1. Introduction

The primary goal of the present paper is to prove an existence-uniqueness of a periodic solution of the two-body system with radiation terms, derived in a recent paper [1]. We have already mentioned in [1] that using the relativistic form of Lienard-Wiechert retarded potentials (cf. [2]) J. L. Synge [3] has formulated the two-body problem of classical electrodynamics and has suggested the idea to generalize the model including Dirac radiation terms [4]. Developing Synge idea we have derived a new form of Dirac radiation terms [1]. So instead of Synge two-body system [2]

we have introduced the system

where the radiation terms have a new form, although are derived on the basis of the original Dirac physical assumptions [4]. In [1] we have proved that the 4th and the 8ht equations are consequences from the rest ones and after some transformations we have reached a system of 6 equations as there are unknown functions.

We emphasize that classical Lorentz-Dirac radiation term and two- body equations of motion do not satisfy the basic relativistic equation

obviously fulfilled in our case [1].

We also note that on the base of Wheeler-Feynman formalism [8], [9] the authors in [10]-[14] have been obtained interesting results.

Our purpose, however, is to show the existence and uniqueness of a periodic solution of the two-body equations of motion obtained in [1]. We present the system in question in a suitable operator form and the fixed point of this operator appears a periodic solution of the system.

Section 1 is an introduction and we rearrange equations of motion in more convenient form. Main results are given in Section 2. In Subsection 2.1 we derive radiation terms in an explicit form and show that they are bounded, that is, free of singularities. In Subsection 2.2 we formulate the main periodic problem and prove some preliminary assertions. In order to apply fixed point theorems from [5] we introduce suitable function spaces and operator whose fixed points are solutions of the periodic problem mentioned and give some lemmas. Supplement 1 contains preliminary estimates that imply the operator maps the solution set into itself. In Supplement 2 Lipschitz estimates of the operator and its derivatives are obtained. Subsection 2.3 contains the main result: two-body system has a unique periodic solution. Subsection 2.4 includes numerical test results. Section 3 is a conclusion and shows the adequacy of the main theorem.

Recall denotations from [1]:

are the space - time coordinates of the moving particles;

- velocities of the moving particles;

;

- dot product in 3-dimensional Euclidian space.

In [1] we have derived the following equations of motion

:

where

Substituting and into , transforming the expressions obtained under Dirac assumption (cf. [4]) we have:

Denoting by the right-hand sides of the above system we obtain

Solving with respect to under assumption

we reach the neutral system of six equations

:

for six unknown functions

, where The summands are called Lorentz terms, while the summand

- radiation terms.

2. Main Results

2.1. Explicit Form of the Radiation Term

In what follows we derive the explicit form of the radiation terms. By we denote the set of all infinitely differentiable -periodic functions. We introduce the space of functions:

(2.1)

where are positive constants and following A. Sommerfeld [7] .

Recall that is assumed to be infinitely small parameter because

(cf. [2]). In fact .

Since we work in spaces of infinitely smooth functions, using the Taylor expansions we obtain

Then

In explicit form the radiation term is:

Lemma 2.1 The radiation term obtained is bounded provided .

Proof: We need an estimate of H. A. Schwartz difference quotient. In view of we obtain

Lemma 2.1 is thus proved.

2.2. Preliminary Assertions, Formulation of Main Periodic Problem and Lemmas

Our main purpose is to obtain an existence-uniqueness of -periodic solution of the equations of motion (1.p).

We consider (1.p) jointly with functional equations

(2.2)

and the initial value problem

(2.3)

if .

Otherwise we take the initial interval .

We have however already proved in [15], [16] that (2.2) has a unique continuous solution for every Lipschitz continuous trajectories. That is why we can consider only (2.3).

Remark 2.1 Here instead of inequality from [15], [16] we can use . It follows immediately from .

Indeed, considering J. Kepler problem we put which implies the relation required. We prove some additional properties of .

