epl draft

An Extended Dynamical Equation of Motion, Phase Dependencyand Inertial Backreaction

Mario J. Pinheiro1,2 and Marcus Buker1

1 Department of Physics,Instituto Superior Tecnico - IST, Universidade de Lisboa - UL,Av. Rovisco Pais, & 1049-001 Lisboa, Portugal2 Department of GeographyWestern Illinois University, & Macomb, IL, USA 61455

PACS 45.20.-d – Formalisms in classical mechanicsPACS 03.65.Vf – Phases: geometric; dynamic or topologicalPACS 04.30.Db – Wave generation and sources

Abstract – Newton’s second law has limited scope of application when transient phenomenaare present. We consider a modification of Newton’s second law in order to take into account asudden change (surge) of angular momentum or linear momentum. We hypothesize that spaceitself resists such surges according to an induction law (related to inertia).

Introduction. – Newton’s second law has limited1

scope of application when transient (i.e., jerk or surge)2

phenomena are present. This law states that “the force is3

proportional to the rate of change of momentum”. In a4

sense, this implies a kind of “straight-line” theory, since5

force and change of momentum are directed along the6

same line. Critics of the conceptual foundation of this ba-7

sic equation of dynamics include Gustav Kirchhoff, Ernst8

Mach, Henri Poincare, Max Jammer and Frank Wilczek9

(see Ref. [1] and Ref. [2] and references therein).10

Tait and Watson [3] delimits the applicability of the con-11

cept embedded into the Newton’s second law of motion,12

“If the motion be that of a material particle, however,13

there can be no abrupt change of velocity, nor of direction14

unless where the velocity is zero, since [...] such would15

imply the action of an infinite force”.16

Any attempt to formulate a dynamical law is faced with17

the problem of inertia, since this property characterizes18

body responses to external forces, or its state of motion19

in the absence of forces. We may find in literature several20

approaches to tackle inertial properties of matter, among21

them we may refer: the Machian viewpoint, linking in-22

ertia with gravitational interactions with the rest of the23

universe [4–6]; the hypothesis that inertial forces result24

from an interaction of matter with electromagnetic fluc-25

tuations of the zero point field [7–9]; and the concept of26

inertia as resulting of the particle interaction with its own27

field [10–12]. We would like to point out that the local 28

approach followed in this paper, does not rule out the im- 29

portance of the interaction of matter with the rest of the 30

universe; simply it means that, seemingly inertial prop- 31

erties are independent of the properties of the universe, 32

and this approach can provide the correct values of the 33

electron rest mass [1]. 34

According to astronomical observations the amount of 35

mass that can be identified in the universe accounts for 36

only a small percentage of the gravitational force that 37

acts on galaxies and other galactic systems. Galaxies’ 38

rotation curves show that they do not spin according to 39

Newtonian dynamics; they spin considerably faster. While 40

the standard cosmological model argues dark matter may 41

explain the discrepancies, modification of Newtonian dy- 42

namics (MOND) has been proposed [13,14] as a means of 43

explaining the discrepancy. 44

Here, we will show the necessity, at a fundamental level, 45

to modify the usual equation of dynamics in order to de- 46

scribe general mechanical and electromagnetic phenom- 47

ena. We prove that it is missing an inductive term in the 48

fundamental equation of dynamics, we discuss the mean- 49

ing of the emergent term, linking it to the back-reaction 50

of the physical vacuum, and illustrate its usefulness to un- 51

derstand phenomena conditioned by rotatory motion. 52

An Extended Equation of Motion. – The con- 53

cept of self-reaction has been hypothesized in field theory, 54

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Mario J. Pinheiro et al.

as early as the classical Abraham-Lorentz, and later the55

Abraham-Lorentz-Dirac equation (see, e.g., Ref. [15] and56

references therein). As the theory of electromagnetism57

provided the framework for the further development of58

quantum field theory, it may provide as well the prerequi-59

sites to properly understand classical mechanics.60

In this context, considerable progress has been attained61

in the study of the electric charge dynamics [16–18], where62

the point charge and the extended charge models of the63

classical electron have been analyzed.64

The Abraham-Lorentz non-relativistic equation of mo-tion for a point charge is given by [20,21]:

ma = Fext +mτ a, (1)

where τ = 2e2/(3mc3), F(t) = e(E + v × B) is theLorentz force and a dot means a derivative relative totime. Dirac later obtained the equation of motion in afour-dimensional formalism:

maµ = Fµext + Γµ = Fµext +2e2

3c3

(aµ − 1

c2vµaαa

α

). (2)

