The Mechanical Roots of Hehl & Obuchov’s Principles of
Electromagnetism - A Contribution to the Unity of
Classical Physics
∗
Peter Enders1
Dedicated to Werner Ebeling
DOI:
ABSTRACT
I examine the relationships of Hehl & Obuchov’s principles of electromagnetism to classical mechanics
and show that the most important ones can be reduced to point-mechanical principles as established
by Newton, Euler, Lagrange and Helmholtz. This means that the Maxwell-Lorentz equations and
the Lorentz force are largely determined by, or at least connected with the laws of classical pointmechanics. This is hopefully a significant step towards the unity of classical physics. A mechanical
model of electromagnetic phenomena is not sought for.
Keywords: Electromagnetism; axiomatics; unity of classical physics; Lorentz force; Helmholtz force;
Lipschitz force; screened Poisson equation; Yukawa equation; Klein-Gordon equation.
1 INTRODUCTION
While classical mechanics and classical electromagnetism are often called ’the two pillars of classical
physics’, quantum mechanics and quantum electrodynamics are not called ’the two pillars of quantum
physics’. The reason for this is, perhaps, that the former two are usually treated on quite different
axiomatic footings, viz. Newton’s Laws and Maxwell’s equations, respectively. Thus, in order to bring
them as close to each other as their quantum counterparts are, one has to found Maxwell’s equations
on principles compatible with Newton’s Laws.1 This has been tackled by Fritz Bopp 60 years ago [1].
Later, Hehl & Obuchov [2] have formulated similar principles, see Section 3. This does not mean to
construct mechanical mechanisms for electromagnetic phenomena [3] [4].
To be definite, let me recall Newton’s ”Axioms, or the laws of motion” [5].
Law 1 ”Every body perseveres in its state of being at rest or of moving uniformly straight forward,
except insofar as it is compelled to change its state by forces impressed.”
~v = const
1
⇐
Fext = 0
Kazakh National Pedagogical Abai University, Almaty, Kazakhstan.
*Corresponding author: E-mail: enders.kbd@gmx.de;
1
The reverse task requires to derive the Lorentz force from Maxwell’s equations.
(1a)
Law 2 ”A change in motion is proportional to the motive force impressed and takes place along the
straight line in which that force is impressed.”
~ext
∆~
p∼F
(1b)
Law 3 ”To any action there is always an opposite and equal reaction; in other words, the actions of
two bodies upon each other are always equal and always opposite in direction.”
~12 = −F
~21 ,
F
∆~
p2 = −∆~
p1
(1c)
Further, recall Lagrange’s equation of motion of second kind in the form [6]
d ∂L
∂L
=
.
∂~r
dt ∂~v
(2)
On the other hand, Maxwell’s (rationalized) equations read (in SI units)
~ r, t) = ρ(~r, t)
∇ · D(~
(3a)
~ r, t) = 0
∇ · B(~
(3b)
~
~ r, t) = − ∂ B (~r, t)
∇ × E(~
∂t
~
~ r, t) = ~j(~r, t) + ∂ D (~r, t)
∇ × H(~
∂t
(3c)
(3d)
They have been praised by Boltzmann (slightly varying Goethe’s Faust) as ”Was it a God who wrote
these equations?” [4] (Epigraph to vol. 2, p. III). Correspondingly, Hertz postulated, ”Maxwell’s theory
is Maxwell’s system of equations” [7].
As a mattter of fact, Hertz’s postulate is at the heart of one of the two standard representations
of classical electromagnetism. The other one consists in the derivation of the basic laws from
the phenomena observed. Both approaches enlighten the origin of the disparity of the historical
developments as well as of the enduring gap between electromagnetism and mechanics. This gap is
strengthened by the quite different mathematical characters of the equations (1) and (2) on the one
hand, and equation (3) on the other hand.
