XYLENE POWER LTD.

BASIC PHYSICAL CONCEPTS

By Charles Rhodes, P. Eng., Ph.D.

INTRODUCTION:
On most of this website it is implicitly assumed that the reader has a good understanding of basic physical concepts.

This web page reviews basic physical concepts relating to position, time, velocity, energy, momentum and work. Understanding of these concepts is essential for understanding the evolution of the local universe and energy processes that can be harnessed to do work for mankind.

VECTOR NOTATION:
On this web page a bold face parameter like (X - Xo) denotes a vector with orthogonal x, y and z components.

ENERGY - AN INTUITIVE DEFINITION:
A high school definition of energy is "capacity to do work". This simple definition is adequate for many commerial transactions involving exchange of capacity to do work but is inadequate for proper description of physical phenomena.

ENERGY - A MORE PRECISE DEFINITION:
Energy is "the basic constituant of the universe". Particles contain energy. Radiation contains energy. Energy exchanges between charged particles and radiation occur in finite amounts known as quanta (singular is quantum). A quantum of electromagnetic radiation is known as a photon.

PARTICLE LOCATION:
Each particle has a nominal location where the particles energy density is very high. The energy density of a particle diminishes sufficiently rapidly with increasing distance from its nominal location that the total energy of the particle is finite. However, the particle's energy density remains non-zero out to infinity.

CONSERVATION OF ENERGY:
Energy can neither be created nor destroyed but can be changed in form. In any isolated system the total amount of energy is constant.

WORK:
Work is transferred energy that causes a cluster of particles to accelerate in a particular direction. Work is required for important functions such as pumping fluids, moving goods and generating electricity.

Capacity to do work comes from a directional flow of energy. Capacity to do work may come from capture of a stream of photons by a solar panel, from local thermal expansion/contraction of a gas (as in a heat engine or wind turbine), from local evaporation/condensation of a vapor in a gravitational field (as in hydroelectric power) or from gravitational shape distortion of the rotating Earth (as in tidal power).

Delivery to a remote location of capacity to do work may be done by closed loop circulation of an energy transport medium such as electrons or a hydraulic fluid.

Work converts to potential energy, kinetic energy, radiant energy or heat. The rate at which steady state work can be done is usually limited by the rate at which the dissipated heat can be removed and radiated into outer space.

POSITION:
All position measurements are of the form (X - Xo) where Xo is the position of the observer.

TIME:
All time measurements are of the form (T - To) where To is an arbitrary fixed initial time and (T - To) is measured by counting ticks of a clock located at Xo. Since To = constant:
dTo = 0
giving:
d(T - To) = dT

VELOCITY:
Let Vo be the velocity of the observer. Velocity measurements with respect to the observer are of the form (V - Vo) where:
(V - Vo) = d(X - Xo) / d(T - To)
= d(X - Xo) / dT.

INERTIAL OBSERVER:
Mathematical description of the observed universe becomes very complicated if the observer is subject to acceleration. To simplify the mathematics it is assumed herein that at the observer's position Xo the acceleration is always zero. Such an observer is known as an inertial observer.

Since the observer is inertial, by definition of inertial:
dVo / dT = 0
or
Vo = dXo / dT
= constant

RELATIVITY:
Position, time and velocity are relative. There is no means of determining absolute values of reference position Xo, reference time To or reference velocity Vo. Position, time and velocity only have meaning when measured relative to Xo, To, Vo. However, Xo, Vo can only be defined with respect to positions and velocities Xi, Vi. It is helpful to carry Xo and Vo through calculations to provide insight into the physical meaning of the mathematical equations.

PARTICLE:
A particle is energy contained in a singularity that is dimensionally sufficiently small that from the perspective of an external inertial observer at Xo, Vo movement within the singularity volume with respect to the nominal singularity position Xi and the nominal singularity velocity Vi is undetectable. Such undetectable movement may include rotation within the singularity about Xi. This limit on the accuracy of resolution of position and velocity is referred to as uncertainty.

ENERGY:
Energy is concentrated at or near particles known as electrons and protons or propagates through the universe as photons of electromagnetic radiation. A neutron may be considered to be a proton and an electron coupled by a low energy entity known as a neutrino. At relative time (T - To) each singularity i has a nominal energy Ei measured relative to a field free vacuum at Xo, Vo, has a nominal relative position vector with respect to Xo given by:
(Xi - Xo)
and has a nominal relative velocity vector at Xi given by:
(Vi - Vo) = d(Xi - Xo) / dT.

