# Moving Manifolds in Electromagnetic Fields

David V. Svintradze<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics, Tbilisi State University, Chavchavadze Ave. 03, Tbilisi 0179, Georgia

###### Abstract

We propose dynamic non-linear equations for moving surfaces in electromagnetic
field. The field is induced by a material body with a boundary of the surface.
Correspondingly the potential energy, set by the field at the boundary, can be
written as an addition of four-potential times four-current to a contraction
of electromagnetic tensor. Proper application of minimal action principle to
the system Lagrangian yields dynamic non-linear equations for moving three
dimensional manifolds in electromagnetic fields. The equations, in different
conditions simplify to Maxwell equations for massless three surfaces, to Euler
equations for dynamic fluid, to magneto-hydrodynamic equations and to Poisson-
Boltzmann equation. To illustrate effectiveness of the equations of motion we
apply the formalism to analyze dynamics of macro-molecules and membranes.

††preprint: APS/123-QED

##  I introduction

Fluid dynamics is one of the most well understood subjects in classical
physics Landau and Lifshitz (1959) and yet continues to be an actively
developing field of research even today. Fluid dynamics can be treated as a
motion of an inviscid fluid, as an indivisible medium of particles or as a
collective motion of many body system particles. In the first case, when the
fluid is inviscid and indivisible, the two conditions allow formulation of the
Euler equation for dynamic fluid and the equation of continuity, where the
Euler equation is a direct consequence of Newton’s second law Landau and
Lifshitz (1959). The second case is the most complicated and is difficult to
treat. There are two possibilities for dealing with the second case: treat
each separate particle as an individual one and propose that each particle
satisfies Newton’s laws of motion,111If one proposes to treat particles as
classical objects, then the framework fits in Newtonian mechanics. The
application of Newton laws and it’s stochastic generalizations in simulations
is commonly known as molecular dynamics simulations. or treat each particle as
a vertex of a geometric figure and search for equations of motion for such
geometries. If smoothed, such geometries for a sufficient number of particles
can be modeled as continuously differentiable two manifolds embedded in
Euclidean space (classical limit), or continuously differentiable three
manifolds embedded in Minkowskian space-time (relativistic limit). Discussion
of fluid dynamics in Minkowskian space-time corresponds to the fully
relativistic formulation of the problem, while fluid dynamics in Euclidean
space corresponds to the non-relativistic limit and is a specific case.

An example of fluid dynamics modeling as moving surfaces embedded in Euclidian
space is moving two dimensional surfaces of fluid films such as soap films.
Another, biologically relevant examples are dynamic fluid membranes, vesicles
and micelles where large body of notable theoretical results had already been
produced Seifert (1997); Lipowsky and Sackmann (1995).

Soap films can be formed by dipping a closed contour wire or by dipping two
rings into the soapy solution. Stationary fluid films, or films in mechanical
equilibrium with the environment, form a surface with minimal surface area.
Usually surfaces such as soap films are modelled as two dimensional manifolds.
Fluid films not in mechanical equilibrium may have large displacements and can
undergo big deformations W. Drenckhan and Elias (2008); M. Brazovskaia and
Pieranski (1996); Kraus  _et al._ (1998); Seychelles  _et al._ (2008);
Grinfeld (2010a); Manor  _et al._ (2015). The order of magnitude of thickness
variations may vary from the nanometer to millimeter scale.

The equations of motion for free liquid films were initially proposed in
Grinfeld (2009) based on the least action principle of the Lagrangian:

$L=\frac{1}{2}\int_{S}{\rho{(C^{2}+V^{2})}}dS-\sigma\int_{S}dS$ (1)

where $\rho$ is the two dimensional mass density of the fluid film, $C$ is
interface velocity, $V$ is tangential velocity, $\sigma$ is surface tension,
$S$ stands for the surface and free means that interactions with ambient
environment are ignored. Numerical solutions of the dynamic nonlinear
equations for free thin fluid films display a number of new features
consistent with experiments Grinfeld (2010a).

As indicated above, fluid dynamics can be described by motion of fluid
surfaces, where the motion can happen in Euclidean ambient space,
corresponding to the non-relativistic case or in Minkowski ambient space,
corresponding to the fully relativistic case. Minkowskian space-time is more
general and we will carry out derivations in Minkowski space that can be
trivially simplified for non-relativistic cases. Instead of motion of free
fluid films, we discuss motion of charged or partially charged material bodies
with the boundary of charged or partially charged surfaces222e.g. bio-
membranes, macromolecular surfaces, lipid bilayers, micelles, etc. in aqueous
solution making hydrophobic-hydrophilic interactions. Hydrophobic-hydrophilic
interactions are represented as electromagnetic interactions for reasons
explained below. Representation of surfaces requires physical modeling and is
illustrated in the physical models subsection for biomacromolecular surfaces.
To be applicable to biological problems, we take the environment to be aqueous
solution, though the medium does not directly enter into the general equations
for free moving surfaces, so the equations can be applied to any moving
surfaces in electromagnetic field. We propose in this paper the modeling of
fluid dynamics as moving surfaces in an electromagnetic field and consequently
show that this concept non-trivially generalizes classical fluid dynamics. We
pursue fully relativistic calculations because for instance for biological
macromolecules, femtosecond observations revealed that surface deformations,
induced by dynamics of hydration at the surface or by charge transfer for
proteins or DNA, usually happening from angstrom to nanometer scale, may occur
as fast as from femtosecond to picosecond Pal  _et al._ (2002); Wan  _et al._
(2000). This sets upper limit for the interface velocity as high as $C\sim
nm/fs=10^{6}m/s$ and should be incorporated in fully relativistic
framework.333The formalism should be relativistic not only because that
relativistic calculations are more general then classical calculations, or
proper electrodynamics description requires relativistic frame work, but
because that the molecular surface dynamics can be very fast Pal  _et al._
(2002); Wan  _et al._ (2000).

The theoretical concept of hydrophobicity is already developed Lum  _et al._
(1999); Chandler and Garrahan (2010) and is used to simulate shape dependence
on hydrophobic interactions Patel  _et al._ (2011); Rotenberg  _et al._
(2011); Patel  _et al._ (2012); Limmer  _et al._ (2013). Although the basic
principles of the hydrophobic effect are qualitatively well understood, only
recently have theoretical developments begun to explain quantitatively many
features of the phenomenon Chandler (2005).

Hydrophobic and hydrophilic interactions can be described as dispersive
interactions between permanent or induced dipoles and ionic interactions
throughout the molecules Chandler (2005); Leikin  _et al._ (1993). Unification
of all these interactions in one is the electromagnetic interaction’s
dependence on the interacting body’s geometries Svintradze (2010, 2015, 2013,
2009). To lay a foundation for the description of such geometric dependence,
we give exact nonlinear equations governing geometric motion of the surface in
an electromagnetic field set up by dipole moments of water molecules and
partial charges of various molecules.

In the paper we discuss motion of compact and closed manifolds induced by
electromagnetic field, where the field is generated by a continuously
distributed charge in the material body. The boundary of the body is a semi-
permeable surface (manifold) with a charge (or partial charge) and the charge
can flow through the surface. Since, the charge in general is heterogeneously
distributed in the body, the charge flow induces time variable electromagnetic
field on the surface of the body, forcing the motion of the manifold.
Consequently, the problem is to find an equation of motion of moving manifolds
in the electromagnetic field. Actuality of the problem may be connected to
many physics sub-fields, for instance fluid dynamics, membrane dynamics or
molecular surface dynamics. For instance the surface of macromolecules in
aqueous solutions is permeable to some ions and water molecules and the charge
on the surface is heterogeneously distributed. Flow of some ions and water
molecules through the surface and uneven distribution of the charge in the
macromolecules induces the surface dynamics. Same happens to biological
membranes, vesicles, micelles and etc. Here we deduce general partial
differential equations of moving manifolds in electromagnetic field and
demonstrate that the equations, in different conditions, simplify to Euler
equation for fluid dynamics, Poisson-Boltzmann equation for describing the
electric potential distribution on the surface and Maxwell equations for
electrodynamics.

The formalism presented in this paper can be easily extended to hypersurfaces
of any dimension. The limitation by three surfaces embedded in four space-
time, which is necessary to describe electromagnetism Vanderlinde (2004), is a
consequence of specificity of the processes that take place on macromolecular
surfaces. The surface of macromolecules in aqueous solutions is permeable to
some ions and water molecules and the charge on the surface is heterogeneously
distributed. Time frame for dynamics of water molecules on the surface can be
of femtosecond range. Therefor the surface can be charged with variable mass
and charge densities and is continuously deformable. Mathematically the
problem formulates as: find equations of motion in electromagnetic field for a
closed, continuously differentiated and smooth two dimensional manifold in
Euclidean space (non-relativistic case) or three manifolds in Minkowski space-
time (relativistic case). Dynamics of the surfaces under the influence of
potential energy arises from four-potential time four-current and contraction
of the electromagnetic tensor. Kinetic energy of the manifolds is calculated
according to the calculus of moving surfaces Grinfeld (2010b). Potential
energy set by the object is modeled by the electromagnetic tensor in the same
way as for Maxwell’s equations. Definition of the Lagrangian Svintradze (2015)
by subtracting potential energy from the kinetic energy and setting the
minimum action principal yields nonlinear equations for moving surfaces in
electromagnetic field.

