To the issue of the physical meaning of the Laplace – Runge – Lenz vector

In the paper there is presented the Laplace – Runge – Lenz vector as physical force parameter in the regard of its dimension.
Based on the expression of the LRL vector the vector equation is generated where each term has force dimension. In this case, the LRL vector is the determinant of the sum of gravitation forces and fictitious producing no work forces.
Failing the gravitation forces or other real ones, the body motion can be considered as the constant motion in the compensated vector field of producing no work forces.
Such an approach can be justified by the viewpoint of the Newton’s laws, the body motion while the forces are absent and the body motion in the compensated field of forces are equivalent and similar to each other.


Introduction
It is known that when a body moves under the influence of the central force which is inversely proportional to the square of distance the Laplace-Runge-Lenz vector A is conserved together with the angular momentum and total energy [1][2]. Vector A is in the orbit plane and its direction coincides with the line of apsidesthe line connecting the force field center and pericenter of the mechanical trajectory of the body. It is also defined as the eccentricity vector.
From this viewpoint, Laplace-Runge-Lenz vector is better considered as a geometric parameter used for the orbit shape and orientation description [2][3][4].
The Laplace-Runge-Lenz vector (LRL vector) has been known for about 300 years.
However, unlike the angular momentum and energy it is less frequently used at the motion description [1][2][3].
In this paper, the possibility to represent the Laplace -Runge -Lenz vector according to its dimension as the force parameter is considered.

The Laplace -Runge -Lenz vector dimension and its physical interpretation
For a body moving in the central field (Newtonian or Coulomb fields) the LRL vector can be noted as follows [2][3][4][5][6]: where v -is the velocity of the body motion, Lis the angular momentum, r -is the distance, α is the parameter determining the central force value.
The vector А has the same dimension as the parameter α used in the formula for the Newtonian or Coulomb forces. In case the expression (1) is divided on r 2 , we obtain a vector equation, each member of which has the dimension of force:  , then one can obtain the following equivalent components of the force F 2 (F II and F  ).
The first member of the right part equation (F II ) coincides with the centrifugal force of momentum by the amount and direction under the assumption that the centrifugal force equals mv  2 /r under rotary motion. The second member (F  ) can be considered equal to a half the Coriolis force. Consequently, the equation (3) can be used to note down in a vectorial form such forces as the centrifugal force, centripetal force and Coriolis force. The given notation may be convenient to solve some problems.
The potential of the components F II and F  is found as: From the expressions (4) follows, as has been said above that the force F 2 does not work when the body movement (component of the total work is equal to zero), and in this respect it is not real.
It turns out that the physical magnitude ) cos ( is the integral of motion.

On the body motion in the space considered as the equilibrated field of vectors
As stated above the expression of the Laplace-Runge-Lenz vector (1) is used to describe a body motion in the central force field. Still failing the force field the vector А can be in use to describe the body motion in the space. In this case, the parameter  is adopted to be equal to zero and the equation is formulated as: By analogy with (3)  In case the motion is considered according to the expression (2) when the parameter  is equal to zero, the vector field can be interpreted as the balanced force vector field: In the given case, the motion is considered as the motion in the balanced fictitious producing no work forces (F 2 ). As indicated above the mechanical trajectory is the direct circuit.
The same result is predicted (when  = 0) if the body motion is formally considered in accordance with the Newton's second law ( ).Thus the use of force F 2 to calculate the motion is quite acceptable because it does not violate the Newton's laws.

Conclusion
The representation of the LRL vector character allows to use it to study the motion not only in the force field but also in case of its failing. In that case, the Laplace-Runge-Lenz vector equally with the power and impulse can be widely used to describe the body motion in the space.
The suggested representation of the LRL vector as the parameter which determines the sum of the gravitation forces and fictitious (producing no work) forces should be regarded as conjectural and hypothetical. The employment of the fictitious (producing no work) forces in the form (7) is admitted as it does not contradict the Newton's laws. Failing the gravitation forces or any other real ones, the body motion can be regarded as the uniform rectilinear motion in the balanced force field. Their use is justified as well by the fact that from the Newton's laws standpoint the body motion in the absence of forces and the body motion in the balanced field of forces are equivalent and indistinguishable.
Finally, supporting methods describing motion from the viewpoint of forces acting there can be cited Newton himself: «For all the difficulty of philosophy seems to consist in thisfrom the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena» [7]. Even nowadays this statement seems not to be dead.