TOTAL ENERGY:
The total energy E of an isolated system as seen by an inertial observer at Xo is given by:
E = Sum of all Ei

If an isolated system contains no singularities (a field free vacuum), then:
E= Sum of all Ei = 0

LAW OF CONSERVATION OF ENERGY:
The law of conservation of energy is one of the most fundamental physical laws. The law of conservation of energy states that, from the perspective of an inertial observer, the total energy E of any isolated system is constant. Hence:
E = constant
or
dE / dT = 0

Application of the law of conservation of energy indicates that, from the perspective of an inertial observer, the energy Ei of an isolated singularity is constant, in which case:
dEi / dT = 0

MOMENTUM OF A PARTICLE:
The momentum (Pi - Poi) of particle i at relative position (Xi - Xo) is defined by:
(Pi - Poi) = (Ei / C^2) (Vi - Vo)

Note that momentum is a relative quantity. There is no means of determining Poi. Only momentum differences of the form (Pi - Poi) can be determined. Note that vector (Vi - Vo) is located at Xi and can point in any direction.

CONSERVATION OF MOMENTUM FOR AN ISOLATED PARTICLE:
For an isolated particle conservation of energy gives:
dEi / dT = 0
and due to isolation there is no acceleration, giving:
d(Vi - Vo) / dT = 0
Hence:
d(Pi - Poi) / dT = 0
indicating that for any isolated particle momentum (Pi - Poi) is conserved.

POTENTIAL AND KINETIC ENERGY OF A PARTICLE:
The energy Ei of a particle at relative position Xi - Xo and relative velocity Vi - Vo with respect to an observer at Xo, Vo can be expressed as the sum of the potential (rest) energy Epi and the kinetic (motion) energy Eki, or:
Ei = Epi + Eki

POTENTIAL ENERGY OF A PARTICLE:
The potential (rest) energy Epi is the energy of particle i as seen by an observer at Xo, Vo when:
Vi - Vo = 0

KINETIC ENERGY OF A PARTICLE:
The kinetic energy Eki = Ei - Epi is the portion of energy Ei that is a function of the relative velocity:
(Vi - Vo)
of the singularity with respect to the inertial observer at Xo, Vo, subject to the constraint that if:
(Vi - Vo) = 0
then:
Eki = 0
and
Ei = Epi

If an external inertial observer cannot detect rotation about Xi within a singularity containing a particle, then any rotational kinetic energy is part of the observed potential (rest) energy.

ENERGY AND VELOCITY:
A fundamental mathematical relationship discovered by Albert Einstein is:
Ei = Epi / [1 - (|Vi - Vo| / C)^2]^0.5
where C = speed of light in a vacuum.
Note that in this expression if |Vi - Vo| = 0, then Ei = Epi and Eki = 0, as required by the above definition of kinetic energy.

This equation is valid for individual particles but requires generalization for proper application to real objects involving spacially distributed particles.

For the special case of |Vi - Vo| << C:
Eki ~ (Ei / 2)(|Vi - Vo| / C)^2
which with the substitution:
Mi = Ei / C^2
becomes the Newtonian expression for kinetic energy:
Eki = (Mi / 2) |Vi - Vo|^2

The discovery that:
Ei = Mi C^2
was key to understanding nuclear energy.

RATIO OF KINETIC ENERGY TO POTENTIAL ENERGY:
The potential energy Eip of particle i is defined as:
Epi = Ei - Eki

If |Vi - Vo| << C
then:
Epi ~ Ei
and the ratio of Eki / Epi becomes:
Eki / Epi ~ (1 / 2)(|Vi - Vo| / C)^2
indicating that Eki << Epi.

ENERGY OF A PARTICLE:
Recall that the energy of a particle is given by:
Ei = Eip / [1 - (|Vi - Vo| / C)^2]^0.5
Rearranging this equation gives:
Ei^2 [1 - (|Vi - Vo| / C)^2] = Eip^2
or
Ei^2 = Eip^2 + Ei^2 (|Vi - Vo| / C)^2

Recall that:
(Pi - Poi) = (Ei / C^2) (Vi - Vo)
Hence:
Ei^2 = Eip^2 + C^2 |Pi - Poi|^2
This equation is a fundamental physical relationship of great importance.

For the special case of a photon which has no rest energy:
Eip = 0
giving:
Ei = Eik = C |Pi - Poi|

KINETIC ENERGY OF A PARTICLE:
The kinetic energy Eki of a particle is given by:
Eki = [Ei - Epi]
= [(Eip^2 + C^2 |Pi - Poi|^2)^0.5 - Epi]
= Epi [1 + (C^2 / Epi^2)|Pi - Poi|^2)]^0.5 - Epi

REAL OBJECTS:
A real object usually consists of a cluster of spacially distributed particles.

