHDANOV State Standards Committee, Moscow 117049, USSR Received 12 June 1981
The motion of particles in the above theory is governed by functional differential equations containing both retarded and advanced times. We prove that at least in the special case of the two-body problem there exists an equivalent newtonian
(i.e. single-time) equation of motion.
1. The Fokker-type equations of motion (see e.g. refs. [1,2] for references) which are well known in classical relativistic dynamics, take into account advanced interactions. This leads to the associated problems of causality [1]which are essentially the questions of existence of well-behaved solutions of the equations of motion. Driver [2] has recently proved the existence and uniqueness of solutions specified by instantaneous initial data in the special case of the two-body problem. In this letter we extend this result to the case of more general weakly relativistic initial data and relate the above questions to the existence of an equivalent (in a certain sense) ordinary differential equation of motion which contains all particle positions and velocities with the same time value. We consider two identical particles (charges of like sign) moving along the x-axis symmetrically about the origin and interacting by means of half-retarded plus half-advanced Liénard—Wiechert potentials. The particle motion is governed by the functional differential system [2]: _________
—
(1
—
q
—
=
~2)3/2
k1
—
~(t
~.2I + ~ (t
x + x(t + q),
rx+x(t—r)
— —
r) k 1 + ~(t + q) r) + 2 1 I (t + q)’ —
0)
(2) (3)
where x is the position of the right-hand particle, the constant k > 0 depends on the charge and the mass 138
of the particles, r is the delay, q is the advance [according to Driver [2] we write x, q, r instead of x(t). q(t), r(t)]. Throughout the paper we suppose that x, q, r are twice differentiable functions of t and ~ <1 ~ >0 4 The velocities are normalized to the light speed. If ~ C: I~Is~ c < 1 (we shall see that this is true in our case), then it follows from the results of refs. [2,3] that solutions of eqs. (2), (3) with respect to q and r exist and that they are unique; therefore q and r may be considered as known functionals depending uponx. We assert that there exists an equation 5(• X
~X, X,,
which specifies uniquely all solutions of the system (I )—(3) satisfying the initial point conditions .
—
x(t0)
—
v0~ x(t0) x0~ 1v01 < 1, xo > 0, (6) provided that the quantity v~+ k/x0 is sufficiently small. It is crucial that we consider only those solutions of eq. (1)—(3) which satisfy this system for all t (not only for t ~ t0). The above statement will be proved on the basis of the existence—uniqueness theorem for the system (1)—(3) with the conditions (6). Driver [2] has proved such a theorem under the assumptions that v = 0, x ~ 4600k. We shall extend 0 case this theorem to the 0. —
—
2. It follows from the results of Driver [2]that
0 031—9163/81 /0000—0000/s 02.50 © North-Holland Publishing Company
Volume 85A, number 3
PHYSICS LETFERS
21 September 1981
there exists a function z (t, y) which gives the unique solution x (t) = z(t, y) of eq. (1) [considered along with eqs. (2), (3), which define q(x, t) and r(x, t)] satisfying the conditions 1(0) = 0, x (0) = y (y 4600
0 denoting the Heaviside function. Using (12) we oh-
k).
where F(y) = [~~(i v~)312
tain the equation = kF(y),
(13)
~‘
—
Introduce the function T(v, y) by means of the
—
+
k/2x
0
—
equation
J(T, y) = u.
(7)
The dot over z means differentiation with respect to the first argument. It is easy to see that ~(t, y) > 0.
Then the solution of eq. (7) (if it exists) is unique. If there is a y such that T(v 0, y) exists and
then the function —
2 defined by the
‘L)(E) be the domain of R
~TD(E)= {(v, x): v2 + k/x
~E},
k > 0.
The following lemma is obtained by means of a slight modification of the estimates in ref. [2].