Since Minkowskian space-time does not follow Riemannian geometry, we need a
small adjustment of definitions. For Minkowski space-time, which fits to
pseudo-Riemannian geometry, we need definitions of arbitrary base pairs of
ambient space, even though the definitions look exactly same as for Riemannian
geometry embedded in Euclidean ambient space Grinfeld (2010b); Weyl (1952);
Svintradze (2016). The summarized relationships about Riemannian geometry
embedded in Euclidean space are given in tensor calculus text books Grinfeld
(2010b); Weyl (1952).

##  II theoretical preliminaries

###  II.1 Embedded manifolds in ambient Minkowksi space

Combination of three ordinary dimensions with the single time dimension forms
a four-dimensional manifold and represents Minkowski space-time. In this
framework Minkowski four-dimensional space-time is the mathematical model of
physical space in which Einstein’s general theory is formulated. Minkowski
space is independent of the inertial frame of reference and is a consequence
of the postulates of special relativity Weyl (1952); Landau and Lifshitz
(1971).

Euclidean space is the flat analog of Riemannian geometry while Minkowski
space is considered as the flat analog of curved space-time, which is known in
mathematics as pseudo-Riemannian geometry. Considerations of four-dimensional
space-time makes embedded moving manifolds three dimensional, where parametric
time $t$, describing motion of manifolds, may not have anything to do with
proper time $\tau$ used in general relativity.

To briefly describe Minkowskian space-time, let us refer to arbitrary
coordinates $X^{\alpha}$, $\alpha=0,...,3$, where the position vector $\bm{R}$
is expressed in coordinates as $\bm{R}=\bm{R}(X^{\alpha})$. Bold letters,
throughout the manuscript designate vectors. Latin letters in indexes indicate
surface related tensors. Greek letters in indexes show tensors related to the
ambient space. All equations are fully tensorial and follow the Einstein
summation convention.

Suppose that $S^{i}$ ($i=0,1,2$) are the surface coordinates of the moving
manifold $S$ (Fig. 1).

Figure 1:  Two dimensional illustration of a curved three dimensional surface
embedded in Minkowski space-time. $X^{\alpha}$ represents analog of Cartesian
coordinates of Minkowski space-time. $\bm{S}_{i}$ is base vectors defined in
tangent space and $X^{\alpha}=X^{\alpha}(t,S^{i})$ is the general equation of
the surface.

Coordinates $S^{i}$, $X^{\alpha}$ are arbitrary chosen so that sufficient
differentiation is achieved in both space and parametric time. The surface
equation in ambient coordinates can be written as
${X^{\alpha}=X^{\alpha}(t,S^{i})}$ and the position vector can be expressed as

$\bm{R}=\bm{R}(X^{\alpha})=\bm{R}(t,S^{i})$ (2)

Covariant bases for the ambient space are introduced as
${\bm{X}_{\alpha}=\partial_{\alpha}\bm{R}}$, where
$\partial_{\alpha}=\partial/{\partial{X^{\alpha}}}$. The covariant metric
tensor is the dot product of covariant bases

$\eta_{\alpha\beta}=\bm{X}_{\alpha}\cdot{\bm{X}_{\beta}}$ (3)

The contravariant metric tensor is defined as the matrix inverse of the
covariant metric tensor, so that
${\eta^{\alpha\nu}\eta_{\nu\beta}=\delta^{\alpha}_{\beta}}$, where
$\delta^{\alpha}_{\beta}$ is the Kronecker delta. From definition (3) follows
that $\eta_{00}=\bm{X}_{0}\cdot\bm{X}_{0}$ consequently if for Minkowskian
space-time, the space like signature is set $(-1,+1,+1,+1)$, then
${\bm{X}_{0}=(i,0,0,0)}$.444here the speed of light is set to be unit $c=1$
and $i$ stands for imaginary number. Therefore, vector components are complex
numbers in general. As far as the ambient space is set to be Minkowskian, the
covariant bases are linearly independent, so that the square root of the
negative metric tensor determinant is unit $\sqrt{-|\eta_{..}|}=1$.
Furthermore, the Christoffel symbols given by

${\Gamma_{\beta\gamma}^{\alpha}=\bm{X}^{\alpha}\cdot\partial_{\beta}\bm{X}_{\gamma}}$

vanish and the equality between partial and curvilinear derivatives follows
$\partial_{\alpha}=\nabla_{\alpha}$. As far as, in Minkowski space-time (later
space), $\partial_{\alpha}$ partial derivative and $\nabla_{\alpha}$
curvilinear derivative are the same, everywhere in calculations we use
$\partial$ letter for the ambient space derivative and keep in mind that when
referring to Minkowski space the derivative has index in Greek letters and, in
that case, it is same as partial derivative. When indexes are mixed Greek and
Latin letters the last statement, as it is shown below, does not hold in
general.

Now let’s discuss tensors on the embedded surface with arbitrary coordinates
$S^{i}$, where $i=0,1,2$. Latin indexes throughout the text are used
exclusively for curved surfaces and curvilinear derivative $\nabla_{i}$ is no
longer the same as the partial derivative
$\partial_{i}=\partial/{\partial{S^{i}}}$. Similar to the bases of ambient
space, covariant bases of an embedded manifold are defined as
$\bm{S}_{i}=\partial_{i}\bm{R}$ and the covariant surface metric tensor is the
dot product of the covariant surface bases:

$g_{ij}=\bm{S}_{i}\cdot\bm{S}_{j}$ (4)

The definition (4) dictates that the surface is three dimensional pseudo
Riemannian manifold, because ambient space is four dimensional Minkowskian
space and the surface in four manifold is three manifold.

analogically to space metric tensor, $g^{ij}$ the contravariant surface metric
tensor is the matrix inverse of the covariant one $g_{ij}$. The matrix inverse
nature of covariant-contravariant metrics gives possibilities to raise and
lower indexes of tensors defined on the manifold. The surface Christoffel
symbols are given by

${\Gamma_{jk}^{i}=\bm{S}^{i}\cdot\partial_{j}\bm{S}_{k}}$

and along with Christoffel symbols of the ambient space provide all the
necessary tools for covariant derivatives to be defined as tensors with mixed
space/surface indexes:

$\displaystyle\nabla_{i}T_{\beta k}^{\alpha j}=$
$\displaystyle\partial_{i}T_{\beta k}^{\alpha
j}+\eta_{i}^{\gamma}\Gamma_{\gamma\nu}^{\alpha}T_{\beta k}^{\nu
j}-\eta_{i}^{\gamma}\Gamma_{\gamma\beta}^{\mu}T_{\mu k}^{\alpha j}$
$\displaystyle+\Gamma_{im}^{j}T_{\beta k}^{\alpha m}-\Gamma_{ik}^{m}T_{\beta
m}^{\alpha j}$ (5)

where $\eta_{i}^{\gamma}$ is the shift tensor which reciprocally shifts space
bases to surface bases, as well as space metric to surface metric. For
instance, $\bm{S}_{i}=\eta_{i}^{\alpha}\bm{X}_{\alpha}$ and

${g_{ij}=\bm{S}_{i}\cdot\bm{S}_{j}=\eta_{i}^{\alpha}\bm{X}_{\alpha}\eta_{j}^{\beta}\bm{X}_{\beta}=\eta_{i}^{\alpha}\eta_{j}^{\beta}\eta_{\alpha\beta}}$

Metrilinic property $\nabla_{i}g_{mn}=0$ of the surface metric tensor is
direct consequence of (4,5) definitions, therefore
${\bm{S}_{m}\cdot\nabla_{i}\bm{S}_{n}=0}$. The $\bm{S}_{m}$ and
$\nabla_{i}\bm{S}_{n}$ vectors are orthogonal, so that $\nabla_{i}\bm{S}_{n}$
must be parallel to $\bm{N}$ surface normal

$\nabla_{i}\bm{S}_{j}=\bm{N}B_{ij}$ (6)

where $B_{ij}$ is the tensorial coefficient of the (6) relationship and is
generally referred as the symmetric curvature tensor. The trace of the
curvature tensor with upper and lower indexes is the mean curvature and its
determinant is the Gaussian curvature. It is well known that a surface with
constant Gaussian curvature is a sphere, consequently a sphere can be
expressed as:

$B_{i}^{i}=const$ (7)

When the constant becomes null the surface becomes either a plane or a
cylinder. (7) is the expression of constant mean curvature (CMC) surfaces in
general. Finding the curvature tensor defines the way of finding covariant
derivatives of surface base vectors and as so, (6,7) provide the way of
finding surface base vectors which indirectly leads to the identification of
the surface.

###  II.2 Differential geometry for embedded moving manifolds

After defining metric tensor for ambient space $\eta_{\mu\nu}$ (3) and metric
tensor for moving surface $g_{ij}$ (4) we now proceed with brief review of
surface velocity, $t$ explicit (parametric) time derivative of surface tensors
and time differentiation theorems for the surface/space integrals. The
original definitions of time derivatives for moving surfaces were given in
Hadamard (1968) and recently extended in Grinfeld (2010b).

