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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

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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

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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

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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.

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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)

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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

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∇

∇

∇=1

𝑅−𝑟

𝜕

𝜕𝜃 1

2𝐼𝑄𝜔(𝜔−2𝜔)−∇𝐹𝐮𝜃.(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

Ω𝐮𝜃(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=2𝜋7(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.

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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=sin𝜃̇

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=𝑚𝑑2𝐫

𝑑𝑡2𝑆′

+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.

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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

2𝐉×∇

∇

∇×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)

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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

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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

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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.

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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

𝜃2𝐮𝑟

−1

𝑟

𝜕

𝜕𝜃 1

2𝑚𝑅2̇

𝜃2𝐮𝜃,(39)

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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=𝜔2(cos 𝜃− cos 𝜃𝑚)=2𝜔sin2(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.

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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

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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.

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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.

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