(8)
z(T(u0,y),y)x0,
x(t) = z (t
3. Let relation
t0 + T(v0, y), y)
(9)
Lemma. There exists a value E >0 (sufficiently small) such that, if (x, q, r) satisfy the system (1)— (3) and conditions (4), (6), where (u0, x0) E ~D(E), then x(t) 00 for ItI-÷o~3 tmi(tm) 0 and 2(t) + k/x(t) 0(E), ~t E A. (14) 1 This lemma allows us to use a method analogous to that of our paper [3] to prove the existence—uniqueness theorem. We also use some intermediate estimates from refs. [2,3]. These allow us to confine our discus-4
gives us the solution of eqs. (1)—(3) with the condi-
tions (6). the transformations pointed out in ref. [3] Using we represent eq. (1) in the form I (1
~
(10)
+Rr(X, t) + Rq(X, t), 12)5/2 — Z~2
—
~
where
.
‘~
sion to the key points of the proof. 1
f ds I(t
kr
Rr(X, t) —
—
rs)
Theorem 1. There exists an E >0 such that if (v 0,
2x(l X r0
—
LTO
(1
—
—
12)[I +I(t
s) (I + \T
[1
—
r)] 0
—
I (t
—
x0) E~D(E), then eqs. (1)—(3), (6) under conditions (4) have a unique solution. hoof For sufficiently small E standard estimates
r)]],
T0
analogous to those of ref. [3] show that for y we have: (a)
2x(1 +I)~.
=
R is obtained fromR,. byreplacingr-~q,r0 ÷q0 ~x(I
+ q).
—I)~,I(t—r)—*—1(t+q),I(t—r)-+I(t
Multiplying eq. (10) by I and integrating for x we have under the conditions (7), (8): 312+ k/2x ~(1 v~) 0 = + k/2y + Q(y), (11) —
where Q(y) =
f ds 0 [v0
—
~(s, y)] ~(s,y)
0 X [Rr(z, s) + Rq(z, s)},
(12)
~‘
~
IF(Y’)
—
F(y”)I ~ k~0(E)Iy’
(15) —
y”i.
Then the function GO’) 1 kF(y) is continuous, strictly increasing for y ~ 2k/E and changes its sign. Then there exists a unique solution i’o of eq. (13). —
(b) In this case the solution T(y0, u0) of eq. (7) also exists [let e.g. U0 > 0, then one obtains from eqs. (10) ~1 2) ~ (00, y~)> V0 ~ 0 = ~(0, Yo)]- Eq. (11) also yields the relation (8). Then the solution of eqs. (1) —(3), (6) is given by formula (9). (c) Let (x(t), q(t), r(t)) be a solution of eqs. (I)—(3), (6), E being sufficiently small. Then according to the lemma,2 tm:
139
Volume 85A, number 3
PHYSICS LETTERS
21 September 1981
0 and k/X(tm) ~ 0(E). In view of (11), must be a solution of eq. (13). Then the uniqueness of the solution of eqs. (l)—(3), (6) follows from the uniqueness of the solution of eq. (13) stated in
the evolution of the particles being uniquely predicted by the instantaneous values of their positions and yelocities. In this sense causality may be treated as in newtonian dynamics. The newtonian analogy is clear-
(a). This ends the proof of the theorem.
ly demonstrated by the existence of the instantaneous
I(tm) X(tm)
=
In virtue of theorem 1 we have the mappingf: 2(R, A) which defines the unique solution ~7’(E) C x(t) =f(t t 0, v0, x0) of eqs. (1)—(3) under the conditions (6) with (v0, x0) E ~1(E),q and r being defined uniquely by eqs. (2), (3). Because (u0, x0) is an arbitrary point of CZ~(E),we have that x(t) =f(t t’ 1(t’) x(t’)), ~ EA
form (5) of the equations of motion. There is, however, a difference from the newtonian case, because we cannot prescribe an arbitrary behaviour of the par-
-+
—
—
~‘
The second derivative of this relation with respect to t for t = t’ gives us the form (5) with S (y, x) = f(0,y, x). Modifying (1 5) for the case when the initial data (u0, x0) also vary, one can prove the Lipschitz continuity of SO’,x)in every compact subset of C~b(E).The above considerations lead to a theorem which shows the existence of an ordinary differential equation equivalent to eq. (1) in the sense stated below, Theorem 2. There exists a value E> 0 and a unique 2, ~D function S: ~D0 A, where C~0c R 0D ~7(E), such that if (x, q, r) is a solution of eqs. (5), (2), (3) satisfying (6) with (v0, x0) E then (x, q, r) satisfies the conditions (4) and the system (1)—(3) and vice versa.
