Dynamic Theory Explains that 
Protons + Electrons form  Neutron, Deuteron and Alpha Particles

James C. Rock

Retired USAF, Retired Texas A&M University

Abstract

This presentation offers a brief overview of Pharis E. Williams’ Dynamic Theory.  In its most general form the theory predicts the geometry and generates equations for Classical and Relativistic Maxwell Equations, for Newton’s Equations, for Einstein’s Special and General Relativity, and for Schrödinger’s Quantum Mechanics.  One of its key predictions is the Non-Singular Potential, a function derived with a parameter that similar to Plank’s Constant, but is unique to each particle rather than a fixed constant of nature that applies to all particles. This potential function is consistent with classical Newton and Maxwell equations for common applications of those theories at human scales.  At Atomic and Nuclear distances, the non-singular potential explains when a proton and electron form a hydrogen atom, when they form a neutron, and the model predicts the half life of a neutron from quantum tunneling considerations.  At atomic distances the model predicts the structure of the deuteron, of the He-4 nucleus and led to experiments and a patent for a low energy barrier deuterium fusion reaction.  Some examples of Dynamic Theory computations that were not possible in the past but may now be possible in Mathematica, will be offered to conclude this introductory presentation.

a=1;Remove["Global`*"]

Thermodynamics underlies Pharis E. Williams’ Dynamic Theory

Steam System Laws of Thermodynamics, per Pharis Williams:

1st Law:   Conservation of path-dependent Energy ( work = (Overscript[force, ⟶]) * (Overscript[distance, ⟶]))  
2nd Law:  Entropy is path-independent and defines Energy not available to do useful work  
3rd Law:   At T = 0, Entropy, (Subscript[S,  0]) = 0.  (If 2nd Law is used for ΔS, S0 is not important)

Dynamic Theory Laws in purely mechanical systems (no heat energy):

1st Law:  Energy is path-dependent, forces are functions of position and velocity

2nd Law:  Entropy is path independent and depends only on mechanical work

Geometry is Important in Both

An isolated system travels the path that maximizes its Entropy
A non-isolated system travels the path that minimizes Energy
Williams found that Quadratic Variational Principle Specifies Geometry

Thermodynamics has 5 independent variables

A Steam System Thermodynamics Course Starts with First Law

dpath:=OverscriptBox[Style["\[DifferentialD]",24,FontFamily->"Arial"],Style["~",Bold,20]]//DisplayForm
Style[Row[{dpath,"Q = \[DifferentialD]U + P \[DifferentialD]V + (OverVector[(Subscript[F, X])]) • \[DifferentialD](OverVector[X]) + (OverVector[(Subscript[F, Y])]) • \[DifferentialD](OverVector[Y]) + (OverVector[(Subscript[F, Z])]) • \[DifferentialD](OverVector[Z])"}],24]//StandardForm

5 equations are needed for solutions, one for each variable on right side

Use Newton’s 3 Laws for Mechanical Work Terms
Use Equation of State for P = f [V]
Use Conservation of Mass for Specific Volume = 1/density

Energy depends on the path of integration.
1st Law:  Conservation of Energy applies to all paths.
Entropy is path independent.
2nd Law: Entropy measures Energy not avaialble to do useful work.

Williams’ Could find no proof that mass is conserved

On the contrary, from Relativity, E = m (\[ConstantC]^2)
Mass not conserved during these processes.

Williams decided to consider Energy as Fundamental

Recall, Einstein introduced his 4-D manifold by making time an independent variable on the same level as 3-D space.

Williams introduces 5-D manifold with mass density, time and 3-D space as independent variables.

“So what are some of the deductions that come from assuming mass is independent of space and time?  There are many ... and I am still finding more.  To date I have not found any ... to be shown wrong by experiment.”
PAO pg 15

Williams’ Dynamic Theory arises from Three Postulates

Three postulates are generalizations of the laws of Classical Thermodynamics

First Postulate:  Energy is Conserved

The portion of Energy defined by work is path dependent.
The forces are functions of position (Newton) and velocity (Einstein).

Second Postulate:  Paths of reversible processes cannot cross

Third Postulate:  An Integrating Factor Exists & d Entropy is Path-Independent

When the integrating factor becomes infinite, S is a constant, and this constant may be taken to be zero. 
P.E. Williams, The Dynamnic Theory, 2011, Ch 1 -  Introduction.

