International Journal of Theoretical and Mathematical Physics

p-ISSN: 2167-6844    e-ISSN: 2167-6852

2015;  5(5): 120-136

doi:10.5923/j.ijtmp.20150505.05

Two-Body Problem of Classical Electrodynamics with Radiation Terms-Derivation of Equations (I)

Vasil G. Angelov

Department of Mathematics, University of Mining and Geology “St. I. Rilski”, Sofia, Bulgaria

Correspondence to: Vasil G. Angelov, Department of Mathematics, University of Mining and Geology “St. I. Rilski”, Sofia, Bulgaria.

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Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved.

Abstract

This paper is the first part of our investigations devoted to the two-body problem of classical electrodynamics. The primary purpose of this first part is to derive equations of motion describing two moving charged mass particles taking into account the radiation. We proceed from the suggestions given by J. L. Synge [1]. He has proposed a formulation of the of relativistic two-body problem with usually accepted Dirac’s radiation terms [2] containing second derivatives of the velocities: , Here we propose a general approach to introduce new equations of motion based on the same Dirac’s physical assumptions from [2]. Instead of the above system of eight equations of motion we derive consider an analogous system , where the classical Lorentz-Dirac radiation terms are replaced by newly derived ones. We show that two equations are consequences of the rest ones and so we have to solve a system of six equations for six unknown velocities, issue that has not been discussed in the literature. Instead of second order differential system (with respect to the unknown velocities) we obtain a first order neutral system with both retarded and advanced arguments depending on the unknown trajectories. In the second part we solve the system and so we give a method for overcoming the singularities arising in Dirac radiation terms.

Keywords: Classical electrodynamics, Two-Body problem, Dirac-Lorentz radiation term, Neutral equations with both delay and advanced arguments, Fixed point theorem

Cite this paper: Vasil G. Angelov, Two-Body Problem of Classical Electrodynamics with Radiation Terms-Derivation of Equations (I), International Journal of Theoretical and Mathematical Physics, Vol. 5 No. 5, 2015, pp. 120-136. doi: 10.5923/j.ijtmp.20150505.05.

Article Outline

1. Introduction
2. Main Results
    2.1. Derivation of Equations of Motion for Two-Body Problem with New Dirac Radiation Term
    2.2. The Fourth Equation Is a Consequence of the First Three Ones
3. Conclusions
ACKNOWLEDGEMENTS

1. Introduction

In [1] J. L. Synge has formulated a two-body problem in the frame of classical electrodynamics and has mentioned at the very end of [1] the possibility to generalize the model including Dirac radiation terms [2]. The Synge’s considerations [1] of two-body problem are based on the Lorentz ponder-motive force derived in a relativistic form by W. Pauli [3] via Lienard-Wiechert retarded potentials and adding the Dirac’s radiation terms. Our goal is, following [1], to give a unified approach of derivation of equations of motion for two-body problem with radiation terms. We propose a new mathematical formulation of the Dirac’s idea based on retarded and advanced potentials. Let us note that Dirac’s derivations lead to the second order differential equations with respect to the unknown velocities which by necessity needs prescribing of initial accelerations.
A lot of papers investigate equations with usually accepted Dirac-Lorentz radiation terms [4]-[27]. In contrast of these papers following the Synge’s formalism [1] here we derive a new form of the radiation term leading to the first order neutral equations with respect to unknown velocities and introduce retarded and advanced arguments depending on unknown trajectories [28]. We note that this new term is applied in the one-dimensional case to overcome P. Ehrenfest paradox [29] and in three-dimensional case for correction of the Lorentz-Dirac equation [30].
We point out that the following difficulty arises that has not been discussed in the literature. The relativistic Lorentz – Dirac equations in the Minkowski’s space are four in number for three unknown functions. In [30] we have proved that 4-th equation is a consequence of the first three ones. An analogous problems arises in the two-body problem.
The equations of motion are eight in number for six un-known functions velocities. Here we prove that the 4-th and the 8-th equation are consequence of the rest ones. In this manner we overcome a mathematical problem generated by overdetermined system. From the physical point of view, solving this system substituting the found functions in the 4-th and 8-th equations we obtain the energy balance of the moving particles.
In the second part we prove an existence-uniqueness of a periodic solution of the system mentioned which means an existence of a closed orbit of two-body problem. In this way, we show that the Bohr-Sommerfeld stationary states (cf. [31], [32]) are rather implicated by the classical electrodynamics than contradict it.
Here we use the technique introduced in [33], [34] and obtain a system of eight equations of motion with new form of the radiation terms:
Main results are given in Section 2. Subsection 2.1 is devoted to the strict mathematical formulation of the original Dirac’s assumptions. In fact, we compare the relative and absolute times assuming that the past and future instants depend only on the present instant. In Subsection 2.2 we show that the 4-th equation is a consequence of the first three ones and the 8-th equation is a consequence of the 5-th, 6-th and 7-th ones. So we obtain a system of six equations for six unknown functions − the velocities of the moving particles. This system is of neutral type with respect to the unknown velocities with both retarded and advanced arguments depending on the unknown trajectories.
Section 3 is Conclusion.
Some cumbersome calculations are separated in Supplement 1 and Supplement 2.

