A reformulation of mechanics and electrodynamics

Abstract

Classical mechanics, as commonly taught in engineering and science, are confined to the conventional Newtonian theory. But classical mechanics has not really changed in substance since Newton formulation, describing simultaneous rotation and translation of objects with somewhat complicate drawbacks, risking interpretation of forces in non-inertial frames. In this work we introduce a new variational principle for out-of-equilibrium, rotating systems, obtaining a set of two first order differential equations that introduces a thermodynamic-mechanistic time into Newton's dynamical equation, and revealing the same formal symplectic structure shared by classical mechanics, fluid mechanics and thermodynamics. The results is a more consistent formulation of dynamics and electrodynamics, explaining natural phenomena as the outcome from a balance between energy and entropy, embedding translational with rotational motion into a single equation, showing centrifugal and Coriolis force as derivatives from the transport of angular momentum, and offering a natural method to handle variational problems, as shown with the brachistochrone problem. In consequence, a new force term appears, the topological torsion current, important for spacecraft dynamics. We describe a set of solved problems showing the potential of a competing technique, with significant interest to electrodynamics as well. We expect this new approach to have impact in a large class of scientific and technological problems.

Full text

Received:
5 February 2017
Revised:
19 June 2017
Accepted:
19 July 2017
Cite as: Mario J. Pinheiro.
A reformulation of mechanics
and electrodynamics.
Heliyon 3 (2017) e00365.
doi: 10 .1016 /j .heliyon .2017 .
e00365
A reformulation of mechanics
and electrodynamics
Mario J. Pinheiro ∗
Department of Physics, Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Av. Rovisco Pais, 1049-001
Lisboa, Portugal
* Corresponding author.
E-mail address: mpinheiro@tecnico.ulisboa.pt.
Abstract
Classical mechanics, as commonly taught in engineering and science, are confined to
the conventional Newtonian theory. But classical mechanics has not really changed
in substance since Newton formulation, describing simultaneous rotation and
translation of objects with somewhat complicate drawbacks, risking interpretation
of forces in non-inertial frames. In this work we introduce a new variational
principle for out-of-equilibrium, rotating systems, obtaining a set of two first
order differential equations that introduces a thermodynamic-mechanistic time into
Newton’s dynamical equation, and revealing the same formal symplectic structure
shared by classical mechanics, fluid mechanics and thermodynamics. The results is
a more consistent formulation of dynamics and electrodynamics, explaining natural
phenomena as the outcome from a balance between energy and entropy, embedding
translational with rotational motion into a single equation, showing centrifugal and
Coriolis force as derivatives from the transport of angular momentum, and offering
a natural method to handle variational problems, as shown with the brachistochrone
problem. In consequence, a new force term appears, the topological torsion current,
important for spacecraft dynamics. We describe a set of solved problems showing
the potential of a competing technique, with significant interest to electrodynamics
as well. We expect this new approach to have impact in a large class of scientific and
technological problems.
Keywords: Mechanics, Nonlinear physics, Plasma physics, Statistical physics
http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
1. Introduction
Classical mechanics, as commonly taught in engineering science are confined to
the conventional Newtonian theory. In this work we propose an alternative theory
that completes the actual paradigm. The new formulation was elaborated [1] using
the tools of conventional vectorial algebra and a variational principle introduced by
Landau &Lifshitz [2], but not fully explored by these authors, since they have not
included the fundamental equation of thermodynamics 𝑑𝑈 =𝑇𝑑𝑆 −𝐅 ⋅𝑑𝐫that
would link energy (motion) with entropy (dissipation). This new formulation leads
to a set of two first-order differential equations that govern all physical processes. At
the time of Sir Isaac Newton, the development of thermodynamics was still faraway
to express the second law of thermodynamics, and naturally, Newton could not
embed his formulation of natural phenomena within the universal balance between
energy and entropy [3, 4]. The second law of thermodynamics has contributed
greatly to the development of theoretical chemistry and physics, including chemical
kinetics and transition states, Boltzmann kinetic theory, Max Planck pioneering
quantum theory, and Einstein theory of stimulated and spontaneous emission, to
mention a few. However, thermodynamics does not give time evolution equations
for thermodynamic states as Hamilton’s equations do. This situation has been
unsatisfactory and Ilya Prigogine in his Nobel Lecture emphasized that “[...] that
150 years after its formulation the second law of thermodynamics still appears to
be more a program than a well-defined theory in the usual sense, as nothing precise
(except the sign) is said about the S production”.
We propose a new variational procedure initially sketched but not fully explored
by Landau/Lifshitz in their Course on Theoretical Physics. It leads to a modified
dynamical equation which integrates the second law of Newtonian mechanics with
the fundamental equation of thermodynamics. In this new framework, external forces
are imposed to a physical system but it reacts back to the external constraint with
rotational, vortical motion, dissipative motion. Engineering problems can be solved
more straightforwardly, including the paradigmatic brachistochrone problem. In
addition, it reveals the existence of a new force term, the topological torsion current,
enlightening the puzzle of the flyby anomaly. The extension to electrodynamics
shows to be useful to spintronics and a large class of scientific and technological
issues.