Lemma 2.2 If are smooth - periodic trajectories with velocities satisfying , then:

1) (3.1) has a unique smooth -periodic solution.

2) The derivative satisfies inequalities .

Proof:

1. The proof can be accomplished as in [15], [16].

2. Differentiating

and solving with respect to we obtain

Using that (2.2) has a unique solution we have

Obviously and besides

Lemma 2.2 is thus proved.

The main difficulty is to define a suitable operator whose fixed points are solutions sought.

Assuming that the initial point is we introduce the function set:

(2.4)

Introduce a family of pseudo-metrics

(2.5)

It follows that the terms are uniformly bounded

Therefore and we put .

Further on we put

and

In fact

Assuming introduce the operator B as a 6-tuple

where

(2.6)

In the right-hand-sides

(p = 1, 2) we substitute the functions with retarded arguments

by the initial functions translated to the right on the interval . By necessity we assume that are such that their translated image on belong to .

We recall some assertions from [17]:

Lemma 2.3 [17] If then

Lemma 2.4 [17] If the translated function satisfies

Lemma 2.5 [17] For every it follows

Lemma 2.6 [17] The function belongs to .

Lemma 2.7 The following inequalities are fulfilled:

Proof:

Lemma 2.7 is thus proved.

Lemma 2.8 The following inequalities are fulfilled:

Proof: We use the inequality

In the first summand of we use the estimate , while in the second and third ones :

In we estimate the first summand via , while in the second one :

Therefore

For the radiation part we obtain

Lemma 2.8 is thus proved.

Lemma 2.9 (Main Lemma) The periodic problem (2.3) has a unique solution iff

the operator B has a fixed point, belonging to.

Proof: Let be a - periodic solution of (2.3). Then after integration in view of we obtain

Therefore operator becomes

We have supposed that the system has a periodic solution. Then changing the order of integration we obtain

But

Therefore the following equality is satisfied

Conversely, let B has a fixed point

that is,

Therefore or

It follows .We show that . Indeed, if for sufficiently large the inequality of Lemma 2.8 might be violated.

Therefore the operator

becomes . Differentiating the last equalities we obtain that the fixed point of the operator is a periodic solution of (2.3).

Lemma 2.9 is thus proved.

Remark 2.2 We use the equality

for further estimations.

2.3. Existence-Uniqueness of a Periodic Solution of the Two-Body System

Here we prove the main result:

Theorem 2.1 (Main result) Let the following conditions be fulfilled:

IN-1) the initial velocities are -periodic infinitely differentiable functions and initial trajectories are such that

IN-2) the translations (to the right) of on are restrictions of some functions from.

Besides the following inequalities are satisfied:

Then there exists a unique -periodic solution

Proof: With accordance of the Main lemma 2.9 we have to prove that operator defined by (2.6) possesses a unique fixed point which means that (2.3) has a unique periodic solution.

We use function spaces M and M₀ defined by (2.1) and (2.4), respectively, and family of pseudo-metrics defined by (2.5).

The set is endowed with a countable family of pseudo-metrics

whose index set is .

The lemmas from Subsection 2.2 imply that the function

is -periodic one.

Estimates from Supplement 1.1 and condition 1) of the Main theorem imply

Estimates from Supplement 1.2 and condition 2) of the Main theorem imply

In a similar way we obtain for derivatives

that is, the operator B maps the set into itself.

It remains to show that operator B is a contractive one. Indeed, in view of the inequalities obtained in Supplement 2 we have

Multiplying by and taking the supremum in t we obtain

where .

For the derivatives we obtain

and therefore

where .

Analogously we have

Taking the supremum in n we obtain

where . This means B is a contractive operator in the sense of definition given in [5]. Its unique fixed point is a periodic solution of two-body system of equations.

Theorem 2.1 is thus proved.