TOTAL LINEAR MOMENTUM:
The total linear momentum (Pc - Po) of a cluster of particles is defined as:
(Pc - Po) = [Sum of all (Pi - Poi)]
= [Sum of all {(Ei / C^2) (Vi - Vo)}]

CENTER OF MOMENTUM VELOCITY:
The point Xc, Vc is known as the Center of Momentum (CM). The CM velocity (Vc - Vo) is defined by:
(Pc - Po) = [Sum of all {(Ei / C^2) (Vi - Vo)}]
= [E / C^2][Vc - Vo]

Hence:
(Vc - Vo) = [C^2 / E][Sum of all {(Ei / C^2) (Vi - Vo)}]
= [Sum of all {(Ei / E) (Vi - Vo)}]

LAW OF CONSERVATION OF LINEAR MOMENTUM:
The law of conservaton of linear momentum is one of the most fundamental laws of physics. The law of conservation of momentum states that from the perspective of an inertial observer the linear momentum (Pc - Po) of any isolated system is constant. Hence:
(Pc - Po) = constant
or
d(Pc - Po) / dT = 0

This conservation law is actually 3 conservation equations because (Pc - Po) has 3 orthogonal components, each of which is conserved.

CENTER OF MOMENTUM OF AN ISOLATED SYSTEM IS INERTIAL:
The linear momentum of an isolated system with respect to an inertial observer can be expressed as:
(Pc - Po) = (E / C^2) (Vc - Vo)

The law of conservation of momentum requires that the term on the right hand side must be constant. The law of conservation of energy requires that E be constant. Hence for an inertial observer:
(Vc - Vo) = constant.
or
d(Vc - Vo) / dT = 0
Since the observer is inertial:
dVo / dT = 0
Hence:
dVc / dT = 0
Hence for an isolated system the Center of Momentum (CM) is inertial.

In many practical situations it is convenient to locate the reference point Xo, Vo at the CM because the CM is inertial.

CHANGE IN REFERENCE POINT:
The momentum (Pi - Poi) with respect to Xo, Vo is given by:
(Pi - Poi) = ((Ei / C^2) (Vi - Vo))

Hence:
|Pi - Poi|^2 = [(Pi - Pci) + (Ei / C^2) (Vc - Vo)] * [(Pi - Pci) + (Ei / C^2) (Vc - Vo)]
= |Pi - Pci|^2 + |(Ei / C^2) (Vc - Vo)|^2 + 2(Ei / C^2) (Pi - Pci) * (Vc - Vo)

VECTOR IDENTITY:
At the web page Vector Identities the vector identity:
|A X B|^2 + (A * B)^2 = |A|^2 |B|^2
is proven,

APPLICATION OF VECTOR IDENTITY:
Make the substitution:
A = (Pi - Pci)

Make the substitution:
B = (Xi - Xc)

Then:
|(Pi - Pci) X (Xi - Xc)|^2 + ((Pi - Pci) * (Xi - Xc))^2
= |Pi - Pci|^2 |Xi - Xc|^2
or
(|(Pi - Pci) X (Xi - Xc)| / |Xi - Xc|)^2
+ (((Pi - Pci) * (Xi - Xc)) / |Xi - Xc|)^2
= |Pi - Pci|^2

Then:
|Pi - Poi|^2 = |Pi - Pci|^2 + |(Ei / C^2) (Vc - Vo)|^2 + 2(Ei / C^2) (Pi - Pci) * (Vc - Vo)
= (|(Pi - Pci) X (Xi - Xc)| / |Xi - Xc|)^2
+ (((Pi - Pci) * (Xi - Xc)) / |Xi - Xc|)^2
+ |(Ei / C^2) (Vc - Vo)|^2
+ 2(Ei / C^2) (Pi - Pci) * (Vc - Vo)

KINETIC ENERGY OF AN ISOLATED CLUSTER OF PARTICLES:
The law of conservation of energy gives the total energy of an isolated cluster of particles as:
E = Sum of all Ei

The kinetic energy Ek of an isolated cluster of particles is given by:
Ek = Sum of all Eki
Sum of all [Ei - Epi]
= Sum of all{Epi [1 + (C^2 / Epi^2)|Pi - Poi|^2)]^0.5 - Epi}

For |Vi - Vo| << C:
Ek ~ Sum of all {(C^2 / 2 Epi)|Pi - Poi|^2}
= Sum of all (C^2 / 2 Epi){(|(Pi - Pci) X (Xi - Xc)| / |Xi - Xc|)^2
+ (((Pi - Pci) * (Xi - Xc)) / |Xi - Xc|)^2
+ |(Ei / C^2) (Vc - Vo)|^2
+ 2(Ei / C^2) (Pi - Pci) * (Vc - Vo)}