It is clear by the selected problems solved that this theory introduces an innovative
and consistent formulation to comprehend the dynamics of physical systems, at the
same time offering a unifying approach. We have shown before that the gravitational
field has the nature of an entropic force [5] and obtained the basic equations of
electrodynamics with a new plasma balance equation [6], predicting the direct
conversion of angular to linear motion by the agency of a topological torsion
2http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
current [1]. Here, we will dwell on its applications in the framework of classical
mechanics, but further development of the theory could lead ultimately to an
explanation of quantum mechanical phenomena, and a fundamental contribution was
advanced by R. Jackiw et al. [7]. The mathematical description of physical systems
are defined by the following system of equations:
• Canonical momentum :
𝑇𝜕𝑆
𝜕𝐩=−𝐩
𝑚+𝑞
𝑚𝐀+𝐯𝑒+[𝜔
𝜔
𝜔×𝐫].(1)
• Fundamental equation of motion :
𝑇𝜕̄
𝑆
𝜕𝐫=∇
∇
∇𝑈+𝑚𝜕𝐯
𝜕𝑡 .(2)
Eq. (1) gives the quantity of motion, a measure of the vector velocity (with geometric
nature) and the quantity of matter (with energetic content). In the case of a charged
particle, it can be shown [8] that the momentum is given by
𝐩=𝑚𝐯𝑒+𝑞𝐀−𝑚𝑇 𝜕𝑆
𝜕𝐩.(3)
However, Eq. (3) represents a radically different kind of momentum when compared
to the Newtonian concept of force and momentum, because it predicts mutual
interaction between interacting systems, self-regulating exchange of energy (see
also, Ref. [8]).
Eq. (2) gives the fundamental equation of dynamics and has the form of a general
local balance equation having as source term the spatial gradient of entropy,
∇𝑎𝑆>0. At thermodynamic equilibrium the total entropy of the body has a
maximum value and the equality holds, a feature in general absent in the usual
dynamical determinism of Newtonian formulation (although Newton recognizes
friction as a source of asymmetry). The mentioned self-regulating mechanism of
angular and linear momentum is represented by the last term of Eq. (2) and extends
the validity of Newton’s fundamental equation of motion to accelerated (non-inertial)
frames of reference (see Section 2.4). In Eqs. (1)–(2) it appears one of the most
important thermodynamic quantity, the temperature 𝑇. However, with the exception
of Section 2.2, in this work we don’t address the extraordinary diversified phenomena
of the conversion of heat into work (and vice-versa).
The set of Eqs. (1)–(2) open a major change in the so-far classical paradigm of
dynamics, and as well electrodynamics (see Ref. [6]), since new physics may be
brought by the set of two first order differential equations, related to the interplay
3http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
Figure 1. System process chain – a physical system has, at least, two ways to process the external action (it
is not enslaved to it), by means of two processes chain: energy tends to a minimum by means of conversion
of linear or transversal motion to rotational motion; Entropy tends to a maximum by means of decreasing
free energy (e.g., ejection of mass, acoustic or electromagnetic radiation).
between the tendency of energy to attain a minimum, whilst entropy seeks to
maximize its value. According to the proposed variational method, Eq. (2) leads
to the fundamental equation of (classical) dynamics, now presented under the form:
𝑚𝑑𝐯
𝑑𝑡 =𝐅𝑒𝑥𝑡 −∇
∇
∇𝐽2
2𝐼−𝜔
𝜔
𝜔⋅𝐉−Δ𝐹.(4)
It is a consequence of requiring that the energy and entropy attain an optimal balance.
This new principle is, therefore, different from Hamilton’s principle since this last
one is equivalent to 𝐹=𝑚𝑎.
The external force term can be a deterministic or fluctuating force, but cannot be
entropy-dependent. A central role is played by the gradient of rotational energy
(and free energy) which drive the system from one state to another. The system
is not enslaved by the external force 𝐅𝑒𝑥𝑡, which we may envisage as an input to
the system, but can respond to changes by converting linear or angular motion into
angular motion, which we may consider the system output, affecting dynamically
the system, as illustrated in Figure 1. Moreover, according to Eq. (4), motion will
not necessarily follow in a straight line and in the direction of the external force (see
Section 2.9). Angular momentum acts as a damper to dissipate a disturbance, a well-
known redressing mechanism in biomechanics and robotics. The free energy term is
essential to study the thermomechanical behavior of any continuous media.
In addition, considering the traditional hierarchy of agencies responsible for the
motion of matter, on a logic and axiomatic point of view we may establish an
operational relationship between linear and angular motion, with possible relevance
for the understanding of the trajectory of Near-Earth Objects [9, 10].
Formulating the problem in this novel way, the temporal evolution of a system is
wholly contained without the need to add the torque equation, separately, and in
particular, giving immediately the theorem work-energy, 𝐸𝑚𝑒𝑐 =𝐾+𝑈+𝐼𝜔2∕2.
4http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
Variational mass systems can be handled by means of the gradient of free energy.
From the selected number of classical problems solved in Section 2, it is clear that
Eq. (4) gives a more complete and simple description of dynamical processes in
nature. This variational framework, as also shown in a previous work, apparently
link the gravitational force to entropic arguments [5].
From discrete particles to the analysis of the kinematics and the mechanical
behavior of materials modeled as a continuous mass, first formulated by the
French mathematician Augustin-Louis Cauchy in the 19th century, an intensive
research has been conducted, and we must cite the work of Clifford Truesdell
on modern rational mechanics [11], or all subsequent work particularly focused
on the development of coupled atomistic/continuum material modeling from the
nanoscale to the macroscale, surface and semi-continuum theories [12, 13, 14, 15,
16, 17, 18, 19], or help to design new technology for space advanced programs [20].
These departures from the standard theories use integral-type formulations with
weighted spatial averaging or by implicit or explicit gradient models, enriching
the theories of elasticity, elastoplasticity and damage mechanics. Our variational
method [5] provides a starting place for the development of new methods that can
be incorporated in theories of continuum mechanics.
2. Results and discussion
The investigation as to whether and how far this new formulation represents usable
technique is exemplified now with practical application problems.