Here, Γ is known as the Schott term and its first term65

includes the time derivative of the acceleration, already66

present in the Abraham-Lorentz equation, Eq. 1.67

The last term of the right-hand side (rhs) is the68

Abraham-Lorentz force term, describing the radiation re-69

coil force due to the self-interaction between the charge70

and its own force fields. This is an important example71

of a transient phenomena that appears as an intermedi-72

ate state between two permanent conditions: in the above73

instance, the transient is the result of the change of the74

amount of stored energy in the system (e.g., the extended75

charge in the above example). This phenomena represents76

its readjustment to the change of stored energy.77

In electrical engineering, when reactive elements arepresent (for example, in an electric circuit comprising ofa solenoid with a given resistance R and inductance L),Kirchoff’s law dictates the form for the voltage at the ter-minals:

VL = Ri+ Ldi

dt. (3)

The change in the stored magnetic energy is thus describedby the last term involving inductance and the time rate ofchange of current. Analogously, we can write:∑

i

Fi = ma + ℵdadt, (4)

where we call ℵ the intractance (in Eq. 4 in units m·s);78

the other terms have the usual meaning. In the foregoing79

treatment it will be shown that the last term of Eq. 4 is the80

time derivative of the “flux” of angular momentum Φ. To81

illustrate the resistance (back-reaction) of space against a82

sudden change of motion, we will start with an algebraic83

description of the gyroscopic effect.84

The equation of motion for a gyroscope can be obtainedby a vectorial method [22–24]. Consider a wheel spinning

around its own instantaneous axis of rotation (and coinci-dent with axis Oz) which is free to move about the fixedpoint O an axisymmetric gyroscope. Let us suppose thatthe axis of spin is displaced by an infinitesimal angle dφin a time dt around axis Oy, such as dL = Ldφ. Theequation for torque (see also Fig. 4) is:

G =dL

dt= LΩ = (Iω)Ω. (5)

We propose the last term of Eq. 4 is equivalent to a back-reaction force resulting from a mechanical induction law.The implication of Fig. 4 is that a sudden torque gives riseto the angular momentum represented by a new vector dL(pointing toward the sheet). However, this sudden action(“surge” or “jerk”), da/dt, through an induction mech-anism, generates a counter-opposed angular momentum,designated Lind, that appears pointing outside the sheet.This counter-angular momentum is related to the inducedtorque through the equation (here, its x-component):

Gx = (Izωz)Ωy +Gx,ind. (6)

Since Gi,ind = dLi,ind/dt, we obtain

Gx = (Izωz)Ωy + IxΩx. (7)

Applying the same argument, for the y-component we ob-tain

Gy = −(Izωz)Ωx + IyΩy. (8)

We may note here, that the above Eqs. 7- 8 are well-known 85

(see any book on classical mechanics, e.g. Corben & Stehle 86

[25]), what differs is our interpretation of the nature of the 87

inductive term since it seems to result from a kind of oppo- 88

site force, a backreaction that comes from the immediate 89

vicinity of space, the physical vacuum. 90

The equations of change of angular momentum can bewritten under the compact form:

Gi =∑jk

[α

∂

∂αk+ ωk

∂

∂ωk

]Iijωj (9)

where α = (α1, α2, α3) represents Eulerian angles. 91

But at this point we may investigate whether this induc-tive term may be hidden inside the governing equation ofangular momentum. Experimental observations of staticelectromagnetic angular momentum done by Graham &Lahoz [19] ”[...] implies that the vacuum is the seat ofsomething in motion whenever static fields are sat up withnon-vanishing Poynting vector, as Maxwell and Poyntingforesaw”. Hence, we may outline here a statistical descrip-tion of a set of rotating bodies according to Curtiss [26](approach that may be applied ultimately to other phys-ical systems, like tornadoes, or microscopic particles in aplasma showing diamagnetic behavior) introducing a dis-tribution function in the generalized space f(r,v,ααα,ωωω, t),such that

f(r,v,ααα,ωωω, t)drdvdαααdωωω (10)

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An Extended Dynamical Equation of Motion

Fig. 1: Gyroscopic motion of a wheel rotating with angularvelocity ωz around its main axis Oz, and angular velocity Ωy.Notice that the induced torque is transverse to the imposeddisplacement.

represents the number of rotating bodies with vector po-sition r between r and r + dr, with velocity v between vand v + dv, with ααα between ααα+ dααα, and with ωωω betweenωωω + dωωω. Here, the symbol ααα denotes simply a functionaldependence on Eulerian angles α, β (see Fig. 2). The dis-tribution function is normalized:

ˆfdvdαααdωωω = n, (11)

with n denoting the number density of rotating bodies.For convenience we may define an average over v, ααα, andωωω and denote it by

A =1

n

ˆˆˆA(v,ααα,ωωω)f(r,v,ααα, ω, t)dvdαααdωωω. (12)

The above statistical treatment (see Ref. [26] and Eq.2-51 therein) leads us to the equation of conservation ofangular momentum, which can be written under the form:

∂

∂t

(nM

)+∇ · (nv0M) = −∇ ·

←→Φ −∇ ·

←→Φ0 + nG, (13)

where we have introduced the average angular momentumassociated with the rotation of the set of particles:

M ≡ (I ·ωωω) =1

n

ˆˆˆ(I ·ωωω)fdvdαdωωω. (14)

Here, I is the moment of inertia tensor in the spacefixed coordinate system. We also introduced the followingdyadics (second order tensor):

←→Φ = nV(I ·ΩΩΩ);←→Φ0 = n ˜V(I ·ωωω0),

(15)

which is the tensor representing the flux of angular mo-mentum ΩΩΩ = ωωω−ω0ω0ω0, with ωωω0 denoting the average angularvelocity of the set of rotating bodies (the same tensor ap-plies to ω0ω0ω0); v0 is the macroscopic stream velocity of allthe rotating bodies, and V = v − v0 is the velocity of arotating body relative to the stream. If we assume thatthe set of rotating bodies is at rest, Eq. 13 simplifies to

∂

∂t

(nM

)+∇ · [nv(I · ω)] = nG. (16)

Integration over a volume yields 92

ˆˆˆV

∂

∂t

(nM

)dr+

ˆˆˆV

∇ · [nv(I ·ωωω)]dr =

ˆˆˆV

nGdr. (17)

The second term on the left hand side can be trans-formed into a surface integral; therefore, Eq. 17 can bewritten as

∂M

∂t+

‹S

←→Φ · n∗dS = G. (18)

If we substitute in the surface integral nV by an operator:

nV· ≡ σ ∂∂t·, (19)

where σ = N/S is the surface density of the rotating bod-ies, then Eq. 18 reduces to the expression:

∂M

∂t+ σ

∂

∂t

‹S

(L · n∗)dS = G. (20)

Here, the second integral was converted to an integral over 93

an inner product of vectors in three-dimensional space, 94

similar to Faraday’s law of induction. We have also used 95

the geometric property: dSj = n?jdS ↔ dS = n?dS. 96

In Eqs. 7- 8, Ix and Iy are the inertial momentum ofthe rotor relative, resp., to the Ox and Oy axis. It is clearthat the last term can be represented by a time rate ofchange of a flux of angular momentum (per unit of area):

Φy =1

S

‹(Lind · dS). (21)

We may use here the concept of the angular momentum 97

field lines threading an hypothetical surface whose bound- 98

ary is an open surface S(t) =˜dS = 2π(1 − cosα) (in 99

fact, the area of the spherial cap), analogously to the pro- 100

cedure used in Faraday’s law of induction. The “angular 101

momentum field” is supposed to have the origin at the ex- 102

tremity coincident with the gyroscope main axis inclined 103

at an angle α relative to the vertical (if subject to the 104

gravity field). This procedure is offered to manipulate 105

mathematically Eq. 21, see also Fig. 3. 106

However, the angular momentum flux threading the sur-face dS is homogeneous, and Eq. 21 gives simply

Φi = Lind,i = IiΩi, (22)

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Mario J. Pinheiro et al.

Fig. 2: A volume V containing a set of rotating bodies, eachwith vorticity ω. In particular, the figure represent the velocitycomponents and Euler’s angles for one of bodies.

for i = 1, 2, 3. This “phase” is mass dependent in away which seems consistent with the quantum mechan-ics prediction that all phase-dependent phenomena (evenin the presence of a gravitational field) depend on the massthrough de Broglie’s wavelength [27], as we will see later.We can rewrite Eq. 6 under the general form:

Gi = εijkΩjAk + ∂tΦi. (23)

where the inferior Latin index i = 1, 2, 3 refers to the107

Cartesian components of the physical quantities along the108

axes of the inertial system 1, 2, 3; εijk denotes the com-109

pletely anti-symmetric unit tensor of Levi-Civita; Ak =110

ωkIk; and the unit vector −→n points along the Ox or Oy-111

axis.112

On the basis of the above described model, we will show113

how this inductive term explains the direction of preces-114

sion kept by a gyroscope, and it may be seen as a simple115

experimental test for the existence of the inductive term.116

Afterward, we show that this same counter (inductive)117

term is identical to the time dependent Berry’s geometri-118

cal phase and related to the persistent current observed in119

mesoscopic rings.120

Gyroscope Lenz’ law. From this analysis we propose a121

new law for the direction of precession: if a moving body122

is acted upon either gradually or with a sudden jerk, the123

resultant angular momentum flux through a closed linear124

momentum isosurface induces an angular momentum to125

compensate such a change, that is, there is a tendency126

to retain a constant value of flux as described in Eq. 21.127

The last term of Eq. 23 is related to the Faraday’s mag-128

netic induction law by considering the analogy between129

the magnetic field and vorticity [28]. Quite interestingly,130

the above construction also implies a relationship similar131

to the Lenz law.132

From this relationship, one can predict the direction of133

precession for an axisymmetric gyroscope with high angu-134

Fig. 3: Axial flux of angular momentum φy,ind through an area∆S.

lar velocity: if we assume that ωωω is pointing up, and we 135

increase the value of a suspended mass, then the preces- 136

sional velocity turning to the right will increase, since the 137

vector weight points down and would reduce the value of 138

the angular momentum. Therefore, there must be a com- 139

pensating angular momentum through precession in the 140

same direction as ωωω, in order to maintain constant flux; 141

on the contrary, if ωωω points down, the precession motion 142

would reverse direction. We encourage the reader to test 143

with a simple ring gyroscope. This phenomena implies a 144

deep connection between inertia, gravity and the physical 145

vacuum. 146

A similar analysis can be done relative to the fundamen-tal equation of dynamics. For this purpose let us examineFig. 4-(a). Consider a particle of mass m with linear mo-mentum p(t) moving along the Ox axis. Let us supposethat it experiences a sudden change of linear momentumdue to action of some agent, δp. Consequently, at timet+∆t obtains a new linear momentum p(t+∆t). There isalways a given point in space distant of ρ relative to whichthe particle has angular momentum. The important pointto note is that this sudden change of motion (i.e., trajec-tory) is in fact equivalent to a succession of infinitesimalrotational displacements and, therefore, related to a sud-den appearance of angular momentum δL = ρδpuy (whereuy is the unit vector), about an instantaneous axis the po-sition of which passes through the point of intersection ofthe linear momentum at previous times t and t+τ . Then,it can be shown (see Fig. 4-(a)) that

δp =1

ρ(δL · uy)ux, (24)

or

∂tδp =1

ρ(∂tδL · uy)ux, (25)

for a given ρ. It derives from the above argument theexistence of a new overlooked source term, that must be

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An Extended Dynamical Equation of Motion

Fig. 4: (a) - A particle with initial momentum at position ρexperiences an imposed acceleration; (b) - the induction lawimplies an induced angular momentum Lind ”threading” theregion ∆S.

necessarily added to an eventual external force Fext, andwe may easily generalize to three dimensions the funda-mental equation of dynamics:

ma = Fext + ∂tδp. (26)

The angular momentum is (through the agency of ωωω) theanalogue of the B-field (see., e.g., Ref. [28]), hence it isnatural to associate with L an induction law. This ismanifested through a resistance to the sudden increaseof angular momentum flux threading an elementary areadS. This is quantified through the relationship:

ma = Fext + ∂t (δp)ind , (27)

with ∂t (δp)ind = ∂t (δp), representing the inertial resis-tance with which a single mass particle opposes the im-posed acceleration. Note that, when performing an in-finitesimal change of time, then we have

δL = L(t+ τ)− L(t) ≈ τ ∂L

∂t, (28)

which gives∂δL

∂t≈ τ ∂

2L

∂t2, (29)

for a given τ , representing in practice the interval of timewhen the “surge” takes place. Assuming that the “surge”directed along the Ox axis induces angular momentumalong the Oy axis, we obtain

∂δL

∂t≈ τIyΩyuy. (30)

Here, Iy = mρ2. Hence, since Ωy = vx/ρ

1

ρ

(∂δL

∂t· uy

)= mτax. (31)

Therefore, we deduce immediately that the following equa-tion holds:

max = F extx +mτax. (32)

We can write it in a general form:

mai = F exti +mτai, (33)

with i = 1, 2, 3. This is an equation similar to the Abra-ham’s equation that explains (classically) the radiationproblem. A relationship can be inferred for τ . The criti-cal action time τ ≡ ℵ/m is dictated by the kind of physicalfields acting over the body. For example, a simple argu-ment based on the charged particle canonical momentumδp = mδv+qδA leads us to a rough estimation of τ whenconsidering radiation problems. The electromagnetic vec-tor potential produced by a charge q in motion with speedv at a distance r is A = qv/4πε0c