tide trajectories in the past (t < t0) but can deal only with the evolution for t ~ t0.
In general, for functional differential equations, there are infinitely many solutions satisfying initial point conditions. (See examples in refs. [2,4].) It is known, however, that the trajectories of two particles in one dimension are sometimes uniquely determined by the intial data at t = t0 [2—5] Theorem I gives one more example of this situation. It is possible to expect that the single-time form of the equations of motion also exists for a more general many-body problem of electrodynamics if we restrict ourselves to weakly relativistic trajectories. A preliminary inspection, however, shows that this requires much more stringent assumptions on the class of allowed motions than in the one-dimensional case. -
-4
References
~
4. The results obtamed show that m spite of the presence of advanced interactions there exist well-behaved trajectories described by the system (1)— (3),
140
[1] J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. (1945) 157; 21(1949) 425. [2] R.D. Driver, Phys. Rev. D19 (1979) 1098. [3] V.1. Zhdanov, mt. J. Theor. Phys. 15 (1976) 157.
[4] R.D. Driver, Phys. Rev. 178 (1969) 2051.
[5] D.K.
Hsing, Phys. Rev. D16 (1977) 974.
17
PHYSICS LETTERS
21 September 1981
ON THE INSTANTANEOUS FORM OF THE TWO-BODY EQUATIONS OF MOTION IN FOKKER—WHEELER—FEYNMAN ELECTRODYNAMICS V.1. ZHDANOV State Standards Committee, Moscow 117049, USSR Received 12 June 1981
The motion of particles in the above theory is governed by functional differential equations containing both retarded and advanced times. We prove that at least in the special case of the two-body problem there exists an equivalent newtonian
(i.e. single-time) equation of motion.
1. The Fokker-type equations of motion (see e.g. refs. [1,2] for references) which are well known in classical relativistic dynamics, take into account advanced interactions. This leads to the associated problems of causality [1]which are essentially the questions of existence of well-behaved solutions of the equations of motion. Driver [2] has recently proved the existence and uniqueness of solutions specified by instantaneous initial data in the special case of the two-body problem. In this letter we extend this result to the case of more general weakly relativistic initial data and relate the above questions to the existence of an equivalent (in a certain sense) ordinary differential equation of motion which contains all particle positions and velocities with the same time value. We consider two identical particles (charges of like sign) moving along the x-axis symmetrically about the origin and interacting by means of half-retarded plus half-advanced Liénard—Wiechert potentials. The particle motion is governed by the functional differential system [2]: _________
—
(1
—
q
—
=
~2)3/2
k1
—
~(t
~.2I + ~ (t
x + x(t + q),
rx+x(t—r)
— —
r) k 1 + ~(t + q) r) + 2 1 I (t + q)’ —
0)
(2) (3)
where x is the position of the right-hand particle, the constant k > 0 depends on the charge and the mass 138
of the particles, r is the delay, q is the advance [according to Driver [2] we write x, q, r instead of x(t). q(t), r(t)]. Throughout the paper we suppose that x, q, r are twice differentiable functions of t and ~ <1 ~ >0 4 The velocities are normalized to the light speed. If ~ C: I~Is~ c < 1 (we shall see that this is true in our case), then it follows from the results of refs. [2,3] that solutions of eqs. (2), (3) with respect to q and r exist and that they are unique; therefore q and r may be considered as known functionals depending uponx. We assert that there exists an equation 5(• X
~X, X,,
which specifies uniquely all solutions of the system (I )—(3) satisfying the initial point conditions .