Choose the point Xc, Vc such that the fourth term of Ek is zero. Hence:
Sum of all {(C^2 / 2 Epi) 2 (Ei / C^2) (Pi - Pci) * (Vc - Vo)} = 0
or
Sum of all {(Ei^2 / Epi C^2) (Vi - Vc) * (Vc - Vo)} = 0
or
Sum of all {(Ei^2 / Epi C^2) [(Vix - Vcx) x * (Vcx - Vox) x]
+ (Ei^2 / Epi C^2) [(Viy - Vcy) y * (Vcy - Voy) y]
+ (Ei^2 / Epi C^2) [(Viz - Vcz) z * (Vcz - Voz) z]} = 0
or
Sum of all {(Ei^2 / Epi C^2) ([(Vix - Vcx)(Vcx - Vox)]
+ [(Viy - Vcy)(Vcy - Voy)]
+ [(Viz - Vcz)(Vcz - Voz)])} = 0

The parameters Vcx, Vcy and Vcz are orthogonal. Each Vcx, Vcy, Vcz is chosen so that its term sums to zero. Hence this equation becomes three equations:
Sum of all {(Ei^2 / Epi C^2) (Vix - Vcx)} = 0
Sum of all {(Ei^2 / Epi C^2) (Viy - Vcy)} = 0
Sum of all {(Ei^2 / Epi C^2) (Viz - Vcz)} = 0

These equations can be rewritten as:
Sum of all {(Ei^2 / Epi C^2) ((Vix - Vox) - (Vcx - Vox))} = 0
Sum of all {(Ei^2 / Epi C^2) ((Viy - Voy) - (Vcy - Voy))} = 0
Sum of all {(Ei^2 / Epi C^2) ((Viz - Voz) - (Vcx - Voz))} = 0
or
(Vcx - Vox) = [Sum of all {(Ei^2 / Epi C^2) ((Vix - Vox)}] / [Sum of all (Ei^2 / Epi C^2)]
(Vcy - Voy) = [Sum of all {(Ei^2 / Epi C^2) ((Viy - Voy)}] / [Sum of all (Ei^2 / Epi C^2)]
(Vcz - Voz) = [Sum of all {(Ei^2 / Epi C^2) ((Viz - Voz)}] / [Sum of all (Ei^2 / Epi C^2)]

These equations fully define:
Vc - Vo = (Vcx - Vox) x + (Vcy - Voy) y + (Vcz - Voz) z

In vector notation these equations can be summarized as:
[Sum of all (Ei^2 / Epi C^2)][Vc - Vo]
= [Sum of all {(Ei^2 / Epi C^2) (Vi - Vo)}]

If |Vi - Vo| << C
then:
Epi ~ Ei
giving:
[Sum of all (Ei^2 / Epi C^2)][Vc - Vo]
~ [Sum of all (Ei / C^2)][Vc - Vo]
= [E / C^2][Vc - Vo]
= (Pc - Po)
and
[Sum of all {(Ei^2 / Epi C^2) (Vi - Vo)}]
~ [Sum of all {(Ei / C^2) (Vi - Vo)}]
= (Pc - Po)

Thus:
[Sum of all {(Ei^2 / Epi C^2) (Vi - Vo)}]
= [Sum of all (Ei^2 / Epi C^2)][Vc - Vo]

Hence for |Vi - Vo| << C:
Sum of all (C^2 / 2 Epi){2(Ei / C^2) (Pi - Pci) * (Vc - Vo)} ~ 0

Hence for a cluster of particles in which |Vi - Vo| << C Ek simplifies to:
Ek = Sum of all (C^2 / 2 Epi){(|(Pi - Pci) X (Xi - Xc)| / |Xi - Xc|)^2
+ (((Pi - Pci) * (Xi - Xc)) / |Xi - Xc|)^2
+ |(Ei / C^2) (Vc - Vo)|^2}
= rotation kinetic energy
+ radial kinetic energy
+ CM kinetic energy

With a gas or a vapor generally only the CM kinetic energy can do useful work. The rotation kinetic energy and radial kinetic energy generally manifest themselves as heat.

RIGID BODY:
For a rigid body rotating around an axis through its CM for every particle i:
|Xi - Xc| = constant
or
d[|Xi - Xc|^2] / dT = 0
or
d[(Xi - Xc) * (Xi - Xc)] / dT = 0
or
(Vi - Vc) * (Xi - Xc) = 0
or
Ei (Vi - Vc) * (Xi - Xc) = 0
or
(Pi - Pci) * (Xi - Xc) = 0

Hence for a rigid body rotating around an axis through its CM the radial kinetic energy is zero.