Many difficulties Maxwell had to overcome stem from the blurring of the fundamental laws in real
media. Actually, the two inhomogenous equations (3a, 3d) can be phenomenologically obtained, see
Section 2. For this, I will concentrate myself onto the Maxwell-Lorentz equations for charged bodies
in vacuo (basically, Newton’s laws deal with bodies in vacuo, too). Here,
~ r, t) = ε0 E(~
~ r, t);
D(~
Therefore,
~ r, t) = B(~
~ r, t)/µ0
H(~
(4)
~ r, t) = 1 ρ(~r, t)
∇ · E(~
ε0
~
∇ · B(~r, t) = 0
(5a)
(5b)
~
~ r, t) = − ∂ B (~r, t)
∇ × E(~
∂t
~ r, t) = µ0~j(~r, t) + µ0 ε0
∇ × B(~
(5c)
~
∂E
(~r, t)
∂t
(5d)
The initial and boundary conditions as well as the sources ρ(~r, t) and ~j(~r, t) be given such, that the
~ and B
~ are uniquely determined.
fields E
Now, for the sake of the unity of (classical) physics, both, mechanics and electromagnetism, must
be put on equal footing. The spirit of Bopp’s principles of electromagnetism [1] is relatively close
to Newton’s [5], Euler’s [8] and Helmholtz’s [9] principles of mechanics. As a matter of fact, these
authors have formulated the foundations of classical mechanics in such a general manner that they
apply well beyond their original scope.
There are axiomatic approaches to pure electromagnetism (Maxwell’s equations) discarding the
potentials, e.g. [2]. However, the potentials seem to inevitably appear in Lagrangians and Hamiltonians.
The Maxwell-Lorentz equations can also be derived through generalizing Coulomb’s law along to the
rules of special relativity [10]. The isomorphism of Newton’s and Coulomb’s force laws suggests that
Coulomb’s law can be derived similarly to Newton’s one. However, that reasoning lacks the axiomatic
justification for applying Einstein’s kinematic foundations of special relativity to electromagnetic interactions.2
Feynman has derived the microscopic Maxwell’s equations through making the commutator between
position and velocity non-vanishing [12]. It’s truly surprising how far this modification of classical
mechanics reaches, but it is unclear whether there is a deeper mechanical reason for this ingenious
step.
2
MASS AND CHARGE CONSERVATION – THE INHOMOGENEOUS MAXWELL EQUATIONS
As usual, I assume that intrinsic extensive properties like mass, m, and (electrical) charge, q, are
distributed such, that there are sufficiently smooth mass and charge densities, ρm (~r, t) and ρq (~r, t).
~ m,q (~r, t), the scalar sources of which they are [13].
Then, there are vector fields, say, D
~ m,q (~r, t) = ρm,q (~r, t)
∇·D
(6)
~ q exhibits a certain physical meaning, viz. as dielectric
Maxwell’s first equation (3a) states that D
displacement.
Now, within non-relativistic physics, mass is a conserved quantity. Within this parallel treatment
2
Suisky and Enders [11] have presented an axiomatic generalization of Euler’s [5] derivation of
Newton’s equation of motion to a special-relativistic equation of motion.
of point mechanics and electromagnetics of charged bodies, is it natural to assume that charge is
conserved, too. As a consequence, for both, the equation of continuity holds true.
∂ρm,q
∇ · ~jm,q (~r, t) +
(~r, t) = 0
∂t
(7)
Combining it with equation (6) yields3
∂ ~
∇ · ~jm,q + D
=0
m,q
∂t
(8)
~ m,q (~r, t), the vortex of which is Maxwell’s respectively
Consequently, there are vector fields, say, H
~ m,q .
Heaviside’s total current, C
~ m,q := ~jm,q (~r, t) + ∂ D
~ m,q (~r, t) = ∇ × H
~ m,q (~r, t)
C
∂t
(9)
~ q (~r, t) has got a certain physical
The fourth Maxwell equation (3d) states that the vector field H
meaning, viz. as magnetic field strength.
In what follows, I will omit the indices ’m’ and ’q’, if not necessary.
