The main purpose of the present paper is to continue the investigation of the 3-body problem of classical electrodynamics in the 3D Kepler formulation from [1,2]. The number of equations of motion for 3 bodies in the Minkowski space is 12, while the unknown functions are 9. It is proved in [2] that the last three equations are consequences from the first 9 once. These three equations (two in the Kepler formulation) give estimations of the energy of the moving particles. Our main goal here is to calculate this energy and compare it with the values from quantum mechanics [3,4]. The results obtained refer to the helium atom where both electrons circle on the first (ground) stationary state.

The paper consists of six sections and References. Section 1 is an Introduction. In Section 2, a derivation of the explicit form of the equations of motion for the three-body problem is made. Every vector equation contains in the right-hand side the Lorentz force and a radiation term. The Lorentz force is a sum of two terms each of which shows the influence of the other two particles on the first one. In Section 3, the formalism of the transition to the Euclidean coordinates is described, introducing delays depending on the unknown trajectories. In Section 4, we obtain the radiation part of the energy terms in the energy equations. In Section 5 we obtain inequalities which allow us to estimate the energy of the moving particles. Section 6 is a Conclusion. We calculate the energy of the moving electrons rounding the nuclei. We establish that with the natural parameters of electron motion, the energies coincide with the values obtained from quantum mechanics [3,4].

The equations of motion for the 3-body problem introduced in [2] are:

As usually Latin indices run from 1 to 4, while Greek – from 1 to 3.

Let us note the number of equations in (1) is 12, while the unknown functions (trajectories) are 9. It is proved in [2] that every 4-th equation is a consequence of the previous three ones. In this way one obtains that the independent equations in (1) are 9 in number.

We follow the procedure introduced in [2]. By $\left(x_1^{(k)}(t),x_2^{(k)}(t),x_3^{(k)}(t),x_4^{(k)}=ict\right),(k=1,2,3)$ we denote the space-time coordinates of the charged particles. The dot product in the Minkowski space is ${〈a,b〉}_4=a_r{b}_r={\displaystyle \sum _{r=1}^4{a}_r{b}_r}$, while in the 3-dimensional subspace ― $〈a,b〉=a_{\alpha }{b}_{\alpha }={\displaystyle \sum _{\alpha =1}^3{a}_{\alpha }{b}_{\alpha }};$ c is the vacuum speed of light; $m_k(k=1,2,3)$ are the proper masses of the particles; $e_k(k=1,2,3)$ ― their charges. The elements of proper times are $d{s}_k(k=1,2,3)$ and the unit tangent vectors to the world lines are ${\lambda }^{(k)}=\left({\lambda }_1^{(k)},{\lambda }_2^{(k)},{\lambda }_3^{(k)},{\lambda }_4^{(k)}\right)=\left({\displaystyle \frac{u_1^{(1)}(t)}{{\Delta }_k}},{\displaystyle \frac{u_2^{(1)}(t)}{{\Delta }_k}},{\displaystyle \frac{u_3^{(1)}(t)}{{\Delta }_k}},{\displaystyle \frac{ic}{{\Delta }_k}}\right)$, where ${\overrightarrow{u}}^{(k)}=\left(u_1^{(k)}(t),u_2^{(k)}(t),u_3^{(k)}(t)\right)=\left({\dot{x}}_1^{(k)}(t),{\dot{x}}_2^{(k)}(t),{\dot{x}}_3^{(k)}(t)\right)$, ${\Delta }_k=\sqrt{c^2-〈{\overrightarrow{u}}^{(k)}(t),{\overrightarrow{u}}^{(k)}(t)〉}$.

To compare denotations with originally accepted ones we recall that (cf. [5]):

The components of accelerations are ${\displaystyle \frac{d{\lambda }^{(k)}}{d{s}_k}}=\left({\displaystyle \frac{{\gamma }_k}{c}}{\displaystyle \frac{d}{dt}}{\displaystyle \frac{{\gamma }_k{\lambda }_{\alpha }^{(k)}}{c}},i{\displaystyle \frac{{\gamma }_k}{c}}{\displaystyle \frac{d{\gamma }_k}{dt}}\right)=\left({\displaystyle \frac{1}{{\Delta }_k}}{\displaystyle \frac{d}{dt}}{\displaystyle \frac{u_{\alpha }^{(k)}(t)}{{\Delta }_k}},{\displaystyle \frac{ic}{{\Delta }_k}}{\displaystyle \frac{d}{dt}}{\displaystyle \frac{1}{{\Delta }_k}}\right)$.

The isotropic vectors ${\xi }^{(kn)}(k=1,2,3;n\ne k)$ are obtained as in [2] (cf. [5,6]). Namely, fix any event on the world line of the k-th particle and draw the light cone into the past. This cone intersects the world line of the n-th particle in any other (past) event. Then

Since ${〈{\xi }^{(kn)},{\xi }^{(kn)}〉}_4=0$, the retarded functions ${\tau }_{kn}(t)$ should satisfy the following 6 functional equations

For the velocities vectors we have where ${\Delta }_{kn}=\sqrt{c^2-〈{\overrightarrow{u}}^{(n)}(t-{\tau }_{kn}),{\overrightarrow{u}}^{(n)}(t-{\tau }_{kn})〉}$; ${\displaystyle \frac{d}{d{s}_k}}={\displaystyle \frac{1}{{\Delta }_k}}{\displaystyle \frac{d}{dt}};{\displaystyle \frac{d}{d{s}_n}}={\displaystyle \frac{1}{{\Delta }_{kn}}}{\displaystyle \frac{d}{d{t}_{kn}}}={\displaystyle \frac{1}{{\Delta }_{kn}}}{\displaystyle \frac{dt}{d{t}_{kn}}}{\displaystyle \frac{d}{dt}},$${\displaystyle \frac{dt}{d{t}_{kn}}}={\displaystyle \frac{c^2{\tau }_{kn}-〈{\overrightarrow{\xi }}^{(kn)},{\overrightarrow{u}}^{(n)}〉}{c^2{\tau }_{kn}-〈{\overrightarrow{\xi }}^{(kn)},{\overrightarrow{u}}^{(k)}〉}}=D_{kn}$.