2.4. Numerical Test Results

Let us show that the inequalities of Theorem 2.1 are satisfied:

We recall that

Since the radius of first Bohr orbit is

and its velocity is (1/137 is Sommerfeld fine structure constant), then

In the above inequalities n could be chosen arbitrarily large because the solution belongs to the space of infinitely differentiable functions. Our estimates require to be a constant. Here and we have to take , for instance . Then

Therefore and, that is condition is satisfied.

We choose. For the inequalities we get

We notice that the initial conditions of Theorem 2.1 exclude the condition from [16] for escape trajectories.

3. Conclusions

It is easy to see that the inequalities are satisfied for every radius of “larger” orbit .

Following [18] and [19] we would like to recall some difficulties of planetary model of the atom.

Difficulty (1): It is known that the properties of an atom are determined by its electric field. The identity of their properties means that all of the hydrogen atoms possess identical electrical fields. In other words the electrons of all these atoms move in strictly identical orbits (for instance, circles or ellipses and so on). The shape of electron orbits must depend on the initial conditions of formation of the atom. It is clear that these initial conditions can be very different. Completely incomprehensible is why at various initial conditions the electron attaches the same orbit.

Difficulty (2): In the planetary model of the hydrogen atom the electron moves in a stable orbit. This contradicts electrodynamics because the radiation of the electron leads to decreasing of its radius as a result it should be collided to the nucleus. This time is [18], [19].

Our conclusion: It is not natural to expect that the applying of the methods of classical mechanics to relativistic objects will give adequate results. Namely:

1) The orbits in question [18], [19] are obtained in the frame of classical mechanics, while we consider two-body problem in the relativistic case;

2) We have found initial conditions (these are the conditions of our main Theorem 2.1) which guarantee an existence-uniqueness of a closed orbit;

3) We have considered equations with radiation terms and nevertheless there is unique periodic orbit for every stationary state without radiation. In other words the radiation is so small that it does not affect the stability of the atom. This fact is confirmed experimentally. In our model the hydrogen atom exists infinitely long.

Finally, we would like to say that basic difficulties of the planetary model are overcome provided the considerations to be relativistic ones.

Supplement 1. The Operator Maps Solution Set into Itself

1.1. Estimates of the Right-Hand Sides

We have to show that the operator

maps into itself.

In view the previous Remark 5.2 we have:

From the proof of the Main Lemma and the assertion of Lemma 2.8 we obtain

We have

Having in mind Lemma 2.7 we use the estimate in the first summand of , while in the second and third ones ̶ . Then in view of we have

In a similar way in the first summand of we use , while in the second one ̶ and get

For we have

Then

In a similar way we obtain

For the radiation term in view of

we obtain

Therefore the integral of the radiation term is bounded.

We have already obtained that .

Finally we obtain

1.2. Estimates of the Derivatives

For the derivative

We use the above estimates and for we obtain

In a similar way in the first summand of we use

, while in the second one ̶̶ and have

For we have.

Therefore

and

For the radiation term we can use the estimate from Lemma 2.1, namely

and then

Therefore

Supplement 2. Lipschitz Estimate

2.1. Lipschitz Estimates of the Right-Hand Sides

Prior to obtain Lipschitz estimates we notice

and for the derivatives we have the similar inequalities.

Therefore we can obtain Lipschitz estimates only for derivatives. They are

In order to simplify the next expressions we introduce the following denotation for H. A. Schwartz difference quotient:

Then

and in view of

we have

Then

For the radiation term we obtain

Since

we have

and

.

Therefore the Lipschitz estimate for the operator is

2.2. Lipschitz Estimates of the Derivatives

In view of we obtain the following inequality:

For the radiation term we have

and

In view of we have

ACKNOWLEDGENTS

1) The author thanks Prof. D-r D.-A. Deckert for his kind invitation to deliver a lecture devoted to the problems of the present paper at the Seminar of International Junior Research Group "Interaction between Light and Matter”, Ludwig Maximilian University, Munich, June 2015.

2) The author thanks the referee for his very useful remarks and suggestions which improved the paper.