Hence, for |Vi - Vo| << C, the kinetic energy of a rigid body simplifies to:
Ek = Sum of all [(C^2 / 2 Epi){|(Pi - Pci) X (Xi - Xc)|^2 / |Xi - Xc|)^2
+ |(Ei / C^2) |Vc - Vo|^2}]
= Sum of all [(C^2 / 2 Epi){|(Pi - Pci) X (Xi - Xc)|^2 / |Xi - Xc|^2}
+ (Ei^2 / 2 Epi C^2) |(Vc - Vo)|^2]
~ Sum of all [(C^2 / 2 Epi) |(Pi - Pci) X (Xi - Xc)|^2 / |Xi - Xc|^2]
+ (E / 2 C^2) |Vc - Vo|^2

SUMMARY:
The velocity (Vi - Vc) of a particle at Xi relative to Xc can be resolved into a radial velocity component along (Xi - Xc) and a tangential velocity component at Xi that is perpendicular to (Xi - Xc).

With proper choice of Xc, Vc for a rigid body rotating around an axis through its CM the sum of the radial momentum components equals zero leaving only rotational momentum and linear momentum components.

For an observer that cannot see the rotation the rotational kinetic energy becomes part of the apparent CM rest energy Ecp. If the invisible rotational kinetic energy is quantized then the apparent rest energy is also quantized.

ANGULAR MOMENTUM OF A PARTICLE:
The angular momentum Lci of a particle is defined by:
Lci = ((Pi - Pci) X (Xi - Xc))

TOTAL KINETIC ENERGY OF A RIGID BODY:
The total kinetic energy of a rigid body becomes:
Ek = Sum of all [(C^2 / 2 Epi) {|(Pi - Pci) X (Xi - Xc)|^2 / |Xi - Xc|^2]
+ {(E / 2 C^2) |Vc - Vo|^2}
= Sum of all [(C^2 / 2 Epi) (|Lci|^2 / |Xi - Xc|^2)]
+ {(E / 2 C^2) |Vc - Vo|^2}

In English what this equation states is that for a rigid body with |Vi - Vo| << C:
Total kinetic energy = kinetic energy of rotation about the CM + kinetic energy of linear motion of energy E as if it were located at the CM at Xc, Vc.

ANGULAR MOMENTUM OF A RIGID BODY:
Recall that:
Ek = Sum of all [(C^2 / 2 Epi) (|Lci|^2 / |Xi - Xc|^2)]
+ {(E / 2 C^2) |Vc - Vo|^2}
The total angular momentum Lc caused by revolution of a rigid body about Xc, Vc is defined by:
Ek ~ [(C^2 / 2 E) (|Lc|^2 / |Xe - Xc|^2)]
+ {(E / 2 C^2) |Vc - Vo|^2}
where |Xe - Xc| is the radius of an equivalent ring containing the same rest energy Ep and the same kinetic energy Ek as the rotating cluster of particles forming the rigid body.

The total momentum of a rigid body is:
(P - Po)
= (Pc - Po) + (P - Pc)
= (E / C^2) (Vc - Vo) + Lc / |Xe - Xc|

Note that the linear and angular momentum terms are orthogonal to each other. Hence the products of linear and angular momentum terms are zero, giving:
|(P - Po)|^2
= |(E / C^2) (Vc - Vo)|^2 + |Lc|^2 / |Xe - Xc|^2
+ 2(E / C^2) (Vc - Vo) * Lc / |Xe - Xc|
= |(E / C^2) (Vc - Vo)|^2 + |Lc|^2 / |Xe - Xc|^2

Hence the generalized form of the Einstein equation for a rigid body is:
E^2 = Ep^2 + C^2 |P - Po|^2
= Ep^2 + [C^2 |P - Pc|^2] + [C^2 |Pc - Po|^2]
where:
|Pc - Po|^2 = |(E / C^2) (Vc - Vo)|^2
= (linear momentum of body}^2
and
|P - Pc|^2 = |Lc|^2 / |Xe - Xc|^2
= (rotational momentum of body)^2,
where:
Lc = angular momentum of body.

If the rotation is invisible in order to keep E constant Ep must increase to absorb the rotational kinetic energy.

ROTATION INCREASES APPARENT REST ENERGY Ep:
The rest energy is determined by measurement of rest mass.

From the perspective of only linear motion the rotation of a charged particle such as an electron or proton about its own CM cannot be seen, so such rotational kinetic energy is included in the particles apparent rest energy.

From the perspective of only linear motion the revolution of the electrons and protons of an atom around their common CM cannot be seen, so the kinetic energy of such revolution is included in the atom's apparent rest energy.