3
HEHL & OBUKHOV’S PRINCIPLES
Hehl & Obukhov [2] (p.5) have introduced the following six principles as a foundation of classical
electromagnetism.
Principle 1 Conservation of electrical charge.
Remark 3.1. For classical electromagnetism (Maxwell’s theory) being a continuum theory, the existence
of the elementary charge plays no role. Hence, for point-like bodies the charge can be treated as a
given property like its rest mass.
Principle 2 The force on moving charges is given by the Lorentz force.
~L = q E
~ + q~v × B
~
F
(10)
Principle 3 Conservation of magnetic flux.
~ =0
∇·B
Principle 4 Localization of energy-momentum.
3
I always presuppose that it is allowed to reverse the sequence of two partial derivatives
(11)
Principle 5 Maxwell-Lorentz spacetime relation.
Principle 6 Splitting of the electric current in a conserved matter piece and an external piece.
The main goal of this contribution is to reduce them to principles that are well founded in classical
mechanics. In contrast to the principles above, the interpretation of the resulting equations is left to
the application under consideration. This allows for including the (Schrödinger-Fock-)Klein-Gordon
and screened Poisson equations as well, see Subsection 9.1.
Principle (1,2,3)’ Beside contact interaction, bodies may act upon each other via ’field forces’. A
field force is the product of (cf. Newton, ’Principia’ [5], Definitions 6...8)
1. a constant depending on the units of measurement used;
2. a body-dependent factor which is proportional to certain persistent properties of the body
under consideration (mass, charge);
3. an external geometric factor which describes the propagation of the force through the
space.
Principle (4)’ Hertz’s interaction principle [14] applies: If system S1 acts upon system S2 , then, in
turn, system S2 also acts upon system S1 .
Principle (5,6)’ Lagrange’s equation of motion (2) applies.
4
CONSERVATIVE AND LIPSCHITZ FORCES
The central point is Helmholtz’s explorations on the relationships between forces and energies. Let
me begin with Newton’s Law 2 (1b) in differential form,
~ dt.
d~
p=F
(12)
Multiplying both sides with ~v · yields (T being the kinetic energy)
~ dt = F
~ · d~r
~v · d~
p = dT = ~v · F
(13)
This becomes the point-mechanical energy law,
T (v) + V (~r) = const = E,
~ =F
~cons := −∇V (~r)
iff F
(14)
Now, as observed by Lipschitz [15] (1881, Add. 3), the same applies to the force
~Helmholtz (t, ~r, ~v , ~a, . . .) := F
~Lipschitz (t, ~r, ~v , ~a, . . .) + F
~cons (~r).
F
I propose to term it ’Helmholtz force’. Here,
(15)
~Lipschitz := ~v × K(t,
~ ~r, ~v , ~a, . . .).
F
(16)
~ being a rather arbitrary vector field) is a force which I propose to term ’Lipschitz force’. Its direction
(K
is always perpendicular to that of the velocity, so it does not affect the kinetic energy. The best known
example is the magnetic part of the Lorentz force (10).
Notice that Maxwell has already the equations (in modern notation)
fx = jy Bz − jz By ,
fy = jz Bx − jx Bz ,
fz = jx By − jy Bz
(17)
for the force per unit volume, where ~j is the convection current density of a charged body. He calls
~
them the ”Equations of Electromagnetic Force (C)” [16]. In modern notation, Lorentz [17] writes q~v × B
for the force itself.
~ ~r, ~v , ~a, . . .) does not depend on
The magnetic Lorentz force corresponds to the restriction that K(t,
the acceleration, ~a, and higher time-derivatives of ~r.
~ ~r, ~v , ~a, . . .) = K
~ L (t, ~r) := q B(~
~ r, t)
K(t,
(18)
It is compatible with Lagrangian and Hamiltonian mechanics. Is the more general Lipschitz force (16),
without that restriction, compatible, too?
5
5.1
COMPATIBILITY WITH LAGRANGIAN MECHANICS. THE
POTENTIALS
General Considerations
~ (t, ~r, ~v , ~a, . . .), a Lagrangian,
For a general force, F
L=
m 2
v − U (t, ~r, ~v , ~a, . . .)