2. Main Results

2.1. Derivation of Equations of Motion for Two-Body Problem with New Dirac Radiation Term

The considerations are in the Minkowski’s space [3]. Roman suffixes run over 1, 2, 3, 4 while Greek – 1, 2, 3 with Einstein summation convention. We use denotations from [1]. By we denote the dot product in the Minkowski space, and by – the dot product in three-dimensional Euclidean subspace. The space-time coordinates of the moving particles are Quantities relating to the particles are denoted in the following way: – world lines; – proper masses; – charges. The components of unit tangent vectors to world lines are
Then
where
c is the vacuum speed of light and
are velocities of the moving particles. The components of the accelerations are
or
Let be a charge describing any curve L1 in space-time (cf. [1]). Let A2 be any event and let A1 be an intersection of L1 with the null-cone drawn into the past from A2. Let be the unit tangent vector to L1 at A1, and let be the null-vector A1A2. Then, by hypothesis from [1], the field at A2 due to L1 is given by the retarded potential 4-vector
(1)
and the corresponding electromagnetic tensor is
(2)
where
(3)
and
is an isotropic vector, i.e. it lies on the light-cone [1]. The world-line L2 of the charge , passing through A2, satisfies the equations of motion
We have to add however the radiation field caused by the particle itself. Then equations of motion for the second particle are the following four ones:
If we interchange the roles of the two world-lines we have the following equations of motion for the first particle:
(4)
or
(5)
where
So we obtain eight equations of motion for two-body problem with radiation terms:
We recall that derivation of the radiation term is based on the physical assumptions from [2].
Consider a charge describing any curve in the space-time. Let
be any event,
be the intersection of with the null-cone drawn into the past from and
be the intersection of with the null-cone drawn into the future from .
The components of the velocity (tangent) vector to the world-line at are
where and let be isotropic vector ,
where
Similarly the components of the velocity (tangent) vector to the world-line at are
where and let be isotropic vector
where
In accordance with Dirac assumptions [2] the radiation term is defined as a half of the difference between both retarded and advanced potentials, that is,
where
So in view of (3) we obtain the following eight equations:
or
Now follow [33] and [34] we obtain

2.2. The Fourth Equation Is a Consequence of the First Three Ones

The system obtained in Supplement 1 can be rewritten as
Proceeding as in [30], [34] we multiply by , summing up in α and dividing into c2 we obtain.
In other words the fourth equation is a consequence of the first three ones and the eighth equation is a consequence of the previous three ones. In this way we obtain 6 equations for 6 unknown functions.
The system from Supplement 2 can be written in the following form :
Denoting by the right-hand sides of we have to solve the following system with respect to
We make the following:
Assumption (C): All velocities satisfy the inequalities
and then Therefore, the determinant of the above system is and
consequently we reach the system:
The right-hand side of is a sum of two terms
Lorentz term
and radiation term

3. Conclusions

Here we introduce a unified approach to derive two-body problem with radiation terms. The system obtained is of neutral type containing both retarded and advanced arguments. The unknown functions are velocities of the moving particles. The deviating arguments depend on the unknown trajectories. We would like to point that we follow physical reasoning due to Dirac [2] which leads to assumption
But with accordance of special relativity theory we have to consider radiation time in the form where . In view of and as then the parameter should be consider as an infinitely small parameter. Extending the technique from [30] and [34] we replace the usually accepted second order Dirac’s system of ordinary differential equations by a first order system of neutral equations with both retarded and advanced arguments.
We would like to comment our basic assumption (C): (cf. also [30]). In the Newton theory − the speed of propagation of the interaction is , but anybody cannot reach this speed . Here the role of is played by c.
Supplement 1
In 2.1 we have obtained system . In order to solve it we have to transform it in a suitable form. First of all we have to find a relation between the relativistic and absolute time. Indeed, following [19], [33], and [34] we assume that
Since and lie on the trajectory we put
where
where
have lengths 0 we obtain
We put
and then
,
can be obtained as solutions of the equations
or
We have to find relations between the derivatives at past, present and future instants. Indeed, extending reasoning from [19], differentiating the relations
and solving with respect to we obtain
Therefore
In a similar way we differentiate
with respect to (considering ) and obtain
Hence
and similarly
Further on we have
Substituting the expressions obtained we reach the system
The last equation should be divided into .
Supplement 2
Introduce denotations
Regrouping the terms in the right-hand sides of we obtain the system in a vector form:

ACKNOWLEDGEMENTS

The author thanks very much the referee for his very useful suggestions which improve the paper.

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