2.1. The rolling body in an inclined plane
One standard example of classical mechanics is this one: a rigid body of mass 𝑀
rolling down an inclined plane making an angle 𝜃with the horizontal (see, e.g., p. 97
of Ref. [21]). Eq. (4) can be applied to solve the problem, with 𝜔 =0(there is no
rotation of the frame of reference) and considering that only the gravitational force
acts on the rolling body, with inertial moment relative to its own center of mass given
by 𝐼𝑐=𝛽𝑀𝑅2. Hence, we obtain:
𝑀̈𝑥=𝑀𝑔sin 𝜃−𝜕𝑥𝑤. (5)
Here, 𝑤 ≡(𝐉𝐜)2
2𝐼𝑐
. Assuming that the x-axis is directed along the inclined plane, and
considering that the angular momentum relative to the rigid body center of mass is
given by 𝐽𝑐=𝐼𝑐𝜔′, with 𝜔′=𝑑𝜃∕𝑑𝑡, and noticing that 𝑑𝑥 =𝑣𝑥𝑑𝑡 (holonomic
constraint), it is readily obtained
𝑀̈𝑥=𝑀𝑎𝑥=𝑀𝑔 sin 𝜃−𝐼𝑐𝜔′𝑑𝜔′
𝑣𝑥𝑑𝑡 =𝑀𝑔sin 𝜃−𝛽𝑀𝑅2𝜔′
𝜔′𝑅𝛼. (6)
5http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
Figure 2. The spherometer.
Since 𝛼=𝑎∕𝑅, then it results the well-known equation
𝑎𝑥=𝑔sin 𝜃
(1 + 𝛽).(7)
The usual approach is overridden by the new framework, since the linear and angular
momenta are embedded in a unified equation, Eq. (4).
The usual approach in textbooks begins by solving the 2nd Law of dynamics for
the center of mass (assuming a point particle) and then, in a separate way, solve the
torque equation for an extended particle, with the intermediate holonomic condition
for rolling. It looks like a collection of theories, inconsistent under the logical point
of view.
2.2. Oscillations of a sphere rolling on concave surface
When a sphere is displaced from its position of equilibrium on the surface of a
concave spherical surface, and if be then set free, it will roll backwards and forwards,
provided the coefficient of friction between the sphere and the surface is sufficiently
great to ensure the sphere is not sliding. This is the problem of a sphere rolling on
concave surface and is a current process used to measure the curvature of a mirror
or lens in astronomy (spherometer).
Let 𝑂be the center of curvature of the mirror, 𝑄the center of the rolling sphere, 𝐴
the point on the mirror vertically below 𝑂, and let 𝑃be the point of contact of the
sphere with the mirror. The angle 𝑃𝑂𝐴is denoted 𝜃, 𝑅is the radius of the mirror, 𝑟
is the radius of the sphere (Figure 2).
We have 𝐽
𝐽
𝐽𝑄=𝐼𝑄Ω
Ω
Ωwith 𝐼𝑄=2
5𝑀𝑟2. The rotational energy is =1
2𝐼𝑄Ω2−
𝐼𝑄𝜔Ω. The gravitational energy is considered an external force. The acceleration of
the CM point 𝑄is given by
𝐚𝑄=−(𝑅−𝑟)̇
𝜃2𝐮𝑟+(𝑅−𝑟)̈
𝜃𝐮𝜃,(8)
and the gradient of the energy in cylindrical coordinates is
6http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
∇
∇
∇=1
𝑅−𝑟
𝜕
𝜕𝜃 1
2𝐼𝑄𝜔(𝜔−2𝜔)−∇𝐹𝐮𝜃.(9)
We haven’t took into account the radial coordinate so far because it is supposed not
to change. It is important to notice that 𝜔and Ωhave different sign since they rotate
in different direction, supposing that the sphere rolls, not slides, and the constraint
equation is
−Ω = 𝑅−𝑟
𝑟𝜔. (10)
Hence,
∇
∇
∇=𝑀𝑟2
5(𝑅−𝑟)2Ω
𝜔
̇
Ω−2Ω
𝜔̇𝜔 −2̇
Ω𝐮𝜃(11)
since −̇
Ω= 𝑅−𝑟
𝑟̇𝜔. Thus we obtain
∇
∇
∇=2𝑀
5(𝑅−𝑟)̈
𝜃𝐮𝜃.(12)
Finally, the equations of motion, when observed in an inertial frame (the mirror) are
the following
𝑀(𝑅−𝑟)̇
𝜃2+𝑀𝑔 cos 𝜃−𝑁=− 𝜕
𝜕𝑅(Δ𝐹)(13)
̈
𝜃+5𝑔
7(𝑅−𝑟)sin 𝜃=0.(14)
Eq. (13) takes into account external forces, such as the weight and the normal
reaction 𝑁. Eq. (14) gives the well-known period of oscillation, on the approximation
of small angles
𝑇=2𝜋7(𝑅−𝑟)
5𝑔.(15)
Highly finished hard steel spheres are more suitable for the experiment and more
accurate results will be obtained with large spheres than with small. In general, the
proposed solution of the present problem is given resorting to Lagrangian or energy
considerations, but we give here a new approach. In addition, notice that Eq. (13)
points to the possible release (or absorption) of heat during the oscillation at the
expense of the gravitational field (or dissipative forces):
𝜕Δ𝐹
𝜕𝑅 ≈12
7𝑀𝑔 (16)
The classical solution of this problem using Newtonian Mechanics includes a friction
force, and again solving separately the extended body by means of the torque
equation. In practice, we do need at the beginning a friction force for the ball start
rolling, but once the rolling condition is met the ball experiences no friction. The
introduction of the friction force can be misleading and within the new framework
it is not taken into account. In textbooks the Lagrangian formulations is frequently
used to solve this problem and also certainly has practical advantages.
7http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
Figure 3. Bead on the hoop.