To derive the explicit form of the radiation terms (cf. [7]), we consider a charge $e_k(k=1,2,3)$ describing any curve $L_k(k=1,2,3)$ in the space-time. Let $A_k\left(x_1^{(k)}(t),x_2^{(k)}(t),x_3^{(k)}(t),ict\right)$ be any event and let

$A_k^{ret}\left(x_1^{(k)}({\stackrel{⌣}{t}}_k),x_2^{(k)}({\stackrel{⌣}{t}}_k),x_3^{(k)}({\stackrel{⌣}{t}}_k),ic{\stackrel{⌣}{t}}_k\right),{\stackrel{⌣}{t}}_k<t$ be the intersection of $L_k(k=1,2,3)$ with the null-cone drawn into the past from $A_k(k=1,2,3)$, and $A_k^{adv}\left(x_1^{(k)}({\stackrel{⌢}{t}}_k),x_2^{(k)}({\stackrel{⌢}{t}}_k),x_3^{(k)}({\stackrel{⌢}{t}}_k),ic{\stackrel{⌢}{t}}_k\right),{\stackrel{⌢}{t}}_k>t$ be the intersection of $L_k(k=1,2,3)$ with the null-cone drawn into the future from $A_k(k=1,2,3)$. Let $A_k^{ret}{A}_k$ and $A_k^{adv}{A}_k$ be the isotropic vectors

${\xi }^{(k)ret}=\left({\xi }_1^{(k)ret},{\xi }_2^{(k)ret},{\xi }_3^{(k)ret},{\xi }_4^{(k)ret}\right)$, ${\xi }^{(k)adv}=\left({\xi }_1^{(k)adv},{\xi }_2^{(k)adv},{\xi }_3^{(k)adv},{\xi }_4^{(k)adv}\right)$. Then we set

${\tau }_k^{ret}(t)=t-{\stackrel{⌣}{t}}_k\Rightarrow {\stackrel{⌣}{t}}_k=t-{\tau }_k^{ret}(t)$ and ${\tau }_k^{adv}(t)={\stackrel{⌢}{t}}_k-t\Rightarrow {\stackrel{⌢}{t}}_k=t+{\tau }_k^{adv}(t)$ and obtain

The velocity vectors to the world-line $L_k(k=1,2,3)$ at $A_k^{ret}$ and $A_k^{adv}$ are:

Following the Dirac physical assumptions we define the radiation term as a half of the difference between both retarded and advanced potentials (cf. [7]) $F_{mn}^{(k)rad}={\displaystyle \frac{1}{2}}\left[\left({\displaystyle \frac{\partial A_n^{(k)ret}}{\partial x_m^{(k)ret}}}-{\displaystyle \frac{\partial A_m^{(n)ret}}{\partial x_n^{(k)ret}}}\right)-\left({\displaystyle \frac{\partial A_n^{(k)adv}}{\partial x_m^{(k)adv}}}-{\displaystyle \frac{\partial A_m^{(n)adv}}{\partial x_n^{(k)adv}}}\right)\right]$, where $A_n^{(k)ret}=-{\displaystyle \frac{e_k{\lambda }_n^{(k)ret}}{{〈{\lambda }^{(k)ret},{\xi }^{(k)ret}〉}_4}}$ and $A_n^{(k)adv}=-{\displaystyle \frac{e_k{\lambda }_n^{(k)adv}}{{〈{\lambda }^{(k)adv},{\xi }^{(k)adv}〉}_4}}$.

Then the equations of motion become $(r=1,2,3,4)$:

In the present paper we investigate only the fourth, eight and twelfth equations $(k=1,2,3)$:

Following reasoning from [7] the relations ${〈{\xi }^{(k)ret},{\xi }^{(k)ret}〉}_4=0,{〈{\xi }^{(k)adv},{\xi }^{(k)adv}〉}_4=0$ yield

For the retarded part of radiation term we differentiate $t-{\stackrel{⌣}{t}}_k={\displaystyle \frac{1}{c}}\sqrt{{\displaystyle \sum _{\gamma =1}^3{\left[x_{\gamma }^{(k)}(t)-x_{\gamma }^{(k)}({\stackrel{⌣}{t}}_k)\right]}^2}}$ with respect to ${\stackrel{⌣}{t}}_k$ and obtain $D_k^{ret}={\displaystyle \frac{dt}{d{\stackrel{⌣}{t}}_k}}=={\displaystyle \frac{c^2{\tau }_k^{ret}-〈{\overrightarrow{\xi }}^{(k)ret},{\overrightarrow{u}}^{(k)}(t-{\tau }_k^{ret})〉}{c^2{\tau }_k^{ret}-〈{\overrightarrow{\xi }}^{(k)ret},{\overrightarrow{u}}^{(k)}(t)〉}}$ and in a similar way for advanced term we obtain $D_k^{adv}={\displaystyle \frac{dt}{d{\stackrel{⌢}{t}}_k}}={\displaystyle \frac{c^2{\tau }_k^{adv}-〈{\overrightarrow{\xi }}^{(k)adv},{\overrightarrow{u}}^{(k)}(t+{\tau }_k^{adv})〉}{c^2{\tau }_k^{adv}-〈{\overrightarrow{\xi }}^{(k)adv},{\overrightarrow{u}}^{(k)}(t)〉}}$.

Further on,