2
(19)
exists, if it can be represented by the (generalized) potential, U (t, ~r, ~v , ~a, . . .), in the form
∂U
d ∂U
!
~ =
F
−
+
.
∂~r
dt ∂~v
(20)
One crucial point of this exploration is the observation that – in view of the distinguished role of the
velocity in the Lipschitz force (16) – the second term on the r.h.s. suggests to expand U in powers of
~v .
~ 1 (t, ~r, ~a, . . .) + 1 ~v · Û2 (t, ~r, ~a, . . .) · ~v + . . .
U (t, ~r, ~v , ~a, . . .) = U0 (t, ~r, ~a, . . .) + ~v · U
2
(21)
~ , in the formula (20) is required to exhibit an expansion in terms of ~v as
Consequently, the force, F
∂ ~
!
~ =
F
−∇U0 +
U1 + Û2 · ~v + . . .
∂t
h
i ~ 1 + Û2 · ~v + . . . + d~a · ∂
~ 1 + Û2 · ~v + . . . + . . .
− ~v × ∇ × U
U
dt ∂~a
This allows for identifying the forces that are compatible with Lagrange’s equation of motion (2).
5.2
Lipschitz Force
Let me first check the Lipschitz force (16) against the expansion (??).
!
~ = −∇U0 +
~v × K
∂ ~
U1 + Û2 · ~v + . . .
∂t
h
i ~ 1 + Û2 · ~v + . . . + d~a · ∂
~ 1 + Û2 · ~v + . . . + . . .
− ~v × ∇ × U
U
dt ∂~a
The inspection term by term leads to
U0 ≡ 0,
~1 = U
~ 1 (~r),
U
Û2 = Û3 = . . . ≡ 0̂
(22a)
The Lagrange-restricted Lipschitz force is static.
~ ~r, ~v , ~a, . . .) = K
~ static (~r) := −∇ × U
~ 1 (~r)
K(t,
~Lipschitz = F
~Lipschitz,static (~r) := ~v × K
~ static (~r)
F
(22b)
(22c)
This case corresponds to the well-known fact that a magnetic field is static, if there is no electric field.
5.3
Helmholtz Force
Second, let me check the expansion (??) for the Helmholtz force (15).
∂ ~
!
~ − ∇V =
~v × K
−∇U0 +
U1 + Û2 · ~v + . . .
∂t
h
i ~ 1 + Û2 · ~v + . . . + d~a · ∂
~ 1 + Û2 · ~v + . . . + . . .
U
− ~v × ∇ × U
dt ∂~a
This time, inspecting term by term results in
U0 = V (~r),
~1 = U
~ 1 (~r),
U
Û2 = Û3 = . . . ≡ 0̂.
(23a)
The Lagrange-restricted Helmholtz force is static, too.
~Helmholtz = F
~Helmholtz,static := F
~cons (~r) + F
~Lipschitz,static (~r)
F
(23b)
This reveals a remarkable result: There are forces that
• depend on time and arbitrary time-derivatives of ~r and
• constitute a system, in which the total energy is conserved, but
• are not Lagrangian.
The question, whether they are also non-Hamilton, lies outside the scope of this contribution.
5.4
Non-conservative Part in the Helmholtz Force
To obtain a time-dependent Lagrange-compatible force, the Helmholtz force (15) has to be generalized.
The least generalization consists in letting its non-Lipschitz component, (14), to be not conservative,
i.e. time-dependent.
~Helmholtz,gen := F
~Lipschitz + F
~non-cons (~r, t)
F
(24)
Then,
∂ ~
!
~ +F
~non-cons (~r, t) =
U1 + Û2 · ~v + . . .
−∇U0 +
~v × K
∂t
h
i ~ 1 + Û2 · ~v + . . . + d~a · ∂
~ 1 + Û2 · ~v + . . . + . . .