For the definition of surface velocity we need to define ambient coordinate
velocity $V^{\alpha}$ first and to show that the coordinate velocity is
$\alpha$ component of the surface velocity. Indeed, by the velocity definition

$V^{\alpha}=\frac{\partial X^{\alpha}}{\partial t}$ (8)

Taking into account (2), the position vector $\bm{R}$ is tracking the material
point coordinate $S^{i}$, therefore by the partial time differentiation of (2)
and definition of ambient base vectors, we find that $\bm{V}$ surface velocity
is

$\bm{V}=\frac{\partial\bm{R}(t,S^{i})}{\partial
t}=\frac{\partial\bm{R}}{\partial X^{\alpha}}\frac{\partial
X^{\alpha}(t,S^{i})}{\partial t}=V^{\alpha}\bm{X}_{\alpha}$ (9)

Consequently $V^{\alpha}$ is ambient component of the surface velocity.
According to (9), normal component of the surface velocity is the dot product
with the surface normal

$C=\bm{V}\cdot\bm{N}=V_{\alpha}\bm{X}^{\alpha}N^{\beta}\bm{X}_{\beta}=V_{\alpha}N^{\beta}\eta_{\beta}^{\alpha}=V_{\alpha}N^{\alpha}$
(10)

The normal component $C$ of the surface velocity is generally referred as an
interface velocity and is invariant in contrast with coordinate velocity
$V^{\alpha}$. Its sign depends on a choice of the normal. The projection of
the surface velocity on the tangent space (Fig. 2) Svintradze (2016) is
tangential velocity and can be expressed as

$V^{i}=V^{\alpha}\eta_{\alpha}^{i}$ (11)

Graphical illustrations of coordinate velocity $V^{\alpha}$, interface
velocity $C$ and tangential velocity $V^{i}$ are given on Fig. 2.

Figure 2:  2D Graphical illustration of the arbitrary chosen three manifold
and it’s local tangent space. $\bm{S}_{0}$, $\bm{S}_{1}$, $\bm{S}_{2}$ and
$\bm{N}$ are local tangent space base vectors and the normal respectively.
$\bm{V}$ is arbitrary chosen surface velocity and $C$, $V_{i},i=(0,1,2)$
display the projection of the velocity to the $\bm{N},\bm{S}_{i}$ directions.

There is a clear geometric interpretation of the interface velocity Grinfeld
(2010b); Svintradze (2016) and can be expressed as

$\bm{V}=C\bm{N}+V^{i}\bm{S}_{i}$ (12)

Let the surface at two nearby moments of time $t$ and $t+\Delta t$ be $S_{t}$,
$S_{t+\Delta t}$ correspondingly. Suppose that $A$ is a point on the $S_{t}$
surface and the corresponding point $B$, belonging to the $S_{t+\Delta t}$,
has the same surface coordinate as $A$ (Fig. 3), then
${\bm{AB}\approx\bm{V}\Delta t}$. Let $P$ be the point, where the unit normal
$\bm{N}\in S_{t}$ intersects the surface $S_{t+\Delta t}$, then for small
enough $\Delta t$, the angle ${\angle APB\approx\pi/2}$ and
$AP\approx\bm{V}\cdot\bm{N}\Delta t$, therefor, $C$ can be defined as

$C=\lim_{\Delta t\rightarrow 0}\frac{AP}{\Delta t}$

and can be interpreted as the instantaneous velocity of the interface in the
normal direction. It is worth of mentioning that the sign of the interface
velocity depends on the choice of the normal. Although $C$ is a scalar, it is
called interface velocity because the normal direction is implied.

###  II.3 Time derivative

In this section we are briefly explaining concept behind the invariant time
derivative for scalar and tensor fields defined on moving manifolds, even
though these concepts are already given Grinfeld (2010b).

Figure 3:  Geometric interpretation of the invariant time derivative
$\dot{\nabla}$ applied to invariant tensor field $T$. $A$ is an arbitrary
chosen point on the $S_{t}$ surface so that it lays on the $T(S_{t})$ curve.
$B$ is corresponding point on the $S_{t+\Delta t}$ surface. $P$ is the point
where $S_{t}$ surface normal, applied on the point $A$, intersects the surface
$S_{t+\Delta t}$. For infinitely small $\Delta t$ $\bm{AB}\approx\bm{V}\Delta
t$ and $AP\approx\bm{V}\cdot\bm{N}\Delta t$. According to geometric
construction the tensor field $T$ in the point $B$ can be estimated as
$T(B)\approx T(A)+\Delta t\partial T/\partial t$, while $T(B)$ can be
estimated by covariant surface derivative $T(B)\approx T(P)+\Delta
tV^{i}\nabla_{i}T$, where $\nabla_{i}T$ shows rate of change in $T$ along the
directed distance $BP\approx\Delta tV^{i}$ on the surface $S_{t+\Delta t}$.

Suppose that invariant tensor field $T$ is defined on the manifold at all
time. To define time invariant derivative of the tensor field it is necessary
to capture the rate of change of $T$ in the normal direction. Physical
explanation of why the deformations along the normal direction are so
important, we give below when discussing integrals. This is similar to how $C$
measures the rate of deformation in the normal direction. Let for a given
point $A\in S_{t}$, find the point $B\in S_{t+\Delta t}$ and $P$ the
intersection of $S_{t+\Delta t}$ and the straight line orthogonal to $S_{t}$
(Fig. 3). Then the geometrically intuitive definition dictates that

$\dot{\nabla}T=\lim_{\Delta t\rightarrow 0}\frac{T(P)-T(A)}{\Delta t}$ (13)

because of (13) is entirely geometric, it mast be free from choice of a
reference frame, therefore it is invariant. On the other hand, from the
geometric construction follows that

$T(B)\approx T(A)+\Delta t\frac{\partial T}{\partial t}$ (14)

$T(B)$ is related to $T(P)$ because $B$,$P$ are nearby points and are situated
on the $S_{t+\Delta t}$ surface $B,P\in S_{t+\Delta t}$, then

$T(B)\approx T(P)+\Delta tV^{i}\nabla_{i}T$ (15)

since $\nabla_{i}T$ shows rate of change in the tensor field along the surface
and $\Delta tV^{i}$ indicates the directed distance $BP$. After few lines of
algebra, taking into account equations (14,15) in (13), we find

$\dot{\nabla}T=\frac{\partial T}{\partial t}-V^{i}\nabla_{i}T$ (16)

Generalization of (16) to any arbitrary tensors with mixed space and surface
indexes is given by the formula

$\displaystyle\dot{\nabla}T_{\beta j}^{\alpha i}=$
$\displaystyle\frac{\partial T_{\beta j}^{\alpha i}}{\partial
t}-V^{k}\nabla_{k}T_{\beta j}^{\alpha
i}+V^{\gamma}\Gamma_{\gamma\mu}^{\alpha}T_{\beta j}^{\mu
i}-V^{\gamma}\Gamma_{\gamma\beta}^{\mu}T_{\mu j}^{\alpha i}$
$\displaystyle+\dot{\Gamma}_{k}^{i}T_{\beta j}^{\alpha
k}-\dot{\Gamma}_{j}^{k}T_{\beta k}^{\alpha i}$ (17)

Where Christoffel symbol $\dot{\Gamma}_{m}^{n}$ for moving surfaces is
${\dot{\Gamma}_{m}^{n}=\nabla_{m}V^{n}-CB_{m}^{n}}$. The derivative commutes
with contraction, satisfies sum, product and chain rules, is metrinilic with
respect to the ambient metrics and does not commute with the surface
derivative Grinfeld (2010b). Also from (13) it is clear that the invariant
time derivative applied to time independent scalar vanishes.

###  II.4 Time derivatives of space/surface integrals

In evaluation of the least action principal of the Lagrangian there is a
central role for time differentiation of the surface and space integrals, from
which the geometry dependence of the potential energy is rigorously clarified.
For any scalar field $T=T(t,S^{i})$ defined on a Minkwoskian domain $\Omega$
with boundary $S$ manifold evolving with the interface velocity $C$, the
evolution of the space integral and surface integral for closed compact
manifolds are given by the formulas

$\displaystyle\frac{d}{dt}\int_{\Omega}Td\Omega$
$\displaystyle=\int_{\Omega}\frac{\partial T}{\partial t}d\Omega+\int_{S}CTdS$
(18) $\displaystyle\frac{d}{dt}\int_{S}TdS$
$\displaystyle=\int_{S}\dot{\nabla}TdS-\int_{S}CTdS$ (19)

The first term in the integral represents the rate of change of the tensor
field, while the second term shows changes in the geometry, therefore properly
takes into account the convective and advective terms due to volume motion. We
are not going to reproduce proof of these theorems here,555proofs about the
time derivative of integrals can be found in tensor calculus text books, see
for instance Grinfeld (2010b) and references therein. but instead we give
intuitive explanation of why only interface velocity has the role and
tangential velocities do not appear in the integration. If the surface
velocity has no interface velocity and has only tangential components, then
the tangent velocity translates each points to it’s neighboring ones and does
not add new area and volume to the surface and space. Therefore, it provokes
rotational movement of the material object and can be excluded from the
integration. This statement becomes obvious for one dimensional motion. If the
material point is moving along some trajectory, then the velocity is
tangential to the curve. However, motion of the material point along the curve
can be understood as the motion of the curve embedded in the plane. If the
curve has no interface velocity then it only slides in the ambient plane
without changing the local length.666same explanation, with more details, is
given in Svintradze (2016).