—
x(t0)
—
v0~ x(t0) x0~ 1v01 < 1, xo > 0, (6) provided that the quantity v~+ k/x0 is sufficiently small. It is crucial that we consider only those solutions of eq. (1)—(3) which satisfy this system for all t (not only for t ~ t0). The above statement will be proved on the basis of the existence—uniqueness theorem for the system (1)—(3) with the conditions (6). Driver [2] has proved such a theorem under the assumptions that v = 0, x ~ 4600k. We shall extend 0 case this theorem to the 0. —
—
2. It follows from the results of Driver [2]that
0 031—9163/81 /0000—0000/s 02.50 © North-Holland Publishing Company
Volume 85A, number 3
PHYSICS LETFERS
21 September 1981
there exists a function z (t, y) which gives the unique solution x (t) = z(t, y) of eq. (1) [considered along with eqs. (2), (3), which define q(x, t) and r(x, t)] satisfying the conditions 1(0) = 0, x (0) = y (y 4600
0 denoting the Heaviside function. Using (12) we oh-
k).
where F(y) = [~~(i v~)312
tain the equation = kF(y),
(13)
~‘
—
Introduce the function T(v, y) by means of the
—
+
k/2x
0
—
equation
J(T, y) = u.
(7)
The dot over z means differentiation with respect to the first argument. It is easy to see that ~(t, y) > 0.
Then the solution of eq. (7) (if it exists) is unique. If there is a y such that T(v 0, y) exists and
then the function —
2 defined by the
‘L)(E) be the domain of R
~TD(E)= {(v, x): v2 + k/x
~E},
k > 0.
The following lemma is obtained by means of a slight modification of the estimates in ref. [2].
(8)
z(T(u0,y),y)x0,
x(t) = z (t
3. Let relation
t0 + T(v0, y), y)
(9)
Lemma. There exists a value E >0 (sufficiently small) such that, if (x, q, r) satisfy the system (1)— (3) and conditions (4), (6), where (u0, x0) E ~D(E), then x(t) 00 for ItI-÷o~3 tmi(tm) 0 and 2(t) + k/x(t) 0(E), ~t E A. (14) 1 This lemma allows us to use a method analogous to that of our paper [3] to prove the existence—uniqueness theorem. We also use some intermediate estimates from refs. [2,3]. These allow us to confine our discus-4
gives us the solution of eqs. (1)—(3) with the condi-
tions (6). the transformations pointed out in ref. [3] Using we represent eq. (1) in the form I (1
~
(10)
+Rr(X, t) + Rq(X, t), 12)5/2 — Z~2
—
~
where
.
‘~
sion to the key points of the proof. 1
f ds I(t
kr
Rr(X, t) —
—
rs)
Theorem 1. There exists an E >0 such that if (v 0,
2x(l X r0
—
LTO
(1
—
—
12)[I +I(t
s) (I + \T
[1
—
r)] 0
—
I (t
—
x0) E~D(E), then eqs. (1)—(3), (6) under conditions (4) have a unique solution. hoof For sufficiently small E standard estimates
r)]],
T0
analogous to those of ref. [3] show that for y we have: (a)
2x(1 +I)~.
=
R is obtained fromR,. byreplacingr-~q,r0 ÷q0 ~x(I
+ q).
—I)~,I(t—r)—*—1(t+q),I(t—r)-+I(t
Multiplying eq. (10) by I and integrating for x we have under the conditions (7), (8): 312+ k/2x ~(1 v~) 0 = + k/2y + Q(y), (11) —
where Q(y) =
f ds 0 [v0
—
~(s, y)] ~(s,y)
0 X [Rr(z, s) + Rq(z, s)},
(12)
~‘
~
IF(Y’)
—
F(y”)I ~ k~0(E)Iy’
(15) —
y”i.
