1. Introduction
It would be difficult to exaggerate the influence of Newton’s theory of gravitation on the subsequent development of physics. As well as explaining Kepler’s laws of planetary motion, Newton’s theory was central to the successful mathematization of physics using the newly-invented calculus and it served as a paradigm for the later theories of electrostatics and magnetostatics. However, new insights into Milky Way satellite galaxies raise awkward questions for cosmologists: Do we have to modify Newton’s theory of gravitation as it fails to explain so many observations? In other words, although Newton’s theory does, in fact, describe the everyday effects of gravity on Earth, things we can see and measure, it is conceivable that we have completely failed to comprehend the actual physics underlying the Newton’s force of gravity. In addition, Newton’s theory does not fully explain the precession of the perihelion of the orbits of the Planets, especially of planet Mercury. Namely, it has been experimentally stated that the perihelion of Mercury’s orbits moves into the plane of its planetary motion around the Sun. In other words, all planetary motions of Sun’s planetary system depart from elliptical orbits obtained from Newton’s gravity theory, [1]. By the strict Schwarzshild-Droste’s solution to the static gravitational field with spherical symmetry, in the general Einstein’s relativity theory, the perihelion problem has been approximately solved, [1]. On the other hand, Einstein’s theory has some difficulties hard to be overcome such as the problem of singularity, that occurs in Newton’s theory too (all relevant physical variables, such as velocity, force, kinetic and potential energy, don’t exist at point of singularity). Accordingly, in order to solve simultaneously these two acutely vexed questions within Newton’s gravity theory, we present, in this research paper, an approximative modification of Newton’s gravity concept itself. The outline of this article is as follows: In the Preliminaries, the space-time continuum (the integral space), as an ambient space, is completely defined. In Section 1 we establish a causal connection between the expression for the kinetic energy of a material point and the Minkowski metric in the four-dimensional space-time continuum. In addition, in two separate subsections of this section we derive Newton’s equations of motion and the relativistic Hamilton-Jacobi equation for a free particle. Since the dynamic (Newton’s) equations of motion are formally derived from geodesic equations in Section 2, this section together with Appendix at the end of the paper provide a possibility of further work on the modification of Newton’s gravity theory in Section 3. In this last section we show that a comprehensive analysis of particle motion under the modified Newton’s gravity force leads to the perihelion motions of a Planet’s elliptical orbit.
2. Preliminaries
By a material point
, introduced for the purpose of an useful idealization, one means a geometrical point, which is spatially no dimensional on the one hand and exactly fixed mass on the other. Closely related to the notion of a geometrical point is the set of values 
of some arbitrary 
variables 
denoting the contravariant co-ordinates of the real 
-dimensional configurative space. The geometrical point, defined by a set of zero values
, is the zero co-ordinate point. If one of 
arbitrary variables 
is the time variable
, then the space aforementioned becomes the space-time continuum (shortly called the integral space), [2]. As it was noted in [3] the value of 
is called moment or instant.
The set of all geometrical points of the spatial subspace of the integral space, to which the mass 
can be joined in some strictly monotonous sequence of permitted instants of the time
, makes an odograph usually referring to the a trajectory (motion path) of
. The time variable 
is taken for a unique independent variable, so that all remaining spatial variables 
are functional variables. Across all the future text Greek indices take values
, and Latin ones
. In the space-time continuum the aforementioned trajectory of 
blossoms into an integral curve. The vectors
and 
defined with respect to the origin are position vectors of 
in the space-time continuum and in the spatial subspace of the integral space, respectively. The concept of a vector in vector hyper-dimensional spaces 
should be conditionally comprehended in the sense of its geometrical presentation in a form of segments. Hence it bears a name linear tensor, [4]. Covariant vectors
, where 
denotes
, form a covariant vector basis 
of the integral space. The vectors
, such that at any point of the space
, where the second order system 
(Kronecker’s delta-symbol, [5]) is the identity 
matrix, form a dual basis 
of
. The differential 
of the position vector 
of 
is defined by
, where the so called Einstein’s convention is applied to a summation with respect to the repetitive indexes (uppers and lowers), herein as well as in the further text of the paper.
3. The Action Metric in the Integral Space
Since the integral space is a metric affine space, whose linearly independent basis (fundamental) co-ordinate vectors reduced to the origin form an 
-hedral basis, it follows that if 
is a line element of the metric affine space of the spatial continuum, then the expression for the kinetic energy 
of 
can be stated in more appropriate form:
(1)
considering the fact that the basic mechanical (kinematics and dynamics) parameters of 
are its velocity
, quantity of motion
and kinetic energy
. A term of
, where 
is nominally equal to the light velocity in vacuum, can be added to both sides of the previous equation, as follows
(2)
For 
let 
be such that
. Then, (2) becomes
(3)
This means that if
, where 
is a line element of the metric affine space of the space-time continuum, then the four-dimensional integral space has the Minkowski metric, [1,4]. So, in this case the Minkowski metric (3) represents the kinetic energy 
of 
in the integral space. Hence, the Minkowski metric 
is the kinetic metric of the integral space, [6].