− ~v × ∇ × U
U
dt ∂~a
This requirement is fulfilled, iff
U0 = U0 (~r, t),
~ 1 = U1 (~r, t),
U
Û2 = Û3 = . . . ≡ 0̂
~ = K(~
~ r, t) := −∇ × U
~ 1 (~r, t)
K
~Lipschitz = F
~Lipschitz (~r, t) := ~v × K(~
~ r, t)
F
~
~non-cons (~r, t) = −∇U0 (~r, t) + ∂ U1 (~r, t).
F
∂t
(25a)
(25b)
(25c)
(25d)
The generalized Helmholtz force,
~Helmholtz,gen (~r, t) = F
~Lipschitz (~r, t) + F
~non-cons (~r, t)
F
~
~ r, t) − ∇U0 (~r, t) + ∂ U1 (~r, t),
= ~v × K(~
∂t
is a time-dependent Lagrange-compatible force.
(26a)
(26b)
6
6.1
CHARGES. FIELDS. POTENTIALS
Charges and Fields
Now I introduce ’charges’ q1,2 according to the Principle 3 above and referring to the two parts in the
generalized Helmholtz force (26a).
~ r, t) = q1 B(~
~ r, t)
K(~
(27a)
~Lipschitz (~r, t) = q1~v × B(~
~ r, t)
F
(27b)
~ r, t)
~non-cons (~r, t) = q2 E(~
F
(27c)
~ =F
~non-cons /q2 is a true field strength, i.e. a force per charge; it will be identified as electric
Here, E
~ is not a force per charge; it will be identified as magnetic induction.
field strength. In contrast, B
6.2
Vector and Scalar Potentials: Minimal Coupling Lagrangian
By the formula (26b), we have
~ = −∇ × U
~1
q1 B
(28a)
~
~ = −∇U0 (~r, t) + ∂ U1
q2 E
∂t
(28b)
~ 1 is proportional to q1 . The second equation implies q2 = q1 and
According to the first equation, U
~ 1 to be proportional to q1 , too. For this, I set q2 = q1 = q and
makes U
~ 1 = −q A
~
U
(29a)
U0 = qΦ
(29b)
~ r, t) and Φ(~r, t) will be identified as vector and scalar
The signs have been chosen such that A(~
potentials, respectively. They represent the electromagnetic field in the Lagrangian (19) as ’minimal
coupling’.
L(~v , ~r, t) =
6.3
m 2
~ r, t) − qΦ(~r, t)
v + q~v · A(~
2
(30)
Lorentz Force
Inserting the equations (27) with q2 = q1 = q into formula (26) shows that the Lagrange-compatible
generalized Helmholtz force equals the Lorentz force (10).
~Helmholtz,gen (~r, t) = q~v × B(~
~ r, t) + q E(~
~ r, t) = F
~Lorentz (~r, t)
F
(31)
In other words, the structure of the Lorentz force (10) is completely determined by the requirement
that the generalized Helmholtz force (24) be compatible with Lagrange’s equation of motion (2).
The generalized Helmholtz force (24) is distinguished by its relationship to energy conservation, see
Section 4.
7
THE HOMOGENEOUS FIELD EQUATIONS
~ 1 into the formulae (28) yields the fields in terms of the
Inserting the expressions (29) for U0 and U
potentials.
~ =∇×A
~
B
(32a)
~
~ = −∇Φ − ∂ A
E
∂t
(32b)
These representations immediately imply the equations.
~ r, t) = 0
∇ · B(~
~
∂B
~
= −∇ × E
∂t
(33a)
(33b)
~ and B
~ are represented through the only four
This is not surprising, because the six components of E
~
components of A and Φ.
~ and B
~ are interpreted as electrical field strength and magnetic induction, respectively, the
If E
equations (33) are the two homogeneous Maxwell-Lorentz equations in SI units. If they represent the
gravito-electromagnetic field quantities, they are Heaviside’s homogeneous gravito-electromagnetic
equations [18].