From the perspective of only linear motion the random thermal vibration of molecules in a solid cannot be seen, so such vibration kinetic energy is included in the solid's apparent rest energy.

From the perspective of only linear motion the rotation of a multi-atom gas molecule around its CM cannot be seen, so such rotation is included in the molecule's apparent rest energy.

From the perspective of an external observer confined random linear motion of gas molecules cannot be seen, so the kinetic energy due to such confined random motion is included in the molecule's apparent rest energy. The kinetic energy due to such confined random molecular motion is known as heat.

From the perspective of only linear motion the rotation of a fly wheel around its CM cannot be seen, so such rotation adds to the fly wheel's apparent rest energy.

ANGULAR MOMENTUM:
For any real object angular momentum about its CM at Xc, Vc contributes to the apparent potential (rest) energy Ecp at the CM as seen by an external inertial observer that observes only linear motion of the CM.

APPROXIMATE CONSERVATION OF ANGULAR MOMENTUM OF AN ISOLATED SYSTEM DURING A CHANGE IN ROTATIONAL KINETIC ENERGY:
For an isolated cluster of singularities and for |Vi - Vo| << C angular momentum is also approximately conserved. However, that apparent conservation arises from application of the law of conservation of energy at non-relativistic speeds. Conservation of angular momentum is not a true conservation law.

Recall that:
Eki = [(C^2 / 2 Epi) (|Lci|^2 / |Xi - Xc|^2)]
+ {(Ei / 2 C^2) |Vc - Vo|^2}
where:
|Lci| = |(Ei / C^2) (Vi - Vc) X (Xi - Xc)|
= (Ei / C^2) Wi Ri^2
Where:
Wi = angular velocity of singularity i around the axis of rotation
Ri = radius from Xi to the axis of rotation.

Hence the rotational part of Eki is:
Ekir = [(C^2 / 2 Epi) (|Lci|^2 / |Xi - Xc|^2)]
= [(C^2 / 2 Epi) (|(Ei / C^2) (Wi Ri^2)|^2 / |Ri|^2)]

For |Vi| << C:
Ekir ~ (Ei / 2 C^2)(Wi Ri)^2

Hence:
dEkir = (Ei / C^2) (Wi Ri) d[Wi Ri]
= (Ei / C^2) (Wi Ri) [dWi Ri + Wi dRi]

Centrifugal force = (Ei / C^2) Wi^2 Ri
Hence:
dEkir = - (centrifugal force) dRi
= - (Ei / C^2) Wi^2 Ri dRi

Use the law of conservation of energy to equate the two expressions for Ekir to get:
(Ei / C^2)(Wi Ri) [dWi Ri + Wi dRi] = - (Ei / C^2) Wi^2 Ri dRi
or
[dWi Ri + Wi dRi] = - Wi dRi
or
dWi Ri = - 2 Wi dRi
or
dWi / Wi = - 2 dRi / Ri
or
Wib / Wia = (Ria / Rib )^2
or
Wib Rib^2 = Wia Ria^2

Hence:
Wi Ri^2 = constant

Recall that:
|Lci| = (Ei / C^2) Wi Ri^2
Since for an isolated system in which |Vi - Vo| << C both Ei and (Wi Ri^2) are constant:
|Lci| = constant

Total angular momentum is given by:
Lc = Sum of all Lci
Hence, for |Vi - Vo| << C, for an isolated cluster of singularities total angular momentum Lc is approximately constant.

An element of momentum dP can transfer from one object to another object within an isolated cluster of interacting objects but the Center of Momentum (CM) Xc, Vc for the cluster remains inertial and subject to |Vi - Vo| << C the total angular momentum remains approximately constant. This issue has major implications in terms of planetary motion and attempting to deflect from planet Earth potentially impacting comets and other projectiles. Even if these projectiles are blown up their CM paths and total angular momenta remain unchanged.

VECTOR FIELDS:
A fundamental issue in the structure of the universe is that each particle is surrounded by a vector field containing potential (rest) energy which is part of Eip. The potential energy / unit volume at X contained in the external vector field due to the singularity at Xi is proportional to the (local vector field strength)^2. This relationship appears to be a fundamental property of the universe. With different proportionality constants this relationship holds for gravitational, electric and magnetic fields.

The total external radial vector flux from a particle is proportional to the contained energy or contained charge and hence is nearly constant because most of the energy or charge is concentrated very close to the nominal position of the particle. The surface area of a sphere of radius |X - Xi| is:
(4 Pi |X - Xi|^2).
Hence, for a single isolated particle the external local vector field strength diminishes approximately in proportion to:
1 / (|X - Xi|^2).
Vector fields from different particles add linearly. Hence, overlap of vector fields from multiple particles changes the local vector field strength and hence the total potential energy.