If the Pfaff form 
is absolute differential, that means that there exists a scalar valued function 
such that
, then 
and
(4)
where
, and 
is the total mechanical energy of
.
Now, we can start with the action 
in the Lagrange sense along a motion path of 
in the integral space [4, 6],
(5)
Since 
it follows from (4) and (5) that
(6)
Let us introduce an action line element
, thoroughly explained in [6], in such a way that
(7)
Accordingly, the action metric is as follows
(8)
where
are the metric tensors of 
and
, respectively.
3.1. Newton’s Equations of Motion
By the well-known Maupertius-Lagrange’s principle [4], the path motion of 
is just the path along which the action is stationary, more precisely along which the following two mutually equivalent conditions
(9)
where 
is the variational operator, are satisfied. By (7), the previous conditions are reduced to
(10)
The second condition in (9) leads to the Euler Lagrange equations
(11)
where 
and
which yield Newton’s equations of motion
(12)
3.2. The Relativistic Hamilton-Jacobi Equation for a Free Particle
Analyze (5) again, but now let 
be a function of
that means that
. As
(13)
we introduce the functional
, nominally equal to
, such that
(14)
as well as the functional 
satisfying the condition
(15)
which together with (15) yields
(16)
since
. Hence, 
is Lagrangian of
. Further, since 
for
, see (14), it follows from (15) that 
and
(17)
The previous equation is the Hamilton-Jacobi one, so that 
is the principal Hamilton’s functional of
. Clearly, the Hamiltonian 
of 
is equal to
, more precisely to the integral of motion, considering the fact that the kinetic energy 
of 
is a homogenous square function of
. Now, by (14) and (15), we have
, so that
(18)
This together with (17) leads to the second form of the Hamilton-Jacobi equation
(19)
In addition,
(20)
which together with (8) yields
(21)
where
. These two equations are obviously analogous to the relativistic Hamilton-Jacobi equation for a free particle, see [7,8].
4. The Binet Differential Equation
As is well-known from the tensorial analysis, see [6], all curves of the integral space, for which the condition (10) is satisfied, are geodesics, and the absolute Bianchi (covariant) derivative 
of the unit tangent vector 
along geodesics is equal to zero (the vector projection of 
onto the tangent hyper-plane of the integral space is equal to zero). Thus, the geodesic equations are as follows
(22)
where 
denotes
, and
are the second kind Christoffel symbols with respect to the action metric space
. Let
. Since
, see (8), it follows that
(23)
where 
are the second kind Christoffel symbols with respect to the Euclidean metric space
. A new form of the geodesic equations (22), for a constrained material point
, is as follows
(24)
which yields
(25)
since
,
and
.
So, (25) represents the Euler-Lagrange differential equations of the extremal curve in the explicit form, and at the same time Newton’s second law of motion under the action of a potential force 
in the contravariant form:
(26)
Accordingly, one may conclude that the dynamic (Newton’s) Equations (26) of motion are formally derived from the geometric Equations (22).
In the case of the free motion of
, when
, both the kinetic and action metric form of the integral space are pseudo-euclidean, while integral curves are straight-lines (see Appendix), as it was thoroughly explained in the monograph by [6]. On the other hand, the well-known Binet differential equation for central force motion of 
(27)
is obtained by differentiating (58) (see Appendix).
5. Modified Newton’s Gravity Concept
For the conservative Newton’s gravity force 
the expression 
is as follows
, where 
is the gravitational radius, so that (27) is reduced to
(28)
where 
and
.
Since, in the limit as
, Newton’s gravitational potential 
tends to infinity, it is logical to assume that 
is the first-order MacLaurin series approximation of the exponential function
, so that 
and
(29)
Accordingly, the modified Binet differential equation for the modified central Newton’s gravity force
(30)
is as follows
(31)
5.1. A motion Under the Action of 
Start with Newton’s second law of motion
(32)
Multiply (32) on the right by the sector velocity vector 
as follows
(33)
Since 
it follows from (33) that
(34)
where the vector 
satisfying the relation 
is no longer an element of Milankovic’s constant vector elements, more precisely is no longer Laplace’s integration vector constant, see [9]. If we now multiply 
by
, we get
(35)
Since
, it follows from (35) that
(36)
where 
is an angle between 
and
. This equation describes the motion of 
under the action of the modified Newton’s gravity force
. Conditionally speaking, there is no formal difference between (36) and its analog in the ordinary Newton’s gravity theory. The key difference lies in the fact that 
is no longer constant vector.