8
8.1
THE INHOMOGENEOUS FIELD EQUATIONS
The Need for Inhomogeneous Field Equations
~ and B;
~ one needs two more equations for
The equations (33) are not yet sufficient for calculating E
them which, in particular, should include their sources.
~ = J~
∇×B
(34a)
~ =P
∇·E
(34b)
~ and E
~
For obtaining the sources, J~ and P , I evoke Hertz’s interaction principle 3. Since the fields B
act upon the charges (charged bodies) and their convection currents via the Lorentz force (10), these
charges and currents act back upon the fields. Consequently, J~ and P depend on the charges and
convection currents. In what follows, I will show that these dependencies are largely determined by
the CPT symmetries of the field quantities.
8.2
CPT Symmetries of the Fields and Potentials
~ B,
~ Φ, and A,
~ can be read off from the equation of
The CPT symmetries of the field quantities, E,
motion4 ,
m
d2~r
~ + q~v × B.
~
= qE
dt2
(35)
They are listed in Table 1 (cf. [19]).
Table 1. CPT symmetries of the field quantities. Notice that solely the inert mass and the
volume have got the symmetry CP T = + + +
Quantity
Relation to ~r, t, q
C
P
T
position
~r
+
-
+
time
t
+
+
-
velocity
~v := d~r/dt
+
-
-
inert mass
m
+
+
+
+
-
+
2
[m ddt2~r ]
force
~] =
[F
charge, gravitating mass
q
-
+
+
volume
V
+
+
+
charge density
[ρ] = [q/V ]
-
+
+
current density
[~j] = [ρ] [v]
-
-
-
electric field strength
~ = [F
~ ]/ [q]
[E]
-
-
+
scalar potential
~ r]
[Φ] = [E][~
-
+
+
magnetic induction
~ = [F
~ ]/ [q] [v]
[B]
-
+
-
vector potential
~ = [B]/
~ [~r] = [E][t]
~
[A]
-
-
-
It is understood that the motion of a system of N bodies does not change if the time or the space
coordinates or the charges are reversed. For the meaning of the notion inter action of two equal
bodies implies that their locations and charges enter the formulae describing the interactions between
them in a symmetric manner.
8.3
The Inhomogeneous Maxwell-Lorentz Equations
According to Table 1, the equations (34) become
~
~ = κBj~j + κBE ∂ E
∇×B
∂t
~ = κE ρ,
∇·E
4
The special-relativistic correction of the l.h.s. through
symmetries.
p
(36a)
(36b)
1 − v 2 /c2 , does not affect its CPT
All coefficients are invariant properties of the vacuum through which the interaction of charges and
fields is transmitted. By the isotropy of the vacuum, they are scalar quantities, not matrices. The
second term on the r.h.s. of the equation (36a) makes it solenoidal by the equation of continuity (7).
This term corresponds to Maxwell’s extension of Ampè’s flux law.
The CPT invariance allows for all odd powers of charges and currents. These, however, are considered
to create the fields, so that the equations (36) are linear.
Building the curl of the equation (36a) and using the homogeneous Maxwell-Lorentz equations
(33) yields
2~
~ = −4B
~ = κBj ∇ × ~j − κBE ∂ B .
∇× ∇×B
∂t2
(37)
Hence, κBE = 1/c2 , c being the speed of (gravito-)electromagnetic waves in vacuo. κBj and κE
depend on the units of measurement. In SI units, κBj = µ0 , κE = 1/ε0 .
9
CPT INVARIANT EQUATIONS FOR THE POTENTIALS
Furthermore, Table 1 allows for rather general CPT invariant equation for the potentials.
9.1
CPT Invariant Equation for the Scalar Potential Φ
Including derivatives up to second order, that equation for the scalar potential, Φ, reads
κΦ,tt
∂2Φ
+ ∆Φ + κΦΦ Φ = κΦρ ρ
∂t2
(38)
The coefficients are CPT-invariant as above. This equation has numerous applications.