FORCE BETWEEN PARTICLES:
Force is the result of change in total potential energy with respect to particle position that causes a corresponding change in kinetic energy with respect to particle position.

The force Fi on singularity i causes a change in kinetic energy dEki during a change in position d(Xi - Xo). Hence:
dEki = Fi * d(Xi - Xo)

The mechanism of interactions between particles that cause force at a distance is discussed at Field Theory

CONSERVATION OF ENERGY FOR A CLUSTER OF PARTICLES:
The law of conservation of energy requires that for any isolated system the total system energy measured by an inertial observer is constant. Hence, if overlap of vector fields causes a change in potential energy the law of conservation of energy requires a corresponding change in kinetic energy to keep the total energy constant.

CONSERVATION OF ENERGY FOR INTERACTING PARTICLES:
If two objects forming an isolated system interact the individual object energies can remain unchanged or an element of energy dE can be transferred from one object to the other or a new particle can be formed and emitted but the total system energy remains unchanged. Note that a process involving creation and emission of a new particle from an interaction between two previously existing particles is usually not reversible because such reversal requires a three body interaction. Except in neutron stars, the probability of occurrence of three body interactions is extremely small.

EXTERNAL OBSERVER:
From the perspective of an external observer the total energy and the net linear momentum of an isolated system can be considered as being located at the CM of that system.

CONVERSION:
Thermal kinetic energy and rotational kinetic energy seen by an observer at the CM but that cannot be resolved by an external inertial observer become components of the rest energy seen by the external inertial observer. For the external inertial observer the velocity of the CM in the external observer's frame of reference establishes the kinetic energy due to linear momentum seen by this external observer. The change of thermal kinetic energy and rotational kinetic energy seen by an observer at the CM into potential energy seen by an external inertial observer is a very important concept. This concept explains how heat and rotation increase the CM rest energy as seen by an external observer.

The planet formation sequence is:
1) Widely separated dust particles contain maximum gravitational potential energy;
2) The gravitational potential energy converts to ordered kinetic energy;
3) The ordered kinetic energy converts to heat via inelastic collisions leaving the dust trapped in a gravitational potential well;
4) The heat converts to thermal electromagnetic radiation;
5) The thermal electromagnetic radiation propagates into space. This radiation reduces the remaining random kinetic energy that would otherwise allow some dust particles to escape from the gravitational potential well.

WORK:
If otherwise isolated objects interact in a manner that causes a transfer of an element of energy dE and an element of momentum dP from one object to another then the object that supplied the transferred energy is said to have done work on the other object.

FORCE THEOREM:
If the potential energy of the object receiving the element of energy dE remains unchanged, as is the case in many practical physical situations, the differential element of work done dE is given by:
dE dT = dP dX
where dT, and dX are respectively corresponding differential elements of time (T - To) and position X - Xo. Rearranging this formula shows that:
dE = F dX
where:
F = dP / dT
= force
This formula is a corrected version of Newtons formula F = M A. The corrected formula can be proven by using the Einstein formula for energy as a function of velocity to show that if Ep = constant then:
dE / dX = dP / dT
or
dE dT = dP dX

PROOF:
Assume that there is no change in rotation while work is done. Then the Einstein formula for Vo = 0 is:
E = Ep / (1 - (|Vc| / C)^2)^0.5
For the special case of Ep = constant, dEp / dT = 0.
Hence:
dE / dX = Ep Vc dVc / [C^2 dX (1 - (|Vc| / C)^2)^1.5]
= E Vc dVc / [C^2 dX (1 - (|Vc| / C)^2)]
= E (dT / dX) Vc dVc / [C^2 dT (1 - (|Vc| / C)^2)]
= E dVc / [C^2 dT (1 - (|Vc| / C)^2)]
= E dVc / [(C^2 - |Vc|^2) dT]

Recall that:
P = (E / C^2) Vc
Hence:
dP / dT = (E dVc + Vc dE) / [dT C^2]
= (E dVc + Vc {E Vc dVc / [C^2 (1 - (|Vc| / C)^2)]}) / [dT C^2]
= (E dVc + Vc {E Vc dVc / [C^2 - |Vc|^2]}) / [dT C^2]
= (E dVc [C^2 - |Vc|^2] + Vc E Vc dVc)/ ([C^2 - |Vc|^2] dT C^2)
= (E dVc C^2) / ([C^2 - |Vc|^2] dT C^2)
= (E dVc) / ([C^2 - |Vc|^2] dT)
Hence:
dE / dX = dP / dT
or
dE dT = dP * dX

This equation is commonly written in the form:
dE = (d(P) / dT) * dX
or
dE = F * dX
or
Work = (force) * (change in position)

ENERGY FLOW:
A net flow of energy along an axis (a flow of momentum) has the capacity to do work. For example: electromagnetic energy flowing along a power transmission line can do work; solar radiation flowing from the Sun to the Earth can do work; water flowing downhill can do work; and wind flowing in a specific direction can do work.