###  II.5 Several useful theorems

In this section we provide several theorems, which will be directly used to
deduce equations of motions. First such theorem is general Gauss theorem about
integration, which gives the rule to vise verse transfer space integral to
surface integral. For a domain $\Omega$ in Minkowski space with the boundary
$S$, for any sufficiently smooth tensor field $T^{\alpha}$, the Gauss theorem
reads

$\int_{\Omega}\nabla_{\alpha}T^{\alpha}d\Omega=\int_{S}N_{\alpha}T^{\alpha}dS$
(20)

Proof is pretty simple if one uses Voss-Weyl formula to deduce the theorem.
For any sufficiently smooth tensor field in Minkowski space, Voss-Weyl formula
Grinfeld (2010b) reads

$\nabla_{\mu}T^{\mu}=\frac{1}{\sqrt{-|\eta_{..}|}}\partial_{\mu}(\sqrt{-|\eta_{..}|}T^{\mu})$
(21)

Using (21) in right part of (20) and designation $\eta=-|\eta_{..}|$, we have

$\displaystyle\int_{\Omega}\nabla_{\alpha}T^{\alpha}d\Omega$
$\displaystyle=\int_{\Omega}\frac{1}{\sqrt{\eta}}\partial_{\mu}(\sqrt{\eta}T^{\mu})d\Omega$
$\displaystyle=\iiiint\frac{1}{\sqrt{\eta}}\partial_{\mu}(\sqrt{\eta}T^{\mu})\sqrt{\eta}dX^{\alpha}$
$\displaystyle=\iiiint\partial_{\mu}(\sqrt{\eta}T^{\mu})dX^{\alpha}$

where $dX^{\alpha}=dX^{0}dX^{1}dX^{2}dX^{3}$. This term is subject to Gauss’s
theorem in the arithmetic space. Since, the arithmetic space and Minkowski
space, which is a pseudo-Euclidean, can be corresponded to the Cartesian
coordinates, the Minkowski space can be identified as arithmetic one and the
Gauss theorem for the arithmetic space can be used. Thus, using unity of
Minkowski space metric tensor determinant one may prove that777details about
the proof, for Euclidean space, can be found in tensor calculus text book
Grinfeld (2010b) and proof for Minkowski space is identical to Euclidean one.

$\displaystyle\iiiint\partial_{\mu}(\sqrt{\eta}T^{\mu})dX^{\alpha}$
$\displaystyle=\iiint N_{\alpha}T^{\alpha}\sqrt{\eta}\sqrt{g}dS^{i}$
$\displaystyle=\int_{S}N_{\alpha}T^{\alpha}dS$

where $g=|g_{..}|$. This proves that generalized Gauss’s theorem holds for
pseudo-Riemannian manifolds embedded in Minkowski space.

Next step is to provide short proofs for Weingarten’s and Thomas formulas by
using the relation between the surface derivative and the interface velocity.

Weingarten’s formula expresses surface covariant derivative of the surface
normal in the product of the shift and mixed curvature tensors. Proof follows
from the definition $N_{\alpha}N^{\alpha}=1$, from where we find
$N_{\alpha}\nabla_{i}N^{\alpha}=0$. On the other hand

$\displaystyle
0=\bm{N}\cdot\bm{S}_{i}=N^{\alpha}\bm{X}_{\alpha}\cdot\eta^{\beta}_{i}\bm{X}_{\beta}$
$\displaystyle=N^{\alpha}\eta_{i}^{\beta}\eta_{\alpha\beta}$
$\displaystyle=N^{\alpha}\eta_{\alpha i}$ (22)

If we apply covariant derivative to (22) and take into account that from (6)
follows $B_{ji}=N^{\alpha}\nabla_{j}\eta_{i\alpha}$ then by the product rule
we find

$\displaystyle 0=\nabla_{j}N^{\alpha}\eta_{\alpha
i}+N^{\alpha}\nabla_{j}\eta_{\alpha i}$
$\displaystyle=\nabla_{j}N^{\alpha}\eta_{\alpha i}+B_{ji}$
$\displaystyle\nabla_{j}N^{\alpha}\eta_{\alpha i}$ $\displaystyle=-B_{ji}$
(23)

Let’s contract both sides of (23) with $\eta^{i\beta}$ and take into account
commonly used relationship in tensor calculus
${N^{\alpha}N_{\beta}+\eta_{i}^{\alpha}\eta_{\beta}^{i}=\delta_{\beta}^{\alpha}}$,
then we find

$\displaystyle-\eta^{\beta}_{k}B_{j}^{k}$
$\displaystyle=-\eta^{i\beta}B^{k}_{j}g_{ki}=-\eta^{i\beta}B_{ji}=\nabla_{j}N^{\alpha}\eta_{\alpha
i}\eta^{i\beta}$
$\displaystyle=\nabla_{j}N^{\alpha}\eta_{\alpha}^{k}g_{ik}\eta^{\beta}_{m}g^{mi}=\nabla_{j}N^{\alpha}\eta_{\alpha}^{k}\eta^{\beta}_{m}\delta_{k}^{m}$
$\displaystyle=\nabla_{j}N^{\alpha}\eta_{\alpha}^{k}\eta^{\beta}_{k}=\nabla_{j}N^{\alpha}(\delta_{\alpha}^{\beta}-N^{\beta}N_{\alpha})$
$\displaystyle=\nabla_{j}N^{\beta}-N^{\beta}N_{\alpha}\nabla_{j}N^{\alpha}$
$\displaystyle\nabla_{j}N^{\beta}$ $\displaystyle=-\eta^{\beta}_{k}B_{j}^{k}$
(24)

Since the second term of the last equality vanishes, we get (24) also known as
Weingarten’s formula.

Now we turn to the Thomas formula allowing to calculate invariant time
derivative of the surface normal. Indeed, using invariant time derivative
formula for surface base vector Grinfeld (2010b)

$\dot{\nabla}\bm{S}_{i}=\bm{N}\nabla_{i}C$ (25)

and doting both sides of (25) with $\bm{N}$,
${\bm{N}\cdot\dot{\nabla}\bm{S}_{i}=\bm{N}\cdot\bm{N}\nabla_{i}C}$ and using
product rule, taking into account that $\bm{N}\cdot\bm{S}_{i}=0$, we find
$\nabla_{i}C=-\dot{\nabla}\bm{N}\bm{S}_{i}$, therefor

$\dot{\nabla}\bm{N}=-\bm{S}^{i}\nabla_{i}C$ (26)

the equation (26) is generally referred as Thomas formula.

##  III Equations of Motions and Physical Models

###  III.1 Equations of motions

Since we have all mathematical preliminaries in hand we can proceed with
derivation of master equations of motions. To deduce the equations we apply
the calculus of moving surfaces to the motion of manifolds in an
electromagnetic field. On this step we only discuss free motion of single
surface, where in ’single’ surface we mean boundary of the single material
body and free means contact with environment is ignored.888the environment is
set to be vacuum. The interaction with environments can be incorporated into
the equations later on.999in the case of taking into account interaction with
environment we no longer have single surface, instead there are double
surfaces where first one is the boundary of the material body and another one
is the surface of the environment at the boundary/environment interface.
Having two surfaces rises the terms related to surface-surface interactions
and may enter into final equations as viscoelastic effect incorporated in
coefficient of viscosity. The surface is treated as continuum media of
material particles (points), where charge and mass distribution is
heterogeneous in general. The boundary of the body is the surface with a
surface mass density $\rho$ and a surface charge density $q$. The surface can
be semipermeable to some material points, meaning the charge can flow through
the surface. Interaction between material points are exclusively
electromagnetic, as far as the mass of each material particles are set to be
infinitely small comparably to unit charges. As far as the ambient space is
set to be Minkowskian, the body is four dimensional and has the surface
boundary of three dimensional manifold. Electromagnetic interaction between
the material particles and heterogeneous distribution of charges throughout
the object induces motion of the surface and the potential energy of the
interaction can be modeled as

$\mathcal{U}=\int_{\Omega}(\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}+A_{\alpha}J^{\alpha})d\Omega$
(27)

where the electromagnetic tensor $F_{\alpha\beta}$ is the combination of the
electric and magnetic fields in a covariant antisymmetric tensor Vanderlinde
(2004); Landau and Lifshitz (1971). The electromagnetic covariant four-
potential is a covariant four vector $\bm{A}_{\cdot}=(-\varphi/c,\bm{a})$
composed by the $\varphi$ electric potential and the $\bm{a}$ magnetic
potential. Contravariant four current $\bm{J}^{\cdot}=(cQ,\bm{j})$ is the
contravariant four vector combining $\bm{j}$ electric current density and $Q$
the charge density, $c$ is a speed of light and $\mu_{0}$ is the magnetic
permeability of the vacuum. Minkowski space metric tensor signature is set to
be space-like $(-+++)$ throughout the paper. This formulation is a fully
relativistic though it can be easily simplified for non-relativistic cases.
Raising and lowering the indexes is performed by the Minkowski metric
$\eta_{\alpha\beta}$. The relation between the four potentials and the
electromagnetic tensor is given by

$F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$ (28)

As far as the boundary of the material body is moving three manifold, the
surface kinetic energy with variable surface mass density $\rho$ and the
surface velocity $\bm{V}$ is

$\mathcal{T}=\int_{S}\frac{\rho\bm{V}^{2}}{2}dS$ (29)

Subtraction of the potential energy (27) from the kinetic energy (29) leads to
the system Lagrangian

$\mathcal{L}=\int_{S}\frac{\rho\bm{V}^{2}}{2}dS-\int_{\Omega}(\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}+A_{\alpha}J^{\alpha})d\Omega$
(30)

where $S$ is the boundary of $\Omega$. Hamilton’s least action principle Johns
(2005) for the given Lagrangian (30) reads

$\frac{\delta\mathcal{L}}{\delta t}=\frac{\delta\mathcal{T}}{\delta
t}-\frac{\delta\mathcal{U}}{\delta t}=0$ (31)

For proper evaluation of the (31) Lagrangian we start from the simplest term
first, it is variation of the potential energy. Since (27) is the space
integral by theorem (18) we have

