International Journal of Theoretical and Mathematical Physics

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

2018;  8(1): 12-27

doi:10.5923/j.ijtmp.20180801.02

 

Spin Two-Body Problem of Classical Electrodynamics with Radiation Terms (II) – Existence of Solution of the Spin Equations

Vasil G. Angelov

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

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

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Abstract

The primary purpose of the present paper is to continue our studies from previous papers where the spin equations were derived. Here we prove an existence of a periodic solution of the spin equations system using fixed point method. As a consequence, we obtain that the general two-body problem of classical electrodynamics with radiation terms and spin is already solved.

Keywords: Two body problem of classical electrodynamics, Spin equations, Periodic solutions, Radiation terms

Cite this paper: Vasil G. Angelov, Spin Two-Body Problem of Classical Electrodynamics with Radiation Terms (II) – Existence of Solution of the Spin Equations, International Journal of Theoretical and Mathematical Physics, Vol. 8 No. 1, 2018, pp. 12-27. doi: 10.5923/j.ijtmp.20180801.02.

Article Outline

1. Introduction
2. Formulation of the Periodic Problem for Spin Equations System
3. Operator Presentation of the Periodic Problem
4. Preliminary Lipschitz Estimates of the Lorentz Term of the Spin Equations
5. Preliminary Lipschitz Estimates of the Radiation Term of the Spin Equations
6. Existence of a Periodic Solution of the General System
7. Numerical Confirmation of the Results Obtained for the Hydrogen Atom
8. Conclusions

1. Introduction

The present paper is an immediate consequence of [1]. In [1] we have derived a general system of equations of motion describing two-body problem with radiation terms and spin. The results obtained rely on the previous papers [2] - [19]. We note another approach based on Wheeler-Feynman ideas and realized by D.-A. Deckert group (cf. [20] - [23]).
The general system describing motion of two mass charged particles with radiation terms and spin in the frame of classical electrodynamics derived in [1] is:
(1)
where Einstein summation convention is valid. This system is overdetermined. In other words, the number of equations is more than the number of unknown functions. In [6] and [1] we have proved that if the system
(2)
(3)
(containing 12 equations for 12 unknown functions) has a solution then the rest ones from (1) possess a solution too.
In [7] we have proved an existence-uniqueness of a periodic solution of equations of motion for two-body problem with corrected radiation terms, namely system (2). It remains to prove an existence of periodic solution of (3). On the right side of the equation (3) are the speeds and trajectories of the moving particles. Their existence is proven in [7]. We will consider them as known functions in (3).
The main goal of the present paper is to prove an existence of periodic spin functions satisfying (3). To do this we transform (3) using the known relations from vector calculus and obtain:
Therefore
Then the system (3) becomes
In coordinate form (3) is (α = 1,2,3):
(4)
Remark. It is easy to verify that if is a solution of the system (4) then is a solution too.

2. Formulation of the Periodic Problem for Spin Equations System

Recall denotations for quantities relating to the particles.
The space-time coordinates of the moving particles are
;
c is the speed of the light; world lines; – proper masses; – charges;
We have But the in- teresting case is Then;
are components of the null vector lying on the light cone; are velocities of the moving particles;
are components of the unit tangent vectors to world lines;
where
where the derivative is calculated from the equation
where
Consequently
We recall the basic assumption .
Finally we write down (4) in the form
(5)

3. Operator Presentation of the Periodic Problem

In this section we formulate an operator presentation of the periodic problem for (5).
First we transform (5) in order to obtain a suitable form of the spin equations system:
or
(7)
where
In what follows we call Lorentz term of the spin equations, while − radiation term.
We recall (cf. [7]) that velocity functions belong to the space
where is the space of all infinite differentiable - periodic functions. Following A. Sommerfeld [9], [10] we denote by . It follows that trajectories and velocities are -periodic functions.
Our goal is to prove an existence of -periodic solution of (6). For that purpose we define an operator on the space consisting of all continuous functions such that by the formulas
(8)
Lemma. Every -periodic solution of (6) is a fixed point of H and vice versa.
The proof can be found in [24] (M. A. Krasnoselskii).
Consequently, we have to prove an existence of a fixed point of the operator H.

4. Preliminary Lipschitz Estimates of the Lorentz Term of the Spin Equations

First we notice that ;
and
Then

5. Preliminary Lipschitz Estimates of the Radiation Term of the Spin Equations

Here we transform the radiation part of the spin equation using some reasoning from [7]. Indeed, we recall assumption with small:
Then
Then we transform the radiation part of (5) with accordance of assumptions from [7]:
Consequently in view of
we obtain
Then
Therefore
It follows
Therefore the operator H is continuous one.

6. Existence of a Periodic Solution of the General System

The main result of the present paper is:
Theorem 1. Let the following conditions be fulfilled:
Then there exists a periodic solution of (5).
Proof: Introduce the set with a metric
where , is a fixed constant.
Our first step is to show that the operator H maps into itself, that is,
Indeed,
Therefore
Consequently the operator H maps into itself.
It remains to show that the set H(M) is equicontinuous. Indeed, for , we have
Theorem is thus proved.

7. Numerical Confirmation of the Results Obtained for the Hydrogen Atom

Here we show that all assertions obtained concerning two-body problem completely confirm the experimental results for the hydrogen atom. Indeed, we recall that
the radiation time is . Since the radius of first Bohr orbit is and its velocity is (1/137 − Sommerfeld fine structure constant [9], [10]), then
Our estimates require to be a constant. Here and we have to take , for instance . Then . Therefore and , that is condition is satisfied. Here .
The following inequalities guarantee an existence of periodic solution of the equations of motion with radiation terms (obtained in [7]):
In the above inequalities n might be chosen arbitrarily large (cf. [7]).
We have to verify the inequality from the main theorem
or in view of
This means that the initial values of the spin functions should satisfy the last inequality.

8. Conclusions

The present paper completes our investigations on the two-body problem of classical electrodynamics. Beginning with Synge model we have corrected Dirac radiation term and extended Corben-Stehle spin equations. We have proved an existence of unique periodic solution of the equations of motion in [7]
and here an existence of a periodic solution of spin equations
In this manner we have proved the stability of hydrogen atom and showed that stationary states introduced by N. Bohr are implied by classical electrodynamics. Introducing spin equations in the frame of relativistic Synge formalism we can investigate 3-body problem (and in general N-body) of classical electrodynamics.

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