All of human civilization is based on harnessing flows of energy to do useful work. Once the work is done the total energy flow becomes a flow of atmospheric temperature heat. This heat is constantly dissipated into outer space by emission of thermal infrared radiation.

ENERGY MEASUREMENTS:
Energy measurements usually take one of the following forms:
1) Measurement of mass of a fuel (eg Tonnes of coal);
2) Measurement of volume of a liquid fuel at a particular temperature (eg litres of oil);
3) Measurement of volume of a gaseous fuel at a particular pressure and temperature (eg m^3 of natural gas);
4) Measurement of capacity to do work (eg kWh of electricity);
5) Measurement of work done (eg Joules of gravitational potential energy formed);
6) Measurement of heat output (eg Joules of heat);
7) Measurement of mass or volume of liquid vaporized at a particular pressure and temperature (eg Kilograms of steam, litres of steam condensate).

Known material parameters such as the density of water at a particular temperature and the heat of vaporization of water at a particular pressure are used to convert these measurements to energy units.

ENERGY EFFICIENCY:
Most energy conversion processes are characterized by an efficiency of the form:
Efficiency = (useful heat, work or capacity to do work output) / (heat, work or capacity to do work input)
However, energy efficiency does not have a unique definition, so a measurement of energy efficiency is only meaningful if the relevant definition of energy efficiency is also specified.

In some cases, such as commercial boilers, energy efficiency is calculated by measuring rejected waste heat as a fraction of the heating capacity of the fuel consumed.

BINDING ENERGY:
Random thermal kinetic (heat) energy can be spontaneously emitted from a cluster of particles via thermal electromagnetic radiation (photons). This radiation emission process reduces the average random kinetic energy per particle remaining in the system to a level below that necessary for individual particles to escape from the potential energy well caused by overlap of the particles radial vector fields. The remaining particles are bound together by the resulting potential energy well. This process explains the aggregation of matter (energy) to form stars, planets, asteroids and comets.

Similarly atoms are bound into molecules by emission of photons that leave the now less energetic atoms trapped in electromagnetic potential energy wells.

Similarly molecules bind together to form liquids and solids.

This binding energy process is explored in much more detail at Energy Composition of Matter.

ENTROPY:
At steady state conditions work can be done when photons are formed so that the energy flow remains the same but there is an increase in the number of photons and a decrease in energy per photon. For example, on Earth work can be done when the flow of absorbed solar radiation has a higher per photon energy than the balancing flow of emitted infrared radiation. Another way of viewing this matter is that work can be done when heat flows from a hot source to a cooler sink. In scientific language, work can be done when there is an increase in entropy. When the energy flow remains constant but photons are created and emitted with a decrease in energy per photon there is an increase in entropy.

All processes that can provide on-going useful work operate by conversion of an energy flow carried by high energy particles or photons into an approximately equal energy flow carried by a greater number of lower energy particles or photons.

CARNOT EFFICIENCY:
Consider an energy conversion system that inputs photons each with energy Ea and for each input photon outputs a photon along one path with energy Eb and a photon along a different path with energy (Ea - Eb). In order to maintain a steady state energy flow Eb is fixed by the ambient atmospheric temperature. The theoretical maximum efficiency of this energy conversion process is given by:
(Ea - Eb) / Ea
Students of thermodynamics will recognize this expression as the Carnot efficiency, which is the maximum theoretical efficiency of a heat engine.

QUANTIZATION:
The actual energy values of atomic electrons have more than one distinct discrete real solution. The probability of occupancy of a particular real solution depends on the amount of random kinetic (heat) energy available to the atomic electrons. The difference in energy between one discrete electron energy state and another is known as a quantum of energy.

Quantization of atomic energy will occur because a charged particle can only emit or absorb a photon if:
E = h F
where:
E = photon energy
= change in electron energy
h = Plancks constant
and
F = photon frequency.

For rigid bodies, each with a large number of atoms, the energy difference between the adjacent discrete real energy solutions is so small compared to the total system energy that from an external observers perspective the discrete changes in energy associated with changes in bulk motion are invisible. However, an important observable effect of quantization of electron energy states is the frequency dependence on temperature of random kinetic (heat) energy loss via thermal electromagnetic radiation.

This web page last updated April 15, 2012.