For 
let 
and
, where 
is the unit vector orthogonal to
. Then, from (34) we get
(37)
which together with (36) yields
(38)
where 
and
.
In addition, the dot product 
leads to
(39)
Thus,
(40)
If 
is the angle between 
and
, it follows from (36) and (40) that
(41)
Hence,
(42)
Note that
, whenever
. If we now multiply (38) by 
we get
(43)
which together with (41) and (42) yields
(44)
whenever
. If
, then
(45)
Since
, see (38), it follows from (36) that
(46)
which together with (45) finally yields
(47)
This result we can also get explicitly from (46). Namely, if 
is the polar angle, then
. Therefore, it follows from (46) that
(48)
Since 
we have
(49)
that is just the same as (47). Hence,
(50)
So, as
, where 
and 
are the semimajor axis and the eccentricity of the orbit, the following angle value
(51)
is a very good approximation for the perihelion regression 
per one revolution 
of the Planets.
5.2. The Modified Perturbing Force
If, in addition to the modified Newton’s gravity force
, we include the modified perturbing force [10]
where 
and 
are the radial vector between the Planet and the perturbing planet (whose orbit is assumed to be circular and coplanar with Mercury’s orbit) and the gravitational radius for the perturbing planet, respectively, then
where 
is the angle between 
and
, and 
is the unit vector perpendicular to
. Thus, the second equation of (35) becomes
where 
and
This vector is the modified Laplace’s integration vector (or more precisely, the modified Laplace-RungeLenz vector). Their original versions come from the ordinary Newton’s gravity theory. If we denote
by
, then we have
6. Conclusion
The mathematical model of a material point motion in the three-dimensional spatial subspace of the four-dimensional space-time continuum and in the field of the action of a conservative active force 
is analogous to Newton’s mathematical model of the classical mechanics. In addition, the metric 
of the integral space, which represents the kinetic energy of a material point from the viewpoint of that space, is the Minkowski metric from Einstein’s relativity theory. Accordingly, it can be said that in the paper a new connection has been established, in contrast to an approximative one, between the classical Newton’s mathematical model and the relativistic Einstein’s mathematical model.
On the other hand the approximately modified Newton’s gravity concept is not, from any point of view, in collision with old Newton’s one. At the same time it solves the acutely vexed questions within old Newton’s gravity concept (the singularity and perihelion problems). Furthermore, analyzing the analytical expression for the modified Newton’s gravity force
, we can separate the four indicative domains of its field of the action (see Figure 1). The first one is a domain of the weak action on finitely small distances. The second one is a domain
Figure 1. Modified Newton’s gravity force.
of the strong action in a neighborhood of the gravitational radius
. The third one is a domain of the action on finitely large distances relative to the gravitational radius 
and with the relatively small velocities relative to the light velocity, and the fourth on finitely large distances relative to the gravitational radius 
and with velocities that are comparable to the light velocity. Previously separated domains of the field of the action of the modified Newton’s gravity force 
it would be desirable to compare to the fields of the action of the four so far non-unified fundamental forces (weak and strong nuclear interactions, gravity and Lorenz’s electromagnetism). Clearly, all of these facts aforementioned could be subject of further analyses. Note at the end that a correction to Newton’s gravity law in the form of the functional dependence 
irresistibly reminding of the modified Newton’s gravity force, and obviously wrongly called the fifth force, has been revealed by a reexamination of the old attraction data and careful new force measurements presented in [11].
Appendix: The Freee Motion of 
in the Integral Space
Let us start with the Euler-Lagrange equations
(52)
where
(53)
as the condition for the action (12) to be stationary. The geodesic Equations (13) are explicitly obtained from it in a known way. If spatial co-ordinates are spherical ones
, then the components of 
depend only on 
and
, so that it follows from (52) that
(54)
and
(55)
that leads to
(56)
and
(57)
Let the polar extension 
and the polar angle 
be intensities of 
and an angle between the position vector 
and the polar axis 
passing through the origin and the perihelial point, respectively. Then, since
, where 
is the so-called sector velocity vector, it follows from the condition (57) that the motion is the plane one 
and
. As 
then we obtain finally from (5), (10) and (57) that
(58)
that just leads to the Binet differential equation for free motion in plane polar co-ordinates
(59)
The solution
, where
is the perihelial distance, to this differential equation, defines a straightline in plane polar co-ordinates.