$\displaystyle\frac{\delta\mathcal{U}}{\delta t}=$
$\displaystyle\int_{\Omega}\frac{\partial}{\partial
t}(\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}+A_{\alpha}J^{\alpha})d\Omega$
$\displaystyle+\int_{S}C(\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}+A_{\alpha}J^{\alpha})dS$
(32)

According to (32) determination of variation of potential energy is to
calculate time differential of the space integrand. Following to standard
algebraic manipulations for classical electrodynamics, we find

$\displaystyle\int_{\Omega}\frac{\partial u}{\partial t}d\Omega=$
$\displaystyle\int_{\Omega}(\frac{\partial u}{\partial
A_{\alpha}}\frac{\partial A_{\alpha}}{\partial t}+\frac{\partial
u}{\partial(\partial_{\beta}A_{\alpha})}\frac{\partial(\partial_{\beta}A_{\alpha})}{\partial
t})d\Omega$ $\displaystyle=$ $\displaystyle\int_{\Omega}(\frac{\partial
u}{\partial A_{\alpha}}\frac{\partial A_{\alpha}}{\partial
t}+\partial_{\beta}(\frac{\partial
u}{\partial(\partial_{\beta}A_{\alpha})}\frac{\partial A_{\alpha}}{\partial
t})$ $\displaystyle-\partial_{\beta}\frac{\partial
u}{\partial(\partial_{\beta}A_{\alpha})}\frac{\partial A_{\alpha}}{\partial
t})d\Omega$ $\displaystyle=$ $\displaystyle\int_{\Omega}(\frac{\partial
u}{\partial A_{\alpha}}\frac{\partial A_{\alpha}}{\partial
t}-\partial_{\beta}\frac{\partial
u}{\partial(\partial_{\beta}A_{\alpha})}\frac{\partial A_{\alpha}}{\partial
t})d\Omega$ $\displaystyle+{\frac{\partial
u}{\partial(\partial_{\beta}A_{\alpha})}\frac{\partial A_{\alpha}}{\partial
t}}|_{\frac{\partial A_{\alpha}}{\partial t}=0}$ $\displaystyle=$
$\displaystyle\int_{\Omega}(\frac{\partial u}{\partial
A_{\alpha}}-\partial_{\beta}\frac{\partial
u}{\partial(\partial_{\beta}A_{\alpha})})\frac{\partial A_{\alpha}}{\partial
t}d\Omega$ (33)

where we used $u=(1/4\mu_{0})F_{\mu\nu}F^{\mu\nu}+A_{\mu}J^{\mu}$ designation
and the fact that $u$ is a function of $A_{\alpha}$ and
$\partial_{\beta}A_{\alpha}$ and at the boundary condition $\partial
A_{\alpha}/\partial t=0$ the last term vanishes. It is easy to show that,

$\partial u/\partial A_{\alpha}=J^{\alpha}$ (34)

To calculate the last integrand (33), we take into account the definition (28)
and note that covariant electromagnetic tensor can be obtained by lowering
indexes in contravariant tensor
$F^{\alpha\beta}=\eta^{\gamma\alpha}\eta^{\kappa\beta}F_{\gamma\kappa}$ and
the electromagnetic tensor is antisymmetric
$F^{\alpha\beta}=-F^{\beta\alpha}$, so that

$\displaystyle\frac{\partial u}{\partial(\partial_{\beta}A_{\alpha})}=$
$\displaystyle\frac{1}{4\mu_{0}}\frac{\partial}{\partial(\partial_{\beta}A_{\alpha})}(F_{\mu\nu}\eta^{\gamma\mu}\eta^{\lambda\nu}F_{\gamma\lambda})$
$\displaystyle=$
$\displaystyle\frac{1}{4\mu_{0}}\eta^{\gamma\mu}\eta^{\lambda\nu}\frac{\partial(F_{\mu\nu}F_{\gamma\lambda})}{\partial(\partial_{\beta}A_{\alpha})}$
$\displaystyle=$
$\displaystyle\frac{1}{4\mu_{0}}\eta^{\gamma\mu}\eta^{\lambda\nu}(F_{\mu\nu}(\delta_{\gamma}^{\beta}\delta_{\lambda}^{\alpha}-\delta_{\lambda}^{\beta}\delta_{\gamma}^{\alpha})$
$\displaystyle+F_{\gamma\lambda}(\delta_{\mu}^{\beta}\delta_{\nu}^{\alpha}-\delta_{\nu}^{\beta}\delta_{\mu}^{\alpha}))$
$\displaystyle=$
$\displaystyle\frac{1}{4\mu_{0}}(F^{\beta\alpha}-F^{\alpha\beta}+F^{\beta\alpha}-F^{\alpha\beta})=\frac{F^{\beta\alpha}}{\mu_{0}}$
(35)

Taking into account (33-35) in (32) we find the variation of the potential
energy

$\displaystyle\frac{\delta\mathcal{U}}{\delta t}=$
$\displaystyle\int_{\Omega}(J^{\alpha}-\frac{1}{\mu_{0}}\partial_{\beta}F^{\beta\alpha})\frac{\partial
A_{\alpha}}{\partial t}d\Omega$
$\displaystyle+\int_{S}C(\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}+A_{\alpha}J^{\alpha})dS$
(36)

Now we turn to the calculation of the kinetic energy variation. To deduce the
variation for the kinetic energy let’s define generalization of conservation
of mass low first. The variation of the surface mass density must be so that
$dm/dt=0$, where

$m=\int_{S}\rho dS$ (37)

is the surface mass with $\rho$ surface mass density. Since, we discuss
compact closed manifolds the boundary conditions $v=n_{i}V^{i}=0$ dictate,
that a pass integral along any curve across the surface must vanish. This
statement formally, taking into consideration (37), can be rewritten as

$\displaystyle 0=$ $\displaystyle\int_{\gamma}v\rho
d\gamma=\int_{\gamma}n_{i}V^{i}\rho d\gamma=\int_{S}\nabla_{i}(\rho V^{i})dS$
$\displaystyle=$ $\displaystyle\int_{S}(\nabla_{i}(\rho V^{i})-\rho
CB_{i}^{i}+\rho CB_{i}^{i})dS$ $\displaystyle=$
$\displaystyle\int_{S}(\nabla_{i}(\rho V^{i})-\rho
CB_{i}^{i})dS+\int_{S}\dot{\nabla}\rho dS-\frac{d}{dt}\int_{S}\rho dS$
$\displaystyle=$ $\displaystyle\int_{S}(\nabla_{i}(\rho V^{i})-\rho
CB_{i}^{i}+\dot{\nabla}\rho)dS$ (38)

where $n_{i}$ is a normal of the curve that lays in the tangent space, $v$ is
the velocity of the $\gamma$ curve. Since last integral from (38) mast be
identical to zero for any integrand, one immediately finds generalization of
conservation of mass low

$\dot{\nabla}\rho+\nabla_{i}(\rho V^{i})=\rho CB_{i}^{i}$ (39)

Incidently, an equation for the surface charge conservetion can be
analogically deduced and it has excatly the same form. The equation (39) was
also reported in Grinfeld (2009). To calculate the variation of the kinetic
energy we use (19,29,39) and after few lines of algebra, we find

$\displaystyle\frac{\delta\mathcal{T}}{\delta t}=$
$\displaystyle\int_{S}(\dot{\nabla}\frac{\rho\bm{V}^{2}}{2}-CB_{i}^{i}\frac{\rho\bm{V}^{2}}{2})dS$
$\displaystyle=$
$\displaystyle\int_{S}(\dot{\nabla}\rho\frac{V^{2}}{2}+\rho\dot{\nabla}\frac{V^{2}}{2}-CB_{i}^{i}\frac{\rho
V^{2}}{2})dS$ $\displaystyle=$ $\displaystyle\int_{S}((\rho
CB_{i}^{i}-\nabla_{i}(\rho
V^{i}))\frac{V^{2}}{2}+\rho\dot{\nabla}\frac{V^{2}}{2}-CB_{i}^{i}\frac{\rho
V^{2}}{2})dS$ $\displaystyle=$ $\displaystyle\int_{S}(-\nabla_{i}(\rho
V^{i}\frac{V^{2}}{2})+\rho
V^{i}\nabla_{i}\frac{V^{2}}{2}+\rho\dot{\nabla}\frac{V^{2}}{2})dS$
$\displaystyle=$
$\displaystyle\int_{S}\rho\bm{V}(V^{i}\nabla_{i}\bm{V}+\dot{\nabla}\bm{V})dS$
(40)

Here we used that at the end of variations the surface reaches the stationary
point and, therefore, by Gauss theorem integral for $\nabla_{i}(\rho
V^{i}V^{2}/2)$ converted to line integral vanishes (as we used it already in
(38)). To deduce the final form of equations of motion we decompose dot
product in the integral (40) by normal and tangential components. After few
lines of algebraic manipulations, we find

$\displaystyle\dot{\nabla}\bm{V}+V^{i}\nabla_{i}\bm{V}=\dot{\nabla}\bm{V}+V^{i}\nabla_{i}\bm{V}+CV^{i}B_{i}^{j}\bm{S}_{j}-CV^{i}B_{i}^{j}\bm{S}_{j}$
$\displaystyle=\dot{\nabla}\bm{V}+V^{i}\nabla_{i}\bm{V}+CV^{i}B_{i}^{j}\eta_{j}^{\alpha}\bm{X}_{\alpha}-CV^{i}B_{i}^{j}\bm{S}_{j}$
(41)