9.1.1
(Schrödinger-Fock-)Klein-Gordon and Proca Equations
The homogenous time-dependent case (κΦ,tt < 0, κΦρ = 0) is known as Klein-Gordon equation
[20] (also Klein-Fock-Gordon [21], Klein-Gordon-Schrödinger), where Φ represents the scalar wave
function of massive spin-0 particles. In the Proca equation [22], Φ represents the 4-vector wave
function of massive spin-1 particles. In both cases, κΦΦ = −(mc/~)2 .
9.1.2
Screened Poisson equation
The time-independent case of the equation (38) is the screened Poisson (inhomogenous Helmholtz)
equation,
∆Φ + κΦΦ Φ = κΦρ ρ.
(39)
Here, κΦρ depends on the units of measurement (as κE above). κΦΦ , the inverse screening length
squared, depends on the application.
• In Yukawa’s [23] meson theory of nuclear forces, κΦΦ is proportional to the meson mass.
• In Neumann’s [24] and Seeliger’s [25] theories of gravity, κΦΦ serves to make the potential of
an infinite homogeneous mass density finite.
Φ=
ρ
κΦρ κΦΦ
(40)
• κΦΦ quantifies the linear screening in systems with a high concentration of charge carriers
(strong electrolytes, metals, highly doped semiconductors).
−1/2
• In Fritz and Heinz London’s theory of superconductors [26], κΦΦ
the magnetic field into the superconductor.
is the penetration depth of
Again, the point is that all those model equations are not only restricted by symmetry (CPT ), but also
related to point-mechanical principles.
9.2
~
CPT Invariant Equation for the Vector Potential A
Analogously to the rather general CPT invariant equation with derivatives up to second order (38)
~ can be established. This time, all
for the scalar potential, Φ, an equation for the vector potential, A,
~
expressions which have got the same CPT symmetry as A are collected. The result is
κ̂A,tt
~
~
∂2A
~ + κ̂Acc ∇ × ∇ × A
~ + κ̂AA A
~ = κ̂Aj~j + κ̂AE ∂ E
+ ∆A
2
∂t
∂t
(41)
I’m not aware of an application of this equation. For this and for the sake of generality, I have allowed
for matrix-valued coefficients.
10
CONCLUSION
(Please check both CONCLUSION sections)
I have examined Hehl & Obukhov’s principles of electromagnetism w.r.t. to their relationship to
classical mechanics and found that they can be traced back to principles adopted by Newton, Lagrange,
Helmholtz and Hertz. This is hopefully a significant step toward the unity of classical physics as well
as understanding the relationship of quantum electrodynamics and quantum mechanics.
The Lorentz force evolves out of the connection of Helmholtz’s investigations of the relationship
between forces and energy with Lagrange’s equation of motion (2). This concerns, in particular,
the force
~Lipschitz := ~v × K(t,
~ ~r, ~v , ~a, . . .)
F
(42)
which acts always perpendicular to the velocilty vector of a body and hence does not change its
kinetic energy. Due to this meaning, I propose to term it ’Lipschitz force’. As a consequence, not only
a conservative force, −∇V (~r), constitutes a system, in which the total energy is conserved, but also
the force
~Helmholtz (t, ~r, ~v , ~a, . . .) = −∇V (~r) + F
~Lipschitz (t, ~r, ~v , ~a, . . .).
F
(43)
Due to this significance, I propose to call it ’Helmholtz force’.
Now, only a static Helmholtz force is compatible with Lagrange’s equation of motion (2). The simplest
time-dependent generalization of the Helmholtz force (43), which is compatible with Lagrange’s equation
of motion (2), is
~Helmholtz,gen (~r, t) = F
~Lipschitz (~r, t) + F
~non-cons (~r, t)
F
~
~ r, t) − ∇U0 (~r, t) + ∂ U1 (~r, t),
= ~v × K(~
∂t
(44a)
(44b)
The fields are introduced analogously to Newton’s factorization of the force of gravity into mass and
~ 1 are written as a product of charge and scalar
acceleration. Accordingly, the auxiliary fields U0 and U
and vector potentials, respectively.