Then the function GO’) 1 kF(y) is continuous, strictly increasing for y ~ 2k/E and changes its sign. Then there exists a unique solution i’o of eq. (13). —
(b) In this case the solution T(y0, u0) of eq. (7) also exists [let e.g. U0 > 0, then one obtains from eqs. (10) ~1 2) ~ (00, y~)> V0 ~ 0 = ~(0, Yo)]- Eq. (11) also yields the relation (8). Then the solution of eqs. (1) —(3), (6) is given by formula (9). (c) Let (x(t), q(t), r(t)) be a solution of eqs. (I)—(3), (6), E being sufficiently small. Then according to the lemma,2 tm:
139
Volume 85A, number 3
PHYSICS LETTERS
21 September 1981
0 and k/X(tm) ~ 0(E). In view of (11), must be a solution of eq. (13). Then the uniqueness of the solution of eqs. (l)—(3), (6) follows from the uniqueness of the solution of eq. (13) stated in
the evolution of the particles being uniquely predicted by the instantaneous values of their positions and yelocities. In this sense causality may be treated as in newtonian dynamics. The newtonian analogy is clear-
(a). This ends the proof of the theorem.
ly demonstrated by the existence of the instantaneous
I(tm) X(tm)
=
In virtue of theorem 1 we have the mappingf: 2(R, A) which defines the unique solution ~7’(E) C x(t) =f(t t 0, v0, x0) of eqs. (1)—(3) under the conditions (6) with (v0, x0) E ~1(E),q and r being defined uniquely by eqs. (2), (3). Because (u0, x0) is an arbitrary point of CZ~(E),we have that x(t) =f(t t’ 1(t’) x(t’)), ~ EA
form (5) of the equations of motion. There is, however, a difference from the newtonian case, because we cannot prescribe an arbitrary behaviour of the par-
-+
—
—
~‘
The second derivative of this relation with respect to t for t = t’ gives us the form (5) with S (y, x) = f(0,y, x). Modifying (1 5) for the case when the initial data (u0, x0) also vary, one can prove the Lipschitz continuity of SO’,x)in every compact subset of C~b(E).The above considerations lead to a theorem which shows the existence of an ordinary differential equation equivalent to eq. (1) in the sense stated below, Theorem 2. There exists a value E> 0 and a unique 2, ~D function S: ~D0 A, where C~0c R 0D ~7(E), such that if (x, q, r) is a solution of eqs. (5), (2), (3) satisfying (6) with (v0, x0) E then (x, q, r) satisfies the conditions (4) and the system (1)—(3) and vice versa.
tide trajectories in the past (t < t0) but can deal only with the evolution for t ~ t0.
In general, for functional differential equations, there are infinitely many solutions satisfying initial point conditions. (See examples in refs. [2,4].) It is known, however, that the trajectories of two particles in one dimension are sometimes uniquely determined by the intial data at t = t0 [2—5] Theorem I gives one more example of this situation. It is possible to expect that the single-time form of the equations of motion also exists for a more general many-body problem of electrodynamics if we restrict ourselves to weakly relativistic trajectories. A preliminary inspection, however, shows that this requires much more stringent assumptions on the class of allowed motions than in the one-dimensional case. -
-4
References
~
4. The results obtamed show that m spite of the presence of advanced interactions there exist well-behaved trajectories described by the system (1)— (3),
140
[1] J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. (1945) 157; 21(1949) 425. [2] R.D. Driver, Phys. Rev. D19 (1979) 1098. [3] V.1. Zhdanov, mt. J. Theor. Phys. 15 (1976) 157.
[4] R.D. Driver, Phys. Rev. 178 (1969) 2051.
[5] D.K.
Hsing, Phys. Rev. D16 (1977) 974.
17