2r. We can expect thata sudden change of velocity leads to r ≈ e2/mc2, wheree2 = q2/4πε0. The critical action time (or transient time)τ is such as

τ =r

c≈ e2

mc3, (34)

which is of the order of value τ = 2e2/3mc3 given in the 147

framework of the Abraham model. And τ measures the 148

significance of the electromagnetic field relatively to the 149

inertial mass. Notice that it is not necessary that the 150

particle (or body) rotates in order to obtain angular mo- 151

mentum; it is enough that there exists a variation of L 152

relative to a given point (or axis). 153

Hannay’s geometric phase and the inductive torque. 154

Our purpose now is to suggest that the Aharonov-Casher 155

flux effect may be the outcome of the inductive term rep- 156

resented at Eq. 27. 157

For this purpose, let us consider a mesoscopic ring ofradius R the AC flux is given by [29]:

ΦACΦo

= gµBRE

c~, (35)

where Φo = hc/e, g is the g-factor, µB is the Bohr magne-ton, E is the electric field actuating upon the particle inthe ring undergoing rotational motion relative to a z axisof revolution draw along its axis of revolution. Insertingthe Bohr magneton µB = eL/2mc into Eq. 35 we obtain

ΦAC2π

=g

2

eRE

emcL. (36)

However, L = IzΩz, with Iz = mR2, since the particleof mass m orbits along the ring of radius R. In addition,note that we can scale eER with the rest energy of theparticle mc2 [29]. This gives

1

cΦ′AC =

g

2mR2Ωz =

g

2IzΩz. (37)

Here, we have made the appropriate change of variableΦ′AC = eΦAC/2π, in order to get back our “phase” vari-able in angular momentum units. Therefore, this expres-sion is identical to the inductive term (here, in Gaussianunits):

1

c

dΦ′ACdt

=g

2IzΩz. (38)

Since space does not offer any resistance to bodies with 158

uniform motion, we can deem the inductive term of Eq. 4 159

as the cause of inertia, implying that space itself resists a 160

change of angular momentum. 161

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Mario J. Pinheiro et al.

Application to Planetary Motion in the Solar System.162

Defining the body velocity by v(t) = (ds/dt)T, we may163

take use of the Frenet-Serret formulas to parametrize the164

trajectory:165

dT

ds= κN

dN

ds= −κT + τB

dB

ds= −τN.

Here, T is the unit tangent vector, N is the unit normalvector, B is the unit binormal vector, τ is the torsion,and κ is the curvature. Using Eq. 4 we obtain a generalclassical equation for a particle motion, written here invectorial notation:

ma = Fext+

ℵ[(a− 1

c2vc2v2

ρ2

)T +

(3vv

ρ− v2

ρ2ρ

)N +

v3

ρτB

].

(39)We may noticed that the Abraham-Lorentz-Dirac equa-166

tion (ALD) is obtained if we equalize c2v2

ρ2 ≡ a2, or 1

ρ = avc .167

If we have made use of the relationship ρ = c2/a, satis-168

fied when the coordinate time identifies with the proper169

time [31], then we would have a = v2/ρ.170

The intractance ℵ may be guessed on the base of the171

already known intractance term ℵ ≡ 2e2

3c3 for the (ALD)172

equation, as previously discussed, and noting that the173

term on the r.h.s. along the tangential is the same as it ap-174

pears in the ALD equation. However, there is a connection175

between the gravitational constant G and the Coulomb176

constant k0 expressed in terms of Planck units of mass177

mPl [32], G = k0e2/(αm2

Pl), which gives an estimate of178

the gravitational intractance ℵ ∼ Gαm2Pl

c3k0∼ 10−65 kg.s,179

and we estimate a relaxation time of about 1068 years for180

a planetary orbit suffer any significative resistance to mo-181

tion due to the intractance term. Due to the smallness of182

ℵ it is expected that the time derivative of the acceleration183

do not play any relevant role on a planetary scale. Notice184

the resemblance of Eq. 39 with the ALD equation (consid-185

ering only the tangent term), attained when β ≡ v/c→ 1.186

Conclusion. – When a system is subject to a sudden187

change of velocity or direction of motion, there appears an188

induction force that opposes such a change. This induction189

force is related to the inertia of bodies and to intrinsic190

properties of the physical vacuum.191

∗ ∗ ∗

This work was supported in part by the National Science192

Foundation (NSF) through award AGS-1137153.193

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