Using Weingarten’s formula (24), metrinilic property of the Minkowksi space
base vectors $\nabla_{i}\bm{X}_{\alpha}=0$ and the definition of the surface
normal ${\bm{N}=N^{\alpha}\bm{X}_{\alpha}}$, the last equation of (41)
transforms

$\displaystyle\dot{\nabla}\bm{V}+V^{i}\nabla_{i}\bm{V}-CV^{i}\bm{X}_{\alpha}\nabla_{i}N^{\alpha}-CV^{i}B_{i}^{j}\bm{S}_{j}$
$\displaystyle=\dot{\nabla}\bm{V}+V^{i}\nabla_{i}\bm{V}-CV^{i}\nabla_{i}\bm{N}-CV^{i}B_{i}^{j}\bm{S}_{j}$
(42)

Taking into account (12) and its covariant and invariant time derivatives in
(42), we find

$\displaystyle\dot{\nabla}\bm{V}+V^{i}\nabla_{i}(C\bm{N})+V^{i}\nabla_{i}(V^{j}\bm{S}_{j})$
$\displaystyle-CV^{i}\nabla_{i}\bm{N}-CV^{i}B_{i}^{j}\bm{S}_{j}$
$\displaystyle=\dot{\nabla}\bm{V}+V^{i}\bm{N}\nabla_{i}C+V^{i}\nabla_{i}(V^{j}\bm{S}_{j})-CV^{i}B_{i}^{j}\bm{S}_{j}$
$\displaystyle=\dot{\nabla}(C\bm{N})+\dot{\nabla}(V^{j}\bm{S}_{j})+V^{i}\bm{N}\nabla_{i}C$
$\displaystyle+V^{i}\nabla_{i}(V^{j}\bm{S}_{j})-CV^{i}B_{i}^{j}\bm{S}_{j}$
(43)

Continuing algebraic manipulations using the formula for the surface
derivative of the interface velocity (25), Thomas formula (26) and the
definition of the curvature tensor (6) in (43), yield

$\displaystyle\dot{\nabla}(C\bm{N})+C\nabla^{j}C\bm{S}_{j}+2V^{i}\bm{N}\nabla_{i}C+V^{i}V^{j}B_{ij}\bm{N}$
$\displaystyle+\dot{\nabla}(V^{j}\bm{S}_{j})-V^{i}\bm{N}\nabla_{i}C+V^{i}\nabla_{i}(V^{j}\bm{S}_{j})-V^{i}V^{j}B_{ij}\bm{N}$
$\displaystyle-C\nabla^{j}C\bm{S}_{j}-CV^{i}B_{i}^{j}\bm{S}_{j}$
$\displaystyle=\dot{\nabla}(C\bm{N})-C\dot{\nabla}\bm{N}+2V^{i}\bm{N}\nabla_{i}C+V^{i}V^{j}B_{ij}\bm{N}$
$\displaystyle+\dot{\nabla}(V^{j}\bm{S}_{j})-V^{j}\dot{\nabla}\bm{S}_{j}+V^{i}\nabla_{i}(V^{j}\bm{S}_{j})-V^{i}V^{j}\nabla_{i}\bm{S}_{j}$
$\displaystyle-C\nabla^{j}C\bm{S}_{j}-CV^{i}B_{i}^{j}\bm{S}_{j}$
$\displaystyle=(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})\bm{N}$
$\displaystyle+(\dot{\nabla}V^{j}+V^{i}\nabla_{i}V^{j}-C\nabla^{j}C-CV^{i}B_{i}^{j})\bm{S}_{j}$
(44)

Doting (44) on $\bm{V}$ and combining it with (40) the last derivation reveals
variation of the kinetic energy

$\displaystyle\frac{\delta\mathcal{T}}{\delta t}=$ $\displaystyle\int_{S}\rho
C(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})dS$
$\displaystyle+\int_{S}\rho
V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i}-C\nabla^{i}C-CV^{j}B_{j}^{i})dS$
(45)

where first part is normal component and second part is tangent component of
the dot product. Combination of (36,45) with (31) reveals

$\displaystyle\int_{S}\rho
C(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})dS$
$\displaystyle+\int_{S}\rho
V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i}-C\nabla^{i}C-CV^{j}B_{j}^{i})dS$
$\displaystyle=\int_{\Omega}(J^{\alpha}-\frac{1}{\mu_{0}}\partial_{\beta}F^{\beta\alpha})\frac{\partial
A_{\alpha}}{\partial t}d\Omega$
$\displaystyle+\int_{S}C(\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}+A_{\alpha}J^{\alpha})dS$
(46)

To find the final form of equations of motion let’s brake dot product of space
integrand from (46) by normal and tangential components. Let’s propose that
the vector $\boldsymbol{\mathscr{F}}$ with contravariant $\alpha$ component is

$\boldsymbol{\mathscr{F}}=(\mathcal{F}^{\alpha})=(J^{\alpha}-\frac{1}{\mu_{0}}\partial_{\beta}F^{\beta\alpha})=\mathcal{F}\bm{N}+\mathcal{F}^{i}\bm{S}_{i}$
(47)

where $\mathcal{F}$ and $\mathcal{F}^{i}$ are normal and tangential components
of $\boldsymbol{\mathscr{F}}$ correspondingly. Analogically, for
$\partial\bm{A}/\partial t$ four vector partial time derivative, we have

$\frac{\partial\bm{A}}{\partial
t}=\mathcal{A}\bm{N}+\mathcal{A}^{i}\bm{S}_{i}$ (48)

where $\mathcal{A},\mathcal{A}^{i}$ are the normal and tangential components
of the partial time derivative of the four vector potential. Using the
definitions (47,48) the dot product of the two vectors is

$\boldsymbol{\mathscr{F}}\cdot\frac{\partial\bm{A}}{\partial
t}=\mathcal{F}\mathcal{A}+\mathcal{F}^{i}\mathcal{A}_{i}$ (49)

Since the equation (46) must hold for every
$\bm{V},\boldsymbol{\mathscr{F}},\partial\bm{A}/\partial t$ vectors in
general, the normal and tangential components of the dot product must be equal
so that, taking into account (47-49) in (46), we find

$\displaystyle\int_{S}\rho
C(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})dS$
$\displaystyle=\int_{\Omega}\mathcal{F}\mathcal{A}d\Omega+\int_{S}C(\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}+A_{\alpha}J^{\alpha})dS$
(50) $\displaystyle\int_{S}\rho
V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i}-C\nabla^{i}C-CV^{j}B_{j}^{i})dS$
$\displaystyle=\int_{\Omega}\mathcal{F}^{i}\mathcal{A}_{i}d\Omega$ (51)

After applying the Gauss theorem to the surface integrals in (50), the surface
integrals are converted to space integral so that one gets

$\displaystyle\int_{\Omega}\partial_{\mu}(\rho
V^{\mu}(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij}))d\Omega$
$\displaystyle-\int_{\Omega}\partial_{\mu}(V^{\mu}(\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}+A_{\alpha}J^{\alpha}))d\Omega=\int_{\Omega}\mathcal{F}\mathcal{A}d\Omega$
(52)

To summarize (39,50-52) equations of moving manifolds in electromagnetic field
read

$\displaystyle\dot{\nabla}\rho+\nabla_{i}(\rho V^{i})$ $\displaystyle=\rho
CB_{i}^{i}$
$\displaystyle\partial_{\mu}(V^{\mu}(\rho(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})-\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta}-A_{\alpha}J^{\alpha}))$
$\displaystyle=\mathcal{F}\mathcal{A}$ (53) $\displaystyle\int_{S}\rho
V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i}-C\nabla^{i}C-CV^{j}B_{j}^{i})dS$
$\displaystyle=\int_{\Omega}\mathcal{F}^{i}\mathcal{A}_{i}d\Omega$

Equations (53) accomplish the search for master equations of motions.

Perhaps the cases that deserves some attention are homogeneous symmetrical
surface, in that case only nonzero ”force” allowed to be is
$\mathcal{F}\mathcal{A}\neq 0$ and $\mathcal{F}^{i}\mathcal{A}_{i}=0$, this
leads to significant simplification of the third equation from (53) and second
equation can be analytically solved for homogeneous, equilibrium surfaces as
we have done it for micelles Svintradze (2016). When
$\mathcal{F}\mathcal{A}\neq 0$ and $\mathcal{F}^{i}\mathcal{A}_{i}\neq 0$ then
motion of the surface induces swimming of the body. The case
$\mathcal{F}\mathcal{A}=0$ and $\mathcal{F}^{i}\mathcal{A}_{i}\neq 0$, as it
is shown below, simplifies to Euler equation for dynamic fluid for free motion
and to Navier-Stokes equation or to magneto-hydrodynamic (MHD) equations if
one takes into account interactions with environment.

Equations (53) are correct for freely moving manifolds of the body in the
vacuum. Generalization can be trivially achieved if instead of electromagnetic
tensor $F_{\alpha\beta}$ one proposes electromagnetic stress energy tensor
$T^{\alpha\beta}$, which is related to electromagnetic tensor by relationship

$\displaystyle
T^{\alpha\beta}=\frac{1}{\mu_{0}}(\eta_{\gamma\nu}F^{\alpha\gamma}F^{\nu\beta}+\frac{1}{4}\eta^{\alpha\beta}F_{\gamma\nu}F^{\gamma\nu})$
(54)

For objects in matter electromagnetic tensor $F^{\mu\nu}$ in (47,53) is
replaced by electric displacement tensor $D^{\mu\nu}$ and by magnetization-
polarization tensor $M^{\mu\nu}$ so that

$\frac{F^{\mu\nu}}{\mu_{0}}=D^{\mu\nu}+M^{\mu\nu}$ (55)

The charge density $Q$ and four current $\bm{J}$ become sum of bound and free
charges and of bound and free four currents respectively. Electric
displacement tensor, magnetization tensor, free and bound charges/currents can
be modeled differently, for different problems, therefore the general
equations (53) can be modified as needed.