2.3. Bead on the hoop
Just to illustrate with another example the proposed method, let us consider a bead
of mass 𝑚, moving without friction on a circular hoop of radius 𝑎. The hoop rotates
about its vertical axis 𝑜𝑧 with constant angular velocity 𝜔
𝜔
𝜔(see Figure 3). The
position of mass 𝑚is given by the pair of angles 𝜃and 𝜙in spherical coordinates,
and its moment of inertia relative to axis Oz is 𝐼𝑧=𝑚(𝑟 sin 𝜃)2. Considering the
𝜃-component of the velocity in the rotating frame (and avoiding the normal reaction
force 𝐍), we have
𝑑𝑣𝜃
𝑑𝑡 =𝑎̈
𝜃=−𝑔sin 𝜃−1
𝑟
𝜕
𝜕𝜃 1
2𝑎2sin2𝜃̇
𝜃2−𝑎2sin2𝜃̇
𝜃2,(17)
and, finally, we obtain the well-known result, but from a new perspective:
̈
𝜃=sin𝜃̇
𝜃2−𝑔
𝑎.(18)
This is the mechanical analogue of an ion inside a Paul Trap with importance
for quantum computation [22]. To solve this problem, the usual procedure uses
the Lagrangian mechanics which considerably simplifies the physical problem.
However, using Newtonian mechanics one would have a complicated set of equations
of motion in the rotating frame, such as
𝑚𝐠+𝐍=𝑚𝑑2𝐫
𝑑𝑡2𝑆′
+2𝜔
𝜔
𝜔×𝑑𝐫
𝑑𝑡 𝑆′+𝜔
𝜔
𝜔×(𝜔
𝜔
𝜔×𝐫),(19)
where 𝐫=𝑎𝐞𝑟is the position vector relative to the center of the hoop, 𝑆′is
the rotating frame of the wire and 𝜔
𝜔
𝜔is the angular velocity vector. After rather
cumbersome calculations, the following result is derived:
𝑚𝐠+𝐍=𝑚[−𝑎̇
𝜃2𝐞𝑟+𝑎̈
𝜃𝐞𝜃+2𝑎̇
𝜃𝜔
𝜔
𝜔×𝐞𝜃+𝑎(𝜔
𝜔
𝜔⋅𝐞𝑟)𝜔
𝜔
𝜔−𝑎(𝜔
𝜔
𝜔⋅𝜔
𝜔
𝜔)𝐞𝑟].(20)
Then, since 𝐍 ⋅𝐞𝜃=0, we obtain
𝑚𝐠⋅𝐞𝜃=𝑚[𝑎̈
𝜃+𝑎(𝜔
𝜔
𝜔⋅𝐞𝑟)(𝜔
𝜔
𝜔⋅𝐞𝜃)],(21)
and then, after a few more steps, the final result, Eq. (18), is obtained.
8http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
2.4. The origin of the so called “pseudo-forces”
Within this new formulation, the transformation from an inertial coordinate system
to a rotating system shows that the so called “fictitious forces”, although certainly
not related to interacting fields, they are the outcome of the transport of angular
momentum, they are real forces in the rotating medium. We should have in mind
that there is active transformations (implying its motion, strongly or weakly coupled
to other systems) and passive transformations (method of describing transitions
to different reference systems, e.g., Galileo or Lorentz transformations) [23]. Any
mathematical procedure can be reliable unless it is completely understood the
purpose and how it should be applied, otherwise becomes a source of confusion.
In order to justify what we have state above, one therefore has to work out the
resultant term ∇𝑟(𝜔
𝜔
𝜔⋅𝐉)in Eq. (4), developing it in the following manner:
1
2∇𝑟
∇𝑟
∇𝑟(𝜔
𝜔
𝜔⋅𝐉)= 1
2[𝜔
𝜔
𝜔×∇
∇
∇×𝐉]+ 1
2(𝜔
𝜔
𝜔⋅∇
∇
∇)𝐉+1
2[𝐉×∇
∇
∇×𝜔
𝜔
𝜔]+ 1
2(𝐉⋅∇
∇
∇)𝜔
𝜔
𝜔. (22)
We may test the above Eq. (22) with the well-known problem of rotating coordinate
frame, without translation, considering a rotating system (𝑥′, 𝑦′, 𝑧′)whose origin
coincides with the origin of an inertial system (𝑥, 𝑦, 𝑧). We suppose in addition, that
the 𝑧and 𝑧′axes always coincide and, therefore, the angular velocity of the rotating
system, 𝜔
𝜔
𝜔, lies along the 𝑧axis. We take 𝐉 =𝐼Ω
Ω
Ωand Ω
Ω
Ω=𝜔
𝜔
𝜔, since we consider
the “body” in rotational motion around the 𝑧axis. An easy calculation (considering
𝐼𝑧=𝑚𝑟2) show that
∇
∇
∇×𝐉=2𝑚𝐯𝑟𝑜𝑡,(23)
where 𝐯𝑟𝑜𝑡 denotes the particle velocity in the rotating frame. Also, following the
analogy
∇
∇
∇×𝐯=Ω
Ω
Ω=2𝜔
𝜔
𝜔, (24)
and using a mathematical identity, we obtain
∇
∇
∇×𝐉
𝑟2=1
𝑟2∇
∇
∇×𝐉+∇
∇
∇1
𝑟2×𝐉,(25)
which gives
∇
∇
∇×𝐉
𝑟2=2𝑚
𝑟2(𝐯𝑟𝑜𝑡)+[𝜔
𝜔
𝜔×𝐫].(26)
Then
1
2𝐉×∇
∇
∇×𝐉
𝑟2=𝑚𝜔
𝜔
𝜔×[𝐯𝑟𝑜𝑡 +[𝜔
𝜔
𝜔×𝐫]],(27)
and, therefore, adding together all the terms, it is finally obtained the well-known
formula of Newtonian dynamics:
𝑚𝑑𝐯
𝑑𝑡 𝑟𝑜𝑡 =𝐅𝑒𝑥𝑡 −2𝑚[𝜔
𝜔
𝜔×𝐯𝑟𝑜𝑡]−𝑚(𝜔
𝜔
𝜔×[𝜔
𝜔
𝜔×𝐫]).(28)
9http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
We stress that in Eq. (28) the force term is calculated in the non-inertial frame
inserting the acceleration force inside the symbol (...)𝑟𝑜𝑡. The term (𝑑𝜔
𝜔
𝜔∕𝑑𝑡 ×𝐫) does
not appear because it was assumed 𝑑𝜔∕𝑑𝑡 =0. We may notice that the well-known
force terms have not a cinematic origin, but are intrinsically dynamical, resulting
from the balance between the minimization of energy and maximization of entropy.