Williams Applies the laws per Weyl (1918) and London (1927)

Min Energy and Max Entropy paths are computed as variational quadratic forms
The quadratic forms define the geometry for both the Energy and Entropy manifolds

Riemann geometry is the manifold required for path-dependent Energy calculations
Weyl geometry the manifold required for path-independent Entropy calculations

The Energy manifold is connected to the Entropy manifold by a Weyl Guage function
When the guage function is unity in an isolated system, variables are quantized
Quantized variables include: charge, photon Energy, atomic and nuclear orbital Energy

Weyl, Hermann. 1918a. “Gravitation und Elektrizit¨at.” Sitzungsberichte der K¨oniglich Preußischen Akademie der Wissenschaften zu Berlin pp. 465–480.
GA II, 29–42, [31]. In (Weyl 1968, II, 29–42) [31], English in (O’Raifeartaigh 1997, 24–37).
    Weyl, Hermann. 1918b. Raum, - Zeit - Materie. Berlin etc.: Springer. Weitere Auflagen: 21919, 31919, 41921, 51923, 61970, 71988, 81993. English and French translations from the 4th ed. in 1922.
    Weyl, Hermann. 1918c. “Reine Infinitesimalgeometrie.” Mathematische Zeitschrift 2:384–411. In (Weyl 1968, II, 1–28).
    Weyl, H.  Raum, Zeit, Materie: Vorlesungen uber allgemeine Relativitatstheorie, Springer, Berlin, 1919.  
         Translated by from 4th Ed. by  H. L. Brose as Space Time Matter, Dover, NY 1952. [This is Weyl’s exposition and development of relativity and guage theory.]

     Heitler, W.; London, F. (1927). “Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik”. Zeitschrift für Physik 44 (6–7): 455. doi:10.1007/BF01397394. 
     Eisenschitz, R.; London, F. (1930). “Über das Verhältnis der van der Waalsschen Kräfte zu den homöopolaren Bindungskräften”. Zeitschrift für Physik 60 (7–8): 491.
     http://www.ebook3000.com/Fritz-London--A-Scientific-Biography_115819.html,  free download. [London connected gravity with Electromagnetic Forces]

Geometry:  Newton, Einstein and Williams

Newton used Euclid’s Geometry in 3-D space
	( gives trajectory when force was applied, but force was function of position, only)
	( gravity was the only defined force )
	( but, where in the universe is the absolute origin? )
	
Einstein used Riemann Geometry in 4-D space
	( speed of light is limit for all EM events,  force replaced by curved space )
	( paths are geodesics in curved Riemann space )
	( but, is space is really curved? are mass & time really functions of velocity? )
	
Williams found Riemann and Weyl Manifolds in 5-D space 
	( speed of light is the speed limit, because all forces -> 0 as speed -> \[ConstantC] )
	( Energy Manifold is Weyl with path dependent integrals ) 
	( Entropy Manifold is Riemann, with path independent integrals )   PAO 23

William’s Compares the Geometries for Dynamic Theory

Euclid 3-D

Rectangular Manifold
As a vector moves through space it retains its length and orientation.

Reimann 4-D

Elliptical Manifold
As a vector moves through space it retains its length and adjusts its orientation

Weyl 5-D

Hyperbolic Manifold
As a vector moves through space, it adjusts both its length and its orientation

Newton’s Laws by restricting Laws of Thermodynamics

Simplify 1st Law of Thermodynamics to a purely mechanical system.
	No heat, no pressure volume work, no internal Energy.
	System Energy due only to motion and one linear work term.

0 = m v ((dv)/(dt)) - F ((dx)/(dt))

F ((dx)/(dt)) = m v ((dv)/(dt))   (* Note that F is a function of velocity *)

F = m a             (* But, Newton's F = F[x], William’s F = F[x,v] *)

Newtons’ 1st and 2nd Laws can be derived vy Restricting 1st and 2nd Laws of Thermodynamics.  
But not his 3rd Law of Equal and Opposite Reactions!
By taking Energy to be fundamental, we learn that F is a function of velocity.

PAO, 15, 23, 72

Newton’s F = ma from