###  III.2 Physical models

To link above formulated problem with real physical surfaces it is necessary
some modeling and for the beginning let’s illustrate macromolecules101010or
surface made from groups of molecules, for instance lipids as a two
dimensional fluid manifolds with the thickness of variable mass density Fig.
4.

Figure 4:  (Color online) Top $(a,b)$: Gaussian map of the protein and Bottom
$(c,d)$: of the DNA dodecamer. $(a,c)$ highlights smoothed Gaussian mapping of
the macromolecules, at 5 Å resolution, indicating crystallographic
distribution of the water molecules on the surface. $(b,d)$ shows the mapping
of the macromolecules at two different resolutions, capturing surface
thickness variation incorporated in $\delta\rho$. Blue surface is mapping at 2
Å resolution and indicates distribution of the atoms on the surface, while
light gray is smoothed Gaussian map indicating how surface thickness may vary
if free diffusion of the solvent molecules on the surface is taken into
account.

Even though molecular surfaces are three manifold in Minkowski space, in some
cases111111especially for relatively slowly moving surfaces, for instance:
cellular surface, which is bio-membrane; vesicles; micelles etc. it can be
modeled as moving two manifolds in Euclidean space. The surface is considered
to be semipermeable against partial charges and water molecules. The
permeability defines the surface mass density as a variable and the volume
charge also becomes variable. The variability of charge and mass densities is
properly taken into account in the equations of motion (53).

Let’s model a bio-macromolecular surface as a Gaussian map contoured at 2 Å to
8 Å resolution. Fig. 4 shows Gaussian maps for the protein (Fig. 4$(a,b)$) and
for the DNA (Fig. 4$(c,d)$). $\Omega$ is the space inside the macromolecules
and the boundary of the space is the surface $S$. $\bm{S}_{i}$ base vectors
are defined in the tangent space of the Gaussian map. $S_{ij}$ is the metric
tensor of the map. These are illustrations of surfaces as two-manifolds
embedded in Euclidian space and are only true for non-relativistic
representations, therefore do not show the shape of three-manifolds in
Minkowski space-time. Fig. 4 $(a)$ and $(c)$ show a Gaussian map of the
polypeptide main chain of a protein and of a polynucleotide double helical DNA
dodecamer respectively.121212For a model protein was taken one of a peroxide
sensitive gene regulator with Protein Data Bank (PDB) ID 3HO7 Svintradze  _et
al._ (2013) and for model DNA was taken polynucleotide double helical
dodecamer generally known as library DNA with PDB ID 1BNA Drew  _et al._
(1981). Fig. 4 $(b)$ and $(d)$ show thickness variations, captured by surface
mass density, of the modeled surfaces for the protein and DNA. Light gray is
the Gaussian map at 5 Å resolution while the blue surface indicates a more
detailed surface contoured at 2 Å resolution. Thickness variation can be
induced by diffusion of solvent molecules at solvent accessible sites; e.g.
sites marked by water molecules obtained from crystal structures as
illustrated in the Fig. 4 $(a)$ and $(c)$ (red and white sticks), or by
thermal fluctuation of amino acids sidechains. In all these cases, the surface
thickness variation, captured by $\rho$ surface mass density, is in the range
of angstrom to nanometer. This range is higher for micelles, cell membranes,
fluid films etc. If the system is in aqueous solution then the surface motion
is determined by so called hydrophobic-hydrophilic interactions.

As we already stated in introduction, hydrophobic and hydrophilic interaction
incorporate dispersive interactions, throughout the molecules, mainly related
to electrostatics and electrodynamics (Van der Waals forces), induced by
permanent (water molecules) or induced dipoles (dipole-dipole interactions)
and possible quadrupole-quadrupole interactions (for instance stacking or
London forces) plus ionic interactions (Coulomb forces) Leikin  _et al._
(1993). The hydrophobic effect can be considered as synonymous with dispersive
interactivity with water molecules and the hydrophilic one as synonymous with
polar interactivity with water molecules Chandler and Garrahan (2010);
Chandler (2005); Leikin  _et al._ (1993). All these interactions have one
common feature and can be unified as electromagnetic interaction’s dependence
on interacting bodies’ geometries, where by geometries we mean shape of the
objects’ surfaces. To model potential energy we note that on the scale of
hydrophobic-hydrophilic interactions, which usually occurs at nanometer
distances Chandler (2005); Leikin  _et al._ (1993), no interactions other than
electromagnetic forces are available. An electromagnetic field is set up by
dipole moments of water molecules and partial charges of molecules. In other
words, we have a closed, smooth manifold in aqueous solution where charge and
water molecules could migrate through the surface Fig. 4. The surface can be
of mixed nature (hydrophobic, hydrophilic or both) with randomly distributed
polar or non-polar groups, can be compressible, continuously deformable and
permeable against water and ionic charges. At the nanometer scale, for small
masses, potential energy can be electromagnetic only. Therefore we have
potential energy density constructed from the electromagnetic tensor plus the
term related to variation of charges as it is defined in (27). Even though
modeling of potential energy as electromagnetic interaction energy is fairly
clear, the dependence of these interactions on the object’s geometry is not.
The geometry dependence becomes visible only after the complete formulation of
the equations of motion (53).

##  IV Results and Discussions

###  IV.1 Poisson-Boltzmann Equation

To demonstrate effectiveness of (53) let’s discuss free motion of two
manifolds embedded in three dimensional Euclidean space for the stationary
surface in electrostatic field. Therefore we have the following conditions
${\bm{V}=0}$ stationary surface in electrostatics field where
${\bm{a}=0,\bm{j}=0}$, ${\bm{A}_{\cdot}=(-\varphi/c,\bm{0})}$,
$\bm{J}^{\cdot}=(cQ,\bm{0})$ and
${\bm{\partial}=(0,\partial_{x},\partial_{y},\partial_{z})}$. Then from second
equation of (53) with the precondition (46), we find

$cQ-\frac{1}{\mu_{0}}\partial_{\beta}F^{\beta 0}=0$ (56)

Taking into account the definition of electromagnetic tensor and that we
discuss electrostatic field, partial derivative of the electromagnetic tensor
in (56) is $(1/c)\partial_{\beta}E^{\beta}$ and therefore

$\partial_{\beta}E^{\beta}=c^{2}\mu_{0}Q$ (57)

By the definition of the electric field $E^{\beta}=-\partial^{\beta}\varphi$
and ${c^{2}\mu_{0}=1/\epsilon_{0}}$ so that (57) transforms in

$\partial_{\alpha}\partial^{\alpha}\varphi=\nabla^{2}\varphi=-\frac{Q}{\epsilon_{0}}$
(58)

The equation (58) is generally known as Poisson-Boltzmann Equation in vacuum
and was proposed to describe the distribution of the electric potential in
solution in the direction of the normal to a charged surface Gouy (1910);
Chapman (1913); Davis and McCammon (1990).

Here we demonstrated that, the Poisson-Boltzmann Equation is a particular case
and can be obtained from the equations of motion (53) for stationary surfaces
in electrostatic field. To support this statement we have generated
electrostatic field lines by the Adaptive Poisson-Boltzmann Solver (APBS)
Dolinsky  _et al._ (2007) software for the protein Svintradze  _et al._ (2013)
and the DNA Drew  _et al._ (1981) (Fig. 5).

Figure 5:  (Color online) Color-coded electrostatic surface where red
indicates negatively charged regions of the surface, white neutrally charged
and blue positively charged one. Simulated electrostatic field lines are
displayed as hairs on the surface of the protein $(a)$ and the DNA $(b)$.
Hairs are generated by adaptive Poisson-Boltzmann solver and highlight how
variations of the field lines on the surface describe the charge distribution
pattern on the surface. Right side of the figure shows position of the surface
normal $\bm{N}$ and electrostatic hairs.

As it is visible from the Fig. 5 field lines follow the surface normal.

###  IV.2 Classical electrodynamics, Maxwell equations

In this subsection we demonstrate that, the equation of motions simplify to
Maxwell equations for stationary interfaces $C=0$ and massless $\rho=0$ three
manifolds embedded in Minkowski space. Indeed, from the second and the third
equations of (53), taking into account that in stationary case second term in
(32) vanishes, we find

$\displaystyle\int_{\Omega}\mathcal{F}\mathcal{A}d\Omega$ $\displaystyle=0$
(59) $\displaystyle\int_{\Omega}\mathcal{F}^{i}\mathcal{A}_{i}d\Omega$
$\displaystyle=0$ (60)

Adding (59) to (60) and taking into account (47) and (49) one obtains

$\mathcal{F}\mathcal{A}+\mathcal{F}^{i}\mathcal{A}_{i}=\boldsymbol{\mathscr{F}}\cdot\frac{\partial\bm{A}}{\partial
t}=0$ (61)

(61) must hold for any partial time derivative of the four vector potential,
therefore

$\boldsymbol{\mathscr{F}}=(\mathcal{F}^{\alpha})=(J^{\alpha}-\frac{1}{\mu_{0}}\partial_{\beta}F^{\beta\alpha})=0$
(62)

and the Maxwell equations with the source in the vacuum follows.