For the Maxwell equations and the Lorentz force, it plays no role that there is an elementary quantum
of charge. This allows for deriving Heaviside’s gravito-electromagnetic equations in parallel with
the Maxwell-Lorentz equations. The differences to the linearized equations of general relativity are
beyond this approach, of course. Moreover, one has to account for the fact that equal charges repel
each other, while equal masses attract each other.
Similar to other approaches, the two homogeneous (gravito-)electromagnetic equations emerge as
existence and consistency conditions, respectively. The two inhomogeneous ones result largely from
CPT symmetry, Hertz’s interaction principle, and conservation of charge and gravitating mass. The
latter one is not conserved within the special and general theories of relativity, but this fact is beyond
the scope of this contribution.
As expressed clearly in his 3rd letter to Bentley [27], more indirectly in the explanations to Definition 8
of the ’Principia’ [5], and also in Query 31 of his ’Opticks’ [28], for Newton, an action between distant
bodies without any medium transferring the action between them was an ”absurdity” (cf. also [29], pp.
61f.). Within (gravito-)electromagnetism, this role is played by the (gravito-)electromagnetic vacuum
[18]. It propagates the fields from their source ”through the sourrounding regions” (’Principia’ [5],
Definition 6).
The rationalized Maxwell equations (3) state that there are four physically different fields. Two of
~ and H,
~ are caused by the charges and currents - the other two, E
~ and B,
~ act back onto
them, D
them. As a consequence, the coefficients ε0 and µ0 are not merely conversion factors of units of
~ H
~ and E/
~ B?
~
measurement. Then, which are the relationships between D/
A modern variant of such thoughts is Hehl & Obukhov’s premetric treatment [2], where the metric
evolves out off the constitutive relations. It is not clear whether this goal can be tackled along
the paths exploited here. The purely dynamic derivation of the Lorentz factor and thus of special-
relativistic mechanics through a natural generalization of Euler’s derivation of Newton’s equation of
motion [11] demonstrates, however, that it is principally possible to make dynamical statements about
the structure of space-time. The special-relativistic correction of the inertial term in the NewtonLorentz equation of motion,
m
d2~r
~ + q~v × B.
~
= qE
dt2
(45)
can also be obtained from the Lorentz transformation of its r.h.s.
ACKNOWLEDGEMENTS
I feel highly indebted to Prof. W.-F. Hehl for pointing my attention to Bopp’s paper and for numerous
discussions about his premetric approach to electromagnetism. I would also like to thank Profs.
C. Kiefer and N. Straumann for helpful correspondences as well as Prof. J. López-Bonilla and
the Deutsche Akademie der Naturforscher Leopoldina for encouraging early stages of this work.
The recent stages have benefited from my discussions with Vladimir Onoochin on various topics of
electromagnetism. Last but not least, I am grateful to the anonymous reviewers for several comments
and suggestions which contributed to improve this paper.
COMPETING INTERESTS
The author has declared that no competing interests exist.
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Biography of author(s)
Peter Enders
Kazakh National Pedagogical Abai University, Almaty, Kazakhstan.
He obtained his PhD from Moscow State Lomonosov University and his Dr. habil. from Humboldt University at
Berlin. Longer visits led him to the University of East Anglia, Norwich, and the University of Illinois at UrbanaChampaign. His experience comprehends physics and quantum chemistry of solids, non-linear phenomena and
phase transitions, modelling semiconductor lasers, and teaching and administration at universities of applied
sciences in Germany and Kazakhstan. His current research includes the axiomatics and unity of, (i), classical
and quantum mechanics, (ii), classical and quantum statistical mechanics, (iii), classical and special-relativistic
mechanics, (iv), classical mechanics and electromagnetism.
—————————————————————————————————————————————————© Copyright (2021): Author(s). The licensee is the publisher (B P International).
DISCLAIMER
This chapter is an extended version of the article published by the same author in the following journal.
Adv. Studies Theor. Phys., 2(5): 199–214, 2008.