The gradient term of the rotational energy acts like a pullback process to obtain
cinematic from dynamics. Within this formulation, the fundamental equation of
dynamics, Eq. (4), is valid in non-inertial frames of reference as well. We may also
remark that on the rhs of Eq. (22), the second and fourth terms points to the possible
existence of an axial particle motion when the “body” has a characteristic of a viscous
fluid, generating phenomena called Taylor structures.
2.5. Newton’s bucket experiment
The Newton’s bucket experiment, described in his Opus Magnum “The Principia”
in 1689, is one of the most important experiments in the history of physics having a
lasting impact in the thinking of physicists. The experiment can be done with a bucket
filled with water, hanging it by a rope, twisting the rope tightly and let it unwind
freely. As we may notice if we do ourselves the experiment, at first the bucket starts
to spin but the surface of the water (or any other regular fluid) remains practically
flat. But as the bucket increases its rotating speed, the water starts gaining momentum
due to friction with the walls and at the end, water and bucket rotates at the same
speed, and the water’s surface acquires a concave shape.
But here we point to another interpretation that we hope to clarify with the
application of our fundamental equation of dynamics written in vectorial form:
𝑚𝑑𝐯
𝑑𝑡 =𝐅𝑒𝑥𝑡 −∇
∇
∇1
2𝐼Ω2−𝐼(𝜔
𝜔
𝜔⋅Ω
Ω
Ω) − Δ𝐹−∇
∇
∇𝑝. (29)
The gradient term shows that differential rotation may be a source of thermodynamic
free energy in a flow. It is the source of transport of angular momentum outward
through the bucket, transferring matter to regions of higher pressure, and this is the
signature of the second law of thermodynamics.
To simplify, we assume that there is a “body” inside a “container” that rotates
with the container at the same angular velocity, like a solid body – the bucket’s
experiment. Hence, 𝐽
𝐽
𝐽=𝐼Ω
Ω
Ωwith Ω =𝜔. To simplify, we doesn’t include the
buoyant force. At the end, when the rotation attained a uniform angular velocity, each
element of fluid has no acceleration, moves at a given constant radial distance from
the polar axis. It behaves like a stone tied to the end of the rope in a circular motion at
constant speed. We invite the reader to perform the experiment, as Newton described
in Scholium to Book I of the Principia. An observer at rest in the rotating coordinate
frame (not withstanding the physiological impressions) will have no acceleration
10 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
although from the point of view of an inertial observer at rest, let’s say, in the Earth’s
coordinate frame, there is a circular motion. But since the “body” has no tangential
acceleration, nor moves radially, nor is in anyway distorted in its form but keeps
simply describing its circular motion, we must state that, dynamically, in the rotating
frame, 𝐚 =0. Then, we obtain from Eq. (29), and inside the rotating coordinate frame
0=−𝜌𝑔𝐮𝑧−∇
∇
∇−1
2𝐼𝜔2−𝜕𝑝
𝜕𝑟 𝐮𝑟−𝜕𝑝
𝜕𝑧𝐮𝑧.(30)
The gradient of the angular momentum represents the transport of angular
momentum across swirling matter, with a crucial role in the formation of planetary
and proto-stellar accretion disks. It gives
∇
∇
∇1
2𝐼𝜔2=1
2𝜔2∇
∇
∇(𝜌𝑟2)=𝜌𝜔2𝑟𝐮𝑟.(31)
We put 𝐼=𝜌𝑟2as the moment of inertia of the “body” swirling around the axis.
Then, considering
𝜕𝑝
𝜕𝜃 =0, we have
0=−𝜌𝑔𝐮𝑧+𝜌𝜔2𝑟𝐮𝑟−𝜕𝑝
𝜕𝑟 𝐮𝑟−𝜕𝑝
𝜕𝑧𝐮𝑧.(32)
This gives for each component:
𝜕𝑝
𝜕𝑟 =𝜌𝜔2𝑟,
𝜕𝑝
𝜕𝑧 =−𝜌𝑔,
𝜕𝑝
𝜕𝜃 =0. Integrating these
three equations, we obtain the well-known solution 𝑝(𝑟, 𝜃, 𝑧) =1
2𝜌𝜔2𝑟 −𝜌𝑔𝑧 +Const.
However, the interpretation we may give to this experiment is now diverse from
the previously given by Newton, Berkeley, Leibniz (see, e.g., Ref. [24] for a clear
presentation of traditional theories). It doesn’t matter if there is rotation relative to
absolute space (Newton) or the background of galaxies (Mach), or if the motion is
relational (Leibniz, Torres Assis [24]). What does matter is the transport of angular
momentum (imposing a balance between the centrifugal force, pushing the element
of fluid to outside) counterbalanced by the fluid pressure. As it is clear from Eq. (31),
the centrifugal forces are the outcome of the gradient of rotational energy. Within the
scope of the present formulation, in the rotating frame it is not necessary to introduce
new, fictitious, forces to account for this curved motion.
As pointed out by Torres Assis [24], even in the instance that all matter around
the swirling body is annihilated, nothing else would change in the mathematical
prediction.