$\frac{1}{\mu_{0}}\partial_{\beta}F^{\beta\alpha}=J^{\alpha}$

We got somewhat unexpected result: any three manifold with stationary
interface $C=0$ and with massless surface mass density $\rho=0$,131313here
surface mass density is same as the mass density of three manifold, because
three manifold is the surface in 4D space. satisfies Maxwell equations.
However, arguably photon is only massless particle which satisfies Maxwell
equation, therefor photon can be interpreted as stationary interface three
manifold embedded in Minkowski space with vanishing surface mass density.

###  IV.3 Classical hydrodynamics, Euler equation

In this section we simplify equations of motion using physical arguments and
demonstrate that the equation system (53) yields the Euler equation for
dynamic fluid for some simplified cases. Indeed, let’s propose that moving
fluid has planar surface $B_{ij}=0$ with stationary interface $C=0$ and is
embedded in Euclidean three space, then simplifications of (53) lead to a
system of equations of motion

$\displaystyle\dot{\nabla}\rho+\nabla_{i}(\rho V^{i})$ $\displaystyle=0$
$\displaystyle\mathcal{F}\mathcal{A}$ $\displaystyle=0$
$\displaystyle\int_{S}\rho V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i})dS$
$\displaystyle=\int_{\Omega}\mathcal{F}^{i}\mathcal{A}_{i}d\Omega$ (63)

The first equation of (63) is continuity equation for the surface mass density
and is conservation of mass at the flat space, the second one yields that
normal component of the dot product
${\boldsymbol{\mathscr{F}}\cdot(\partial\bm{A}/\partial t)}$ vanishes. To
simplify the last equation of (63) we note that the total ’force’ acting on
the volume is equal to the integral $-\int_{S}pdS$ of the total pressure $p$,
taken over the boundary (surface) of the volume. Applying the Gauss theorem to
the surface integral by taking into account that pressure across the surface
acts on normal direction so that it can be written as
$p=p_{\alpha}N^{\alpha}$, then

$-\int_{S}pdS=-\int_{S}p_{\alpha}N^{\alpha}dS=-\int_{\Omega}\nabla^{\alpha}p_{\alpha}d\Omega$
(64)

On the other hand $\mathcal{F}^{i}\mathcal{A}_{i}$ is a cause of the gradient
of the tangential velocity and the tangential gradient of the pressure,
therefor

$\mathcal{F}^{i}\mathcal{A}_{i}=-\nabla^{\alpha}(V_{i}\nabla^{i}p_{\alpha})$
(65)

Taking into consideration (64, 65) in (63) and applying Gauss theorem to the
space integral, we find

$\displaystyle\int_{S}\rho V_{i}(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i})dS$
$\displaystyle=\int_{\Omega}\mathcal{F}^{i}\mathcal{A}_{i}d\Omega$
$\displaystyle=-\int_{\Omega}\nabla^{\alpha}(V_{i}\nabla^{i}p_{\alpha})d\Omega$
$\displaystyle=-\int_{S}V_{i}N^{\alpha}\nabla^{i}p_{\alpha}dS$ (66)

According to Weingarten’s formula (24) $N^{\alpha}$ is invariant vs the
surface derivative for flat manifolds and, therefore, can be taken into the
surface covariant derivative, so that
$V_{i}N^{\alpha}\nabla^{i}p_{\alpha}=V_{i}\nabla^{i}p$.141414more information
about how the term $\boldsymbol{\mathscr{F}}\cdot\partial\bm{A}/\partial t$
can be modeled as gradient of pressure times velocity can be found in
Svintradze (2016). Then (66) after subtracting $V_{i}$ yields

$\rho(\dot{\nabla}V^{i}+V^{j}\nabla_{j}V^{i})=-\nabla^{i}p$ (67)

Taking into account that for flat surfaces ${\dot{\nabla}=\partial/\partial
t}$ and ${\nabla_{j}=\partial_{j}}$ one immediately recognizes the last
equation (67) is the exact analog of the classical Euler equation of fluid
dynamics.

As we stated above the equations of motion (53) is formulated for freely
moving manifolds i.e. interaction with environment is ignored and the matter
is set to be a vacuum. Though it can be trivially generalized for the matter
and then simplifications, instead of giving Euler equation, will lead to more
complete Navier-Stokes equation and or magnetohydrodynamic equations. For
instance, in the matter, according to (55), electromagnetic tensor becomes sum
of electric displacement and magnetization tensors. Therefor, in (67), instead
of pure pressure gradient we will have additive term coming from magnetic
field so that (67) will transform in ideal magneto hydrodynamic equation.

Analogically, if interaction with an environment is taken into account, then
instead of single surface we have two surfaces at the surface/environment
interface and the Lagrangian (30) is split by two kinetic energy terms, one
for surface and another one for environmental interface. All these will rise
additive terms in the third equation of (53) so that the equation (67) will
transform in Navier-Stokes equation.

###  IV.4 Equilibrium shapes of micelles

Let’s answer the question: what is a shape of micelles, formed from lipid
molecules, when they are in mechanical equilibrium with solvent. Lipids have
hydrophilic heads and hydrophobic tails, so that, in solutions, they tend to
form the surface with heads on one side and tails on the other. Since the
tails disperse the water molecules, the surface made is closed and has some
given volume. Such structures are called micelles Tanford (1973). Since lipids
form a homogeneous surface, in equilibrium conditions we must have

$\mathcal{F}\mathcal{A}=const\neq 0$ (68)

and $\mathcal{F}^{i}\mathcal{A}_{i}=0$. Usually speed of micelle interface
dynamic is in the range of $~{}nm/ns$ and, therefore, there is no necessity of
discussion relativistic formalism so that the surface is two dimensional and
the space is Euclidean. As far as the surface dynamics is slow magnetic field
is much smaller then electric field $B^{2}<<E^{2}$ and the potential energy
becomes

$\mathcal{U}=-\int_{\Omega}(\frac{\epsilon_{0}}{2}E^{2}+\varphi Q)d\Omega$
(69)

Using first low of thermodynamics, (69) can be modeled as volume integral from
the surface pressure Svintradze (2016), therefore

${p=\frac{\epsilon_{0}}{2}E^{2}+\varphi Q}$ (70)

On the other hand, taking into account the conditions (68,69), the total
potential energy of the surface can be modeled as

$\mathcal{U}=\sigma\int_{S}dS$ (71)

Taking into account (71), the system Lagrangian becomes same as it is in (1)
and its variation leads to the equation

$\rho(\dot{\nabla}C+2V^{i}\nabla_{i}C+V^{i}V^{j}B_{ij})=\sigma B_{i}^{i}$ (72)

(72) was first reported in Grinfeld (2009). Using (69,70,72) in the equations
of motions (53), after simple algebra we fined

$\partial_{\alpha}(\sigma
V^{\alpha}B_{i}^{i}+pV^{\alpha})=-V^{\alpha}\partial_{\alpha}p$ (73)

When the homogeneous surface, such is micelle, is in equilibrium with the
environment then the solution of the (73)151515shorter alternative way to
deduce (73) and about it’s solution in equlibrium conditions is given in
Svintradze (2016). is

$B^{i}_{i}=-\frac{p}{\sigma}$ (74)

From the equation (74) immediately follows generalized Young-Laplace relation
which connects surface pressure to Gaussian curvature and surface tension.
(74) dictates that the homogeneous surfaces in equilibrium with environment
adopts the shape with constant mean curvatures (CMC), therefore explains well
anticipated lamellar, cylindrical and spherical shapes of micelles. This is
another unexpected and surprisingly simple solution to the equations of
motions (53).

##  V conclusions

We have proposed equations of moving surfaces in an electromagnetic field and
demonstrated that the equations simplify to: 1) Maxwell equations for massless
three manifolds with stationary interfaces; 2) Euler equations for dynamic
fluid for planar two manifolds with stationary interface embedded in Euclidean
space which can be generalized to Navier-Stokes equations and to magneto-
hydrodynamic equations; 3) Poisson-Boltzmann equation for stationary surfaces
in electrostatic field.

We have applied the equation to analyze motion of hydrophobic-hydrophilic
surfaces and explained ’equilibrium’ shapes of micelles. The application was
done on a protein, DNA dodecamer and micelles. In all cases analyses were in
good qualitative as well as quantitative agreement with known experimental
results for micelles Svintradze (2016) and with simulations for the protein
and the DNA. Analytic solutions to simplified equations for homogeneous
surfaces in equilibrium with environment produced generalized Young-Laplace
law and explained why mean curvature surfaces are such abundant shapes in
nature.

Also we have showed that hydrophobic-hydrophilic effects are just another
expression of well known electromagnetic interactions. In particular,
equations of motion for moving surfaces in hydrophobic and hydrophilic
interactions, together with the analytic solution, provide an explanation for
the nature of the hydrophobic-hydrophilic effect. Hydrophobic and hydrophilic
interactions are dispersive interactions throughout the molecules and conform
to electromagnetic interaction’s dependence on surface morphology of the
material bodies.

###### Acknowledgements.

We were partially supported by personal savings accumulated during the visits
to Department of Mechanical Engineering, Department of Chemical Engineering,
OCMB Philips Institute and Institute for Structural Biology and Drug Discovery
of Virginia Commonwealth University in 2007-2012 years. We thank Dr. Alexander
Y. Grosberg from New York University for comments on an early draft of the
paper and Dr. H. Tonie Wright from Virginia Commonwealth University for
editing the English of an early draft of the paper. Limited access to Virginia
Commonwealth University’s library in 2012-2013 years is also gratefully
acknowledged.

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