2.6. Electrodynamical equilibrium and the dynamic effect of the
topological torsion current: rotating plasma and flyby anomaly
Extremum conditions imposed on entropy or internal energy, does not only provide
criteria for the evolution of the system. They also determine a novel condition for
the stability of thermodynamic systems. With the new formulation, we have shown
its capability to work out the usual sources and fields of the electromagnetic theory
11 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
in a previous work [6], but we have obtained a general equation of dynamics for
electromagnetic-gravitational systems with a new term that we called the topological
torsion current (TTC), see Ref. [1]:
𝜌𝑑𝐯
𝑑𝑡 =𝜌𝐄+[𝐉×𝐁]−∇
∇
∇𝜙𝑔−∇
∇
∇𝑝+𝜌[𝐀×𝜔
𝜔
𝜔].(33)
The symbols have the usual meaning. The last term in Eq. (33) is the topological
torsion current and to our best knowledge it is a sofar unforseen force, producing
work by means of a direct conversion of angular motion into linear motion. As a
consequence, it appears an electric field induced by motion
𝐄𝑚=−∇
∇
∇𝜙−𝜕𝑡𝐀−[𝜔
𝜔
𝜔×𝐀],(34)
where 𝜙represents an electric potential, 𝐀the vector potential, and 𝜔
𝜔
𝜔is the angular
velocity, playing the role of a spin connection, a signature of the torsion field.
For a charged particle 𝑞, its Liénard–Wiechert potentials are given by 𝜙(𝐫, 𝑡) =
𝑞
4𝜋𝜖0
1
(𝑟−𝐫⋅
⋅
⋅𝐯∕𝑐)where 𝐫is the vector from the retarded position to the field point 𝐫and
𝐯is the velocity of the charge at retarded time, and 𝐀(𝐫, 𝑡) =𝐯
𝑐2𝜙(𝐫, 𝑡). Notice that
we may write [𝜔
𝜔
𝜔×𝐀] =̃𝜔
̃𝜔
̃𝜔𝜙. Inserting into Poisson’s equation, which is a second-
order elliptical differential equation describing the electrostatic potential caused by
a charge distribution, ∇
∇
∇⋅
⋅
⋅𝐄 =𝜌
𝜖0
, it gives back a new form of the Poisson’s equation
which feature the possibility of resonant phenomena (e.g., in dusty-plasma medium):
∇2𝜙+(∇
∇
∇𝜙⋅
⋅
⋅̃𝜔
̃𝜔
̃𝜔)+(∇
∇
∇⋅
⋅
⋅̃𝜔
̃𝜔
̃𝜔)𝜙=−𝜌
𝜖0
.(35)
Eq. (35) is structurally intermediate between the Poisson and Helmholtz equations,
having an additional damping term that can be reduced to a one-dimensional
differential equation of the forced, damped linear harmonic oscillator. Poisson’s
equation has extensive applications to engineering problems dealing, e.g., with
steady state heat conduction in heat generating media, theory of torsion of prismatic
elastic bodies, quantum chemistry, gravitational phenomena, and electrostatics.
Evans [25] obtained the same result using a different approach.
Assume to simplify, only radial dependency, such as, ̃𝜔 =̃𝜔(𝑟). Then
𝜕2
𝑟𝑟Φ+2
𝑟+̃𝜔𝑟𝜕𝑟Φ+1
𝑟2𝜕𝑟(𝑟2̃𝜔𝑟)Φ=−
𝑍𝑒
𝜖0
(36)
The analytical solution of Eq. (36) is represented in Figure 4 for the special case
̃𝜔 =1.
It is similar to the potential interatomic potential or the effective potential for the one-
dimensional radial Schrödinger equation for a system with total angular momentum
𝑙, the result of an optimal arrangement between repulsive and attractive forces.
12 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
Figure 4. Set of potential curves similar to the interatomic potential or hydrogen atom potential vs.
distance, in arbitrary units.
The relevance of Eq. (33) in phenomena as the plasma rotation mechanism in
thermonuclear fusion reactors [1] and the possible role of TTC on the flyby anomaly
of a spacecraft [9] was elucidated.
2.7. The spin-orbit planetary coupling
The rotational and orbital period of a planet or a satellite is known as spin-orbit
coupling. From Eq. (4) another effect will result from the gradient of the energy,
representing the transport of angular momentum. This energy is given by (now
omitting the free energy term)
𝐸=1
2𝐼Ω2−𝐼(𝜔
𝜔
𝜔⋅
⋅
⋅Ω
Ω
Ω).(37)
Seeking for the radial minimum of transport of angular momentum
𝑑𝐸
𝑑𝑟 =𝐼(Ω − 𝜔)𝜕Ω
𝜕𝑟 −Ω𝜕𝜔
𝜕𝑟 .(38)
It can be shown (see, e.g., Ref. [26]) that
𝜕𝜔
𝜕𝑟 ≪𝜕Ω
𝜕𝑟 . Therefore, from the equilibrium
condition, it follows 𝜔 =Ω. This is the well-known 1 ∶1resonance verified
by most of the outer planet satellites, with the puzzling exception of Mercury’s
3 ∶2spin-orbit resonance [27]. One possible cause of the mysterious Mercury’s
commensurability is the intense solar radiation actuating by means of the Δ𝐹
term [28, 29].
2.8. Period of oscillation of the simple pendulum
The calculation of the natural frequency of the simple pendulum of mass 𝑚, fixed
length 𝑅, for small amplitude oscillations is done by solving the equation:
𝑚(𝑅̈
𝜃𝐮𝜃−𝑅̇
𝜃2𝐮𝑟)=𝑚𝑔 cos 𝜃𝐮𝑟−𝑚𝑔 sin 𝜃𝐮𝜃−𝑇𝐮𝑟−𝜕
𝜕𝑟 1
2𝑚𝑅2̇
𝜃2𝐮𝑟
−1
𝑟
𝜕
𝜕𝜃 1
2𝑚𝑅2̇
𝜃2𝐮𝜃,(39)
13 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
where we have put 𝐼0=𝑚𝑅2and 𝐽=𝐼0̇
𝜃and denoting by 𝑇the tension of the rope
of length 𝑅. It follows immediately the well known equations:
̈
𝜃+𝑔
𝑅sin 𝜃=0 (40)
and
𝑇(𝜃)=𝑚𝑔 cos 𝜃+𝑚𝑅 ̇
𝜃2.(41)
The exact solution of the Eq. (39) is obtained by the usual method, multiplying by
2̇
𝜃𝑑𝑡 and integrating, obtaining
̇
𝜃2=2𝜔2cos 𝜃+𝐶(42)
where 𝐶is the integration constant. At the maximum angular displacement 𝜃𝑚,
̇
𝜃=
0, and then 𝐶=−2𝜔2cos 𝜃𝑚. Hence
̇
𝜃=𝜔2(cos 𝜃− cos 𝜃𝑚)=2𝜔sin2(𝜃𝑚
2)−sin
2(𝜃
2),(43)
where we have made use of the trigonometric identity cos 𝜃=1 −2 sin2(𝜃∕2).
Separating the variables and integrating between 𝑡 =0and 𝑡, we obtain
𝜃(𝑡)=2arcsin(𝑘JacobiSN(𝜔, 𝑡, 𝑘)) (44)
with 𝑘 = sin(𝜃𝑚∕2). The pendulum period follows:
𝑇=4
𝜔𝐾(𝑘),(45)
with
𝐾(𝑘)=
𝜋∕2
∫
0
𝑑𝜃
1−𝑘2sin2𝜃
,(46)
where JacobiSN(𝑢, 𝑘)is the Jacobian elliptic sine function, and 𝐾(𝑘)is the complete
integral of the first kind.
2.9. Additional remarks on the variational problem according to
the proposed reformulation
In 1686, Newton solved the problem of determining the shape of a rotationally
symmetric body of least resistance, but the beginning of the calculus of variations is
set in the year 1696 with the formulation of the brachistochrone problem by Johann
Bernoulli in the Acta Eruditorum. Without recurring to minimum-time assumptions,
or Euler–Lagrange equations, this reformulation has the advantage to yield an answer
to the problem of the brachistochrone for uniform (e.g., gravitational) field and be
adequate to handle variational problems.
14 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
We now pass on to a description of the brachistochrone. Let us consider a Cartesian
reference frame and a particle with mass 𝑚accelerated by gravity from one point
to another with no friction and solve the problem using Eq. (4). First, notice that
the Cartesian frame is not rotating, hence, 𝜔 =0. Noting that Ωis independent from
coordinates (𝑥, 𝑦), we may assume 𝑥(0) =−𝑅,
𝑑𝑥
𝑑𝑡 =𝑅Ω𝑡and
𝑑𝑦
𝑑𝑡 (0) =0, and 𝑦(0) =
𝑅. It gives
𝑑2𝑦
𝑑𝑡2=−𝑔−𝑦Ω2(47)
𝑑2𝑥
𝑑𝑡2=−𝑥Ω2(48)
since, ∇𝑦(1
2𝑚𝑟2Ω2) =𝑚𝑦Ω2(with similar result for 𝑥component), and plugging the
identity 𝑑𝑟 =𝑑𝑥2+𝑑𝑦2. The solutions of the above system of equations are the
well-known equations of a cycloid:
𝑥(𝑡)=𝑅(Ω𝑡−sinΩ𝑡)(49)
𝑦(𝑡)=− 𝑔
Ω2(1 − cos Ω𝑡)+𝑅cos Ω𝑡. (50)
The cycloid equations were straightforwardly obtained because the fundamental
equation of dynamics, Eq. (4), retains consistently the two extremal principles of
nature, the tendency to minimize energy and maximize entropy. There is no need to
make further hypothesis.
The standard solution of one of the earliest problems in the calculus of variations
starts with the calculation of the time to travel from a point 𝑃1to another point 𝑃2:
𝑡12 =
𝑃2
∫
𝑃1
𝑑𝑠
𝑣,(51)
obtaining the speed at any point by using the equation of conservation of energy, 𝑣 =
2𝑔𝑦. Plugging the arc length 𝑑𝑠 =1+𝑦′2
𝑑𝑥, the integral of 𝑡12 is converted
into
𝑡12 =
𝑃2
∫
𝑃1
1+𝑦′2
2𝑔𝑦 𝑑𝑥 =
𝑃2
∫
𝑃1
𝑓(𝑦, 𝑦′)𝑑𝑥. (52)
Normally, the Euler–Lagrange differential equation is used, and, after some maths,
another differential equation is obtained
[1 + ( 𝑑𝑦
𝑑𝑥)2]𝑦=𝑘2,(53)
where 𝑘2is a positive constant. The solution is given by the parametric equations,
Eqs. (49)–(50). Our foundational project aims at finding a coherent and possibly
true representation of what is physically possible without involving extraneous
15 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
idealizations or empirical generalizations. In this sense, our formulation seems more
natural than the Lagrangian formulation, since it represents a new synthesis of two
different propensities in nature (provided by energy and entropy).
3. Conclusion
It follows, as a result of theory, that dynamics needs irreversibility as an elementary
inherent mechanism to consistently describe and comprehend physical systems. The
theory represents a fundamental conceptual departure from Newtonian mechanics,
showing that a physical system is not only actuated by “forces”, acting as a
“slave system”. It can have mutual interaction with sub-systems and self-regulated
mechanisms. The present approach offers consistent solutions to well-known
problems from different perspectives, revealing leading principles of Nature’s
operating system.
The integrations of the gradient of a new term that contains the rotational energy
and thermodynamical free energy, besides the usual force term which has a non-
entropic character, plays a vital role for an adequate understanding of dynamical
and electrodynamical processes in nature. The set of two first order differential
equations, related to the interplay between the tendency of energy to attain a
minimum, whilst entropy seeks to maximize its value, opens a major change
in the description of dynamical and electrodynamical processes in nature. The
description of physical phenomena by ergontropic dynamics is remarkably simple
and these foundations support applications that extend well beyond the starting
concepts, in particular integrating in one equation translational and rotational
motion. Considering the embryonic stage of thermodynamics at the 17th century,
this synthesis was an impossible task at the time of Sir Isaac Newton. But this
formulation embraces Newton’s law of dynamics with entropy in a novel way.
Of course, at the present stage, this formalism and concepts constitute a research
program needing further development. It does not replace Hamiltonian formulation
but, depending on the nature of the problem, its approach can be an essential tool
for investigation.
Declarations
Author contribution statement
Mario Pinheiro: Conceived and designed the analysis; Analyzed and interpreted the
data; Contributed analysis tools or data; Wrote the paper.
16 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
Funding statement
This work was supported by LaserLab Europe IV – GA No654148 (H2020-INFRAIA-
2014-2015) and EuPRAXIA – GA No653782 (H2020-INFRADEV-1-2014-1).
Competing interest statement
The authors declare no conflict of interest.
Additional information
No additional information is available for this paper.
Acknowledgements
The author gratefully acknowledges the International Space Science Institute (ISSI,
at Bern and Beijing) where he benefited from a friendly environment as a visiting
scientist.
References
[1] Mario J. Pinheiro, A variational method in out-of-equilibrium physical systems,
Sci. Rep. 3 (2013) 3454.
[2] L. Landau, E.M. Lifshitz, Course of Theoretical Physics: Statistical Physics,
vol. 5, 2nd ed, Pergamon Press, Oxford, 1970, p. 32.
[3] R.M. Kiehn, Plasmas and Non-Equilibrium Electrodynamics, vol. 4, Non-
Equilibrium Systems and Irreversible Processes, Lulu Enterprises, Morrisville,
2009.
[4] Rodrigo de Abreu, arXiv:physics/0207022v1, 2012.
[5] M.J. Pinheiro, An information-theoretic formulation of Newton’s second law,
Europhys. Lett. 57 (2002) 305.
[6] M.J. Pinheiro, Information-theoretic determination of ponderomotive forces,
Phys. Scr. 70 (2004) 86.
[7] O. Éboli, R. Jackiw, So-Young Pi, Quantum fields out of thermal equilibrium,
Phys. Rev. D 37 (1988) 3557–3581.
[8] Mario J. Pinheiro, On Newton’s third law and its symmetry-breaking effects,
Phys. Scr. 84 (5) (2011) 055004.
17 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
[9] M.J. Pinheiro, The flyby anomaly and the effect of a topological torsion current,
Phys. Lett. A 378 (41) (2014) 3007–3011.
[10] M.J. Pinheiro, Some effects of topological torsion currents on spacecraft
dynamics and the flyby anomaly, Mon. Not. R. Astron. Soc. 462 (1) (2016)
3948–3953.
[11] C. Truesdell, W. Noll, The Nonlinear Field Theories of Mechanics, Springer-
Verlag, Berlin, 1965.
[12] A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane
waves, Int. J. Eng. Sci. 10 (1972) 425–435.
[13] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of
screw dislocation and surface waves, J. Appl. Phys. 54 (1983) 4703–4710.
[14] J.P. Shen, C. Li, A semi-continuum-based bending analysis for extreme-thin
micro/nano-beams and new proposal for nonlocal differential constitution,
Compos. Struct. 172 (2017) 210–220.
[15] J.P. Shen, C. Li, X.L. Fan, C.M. Jung, Dynamics of silicon nanobeams
with axial motion subjected to transverse and longitudinal loads considering
nonlocal and surface effects, Smart Struct. Syst. 19 (1) (2017) 105–113.
[16] J.J. Liu, C. Li, X.L. Fan, L.H. Tong, Transverse free vibration and stability
of axially moving nanoplates based on nonlocal elasticity theory, Appl. Math.
Model. 45 (2017) 65–84.
[17] C. Li, S. Li, L.Q. Yao, Z.K. Zhu, Nonlocal theoretical approaches and
atomistic simulations for longitudinal free vibration of nanorods/nanotubes
and verification of different nonlocal models, Appl. Math. Model. 39 (2015)
4570–4585.
[18] C. Li, L.Q. Yao, W.Q. Chen, S. Li, Comments on nonlocal effects in nano-
cantilever beams, Int. J. Eng. Sci. 87 (2015) 47–57.
[19] C. Li, Torsional vibration of carbon nanotubes: comparison of two nonlocal
models and a semi-continuum model, Int. J. Mech. Sci. 82 (2014) 25–31.
[20] P.A. Murad, New frontiers in space propulsion science, part II – approaches
to push the new frontiers, in: Takaaki Musha (Ed.), New Frontiers in Space
Propulsion Science, Chapter 2, Nova Science Publishers, New York, 2015.
[21] H. Lamb, Higher Mechanics, Cambridge University Press, London, 1920.
[22] J.I. Cirac, P. Zoller, Quantum computation with cold, trapped ions, Phys. Rev.
Lett. 74 (20) (1995) 4091.
[23] V. Bargman, Relativity, Rev. Mod. Phys. 29 (1957) 161.
18 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Article No~e00365
[24] Andre Koch Torres Assis, Relational Mechanics and Implementation of Mach’s
Principle with Weber’s Gravitational Force, Apeiron, Montreal, 2014.
[25] Myron W. Evans, Spin connection resonance in gravitational general relativity,
Acta Phys. Pol. B 38 (6) (2007) 2201–2220.
[26] C.L. Coughenour, A.W. Archer, K.J. Lacovara, Tides, tidalites, and secular
changes in the Earth-Moon system, Earth-Sci. Rev. 97 (2009) 59.
[27] Robert G. Strom, Ann L. Sprague, Exploring Mercury, Springer, Berlin, 2003.
[28] H-S. Liu, Thermal and Tidal effect on the rotation of Mercury, Celest. Mech. 2
(1970) 4–8.
[29] M.A. Siegler, B.G. Bills, D.A. Paige, Orbital eccentricity driven temperature
variation at Mercury’s pole, J. Geophys. Res. E 118 (2013) 930–937.
19 http://dx.doi.org/10.1016/j.heliyon.2017.e00365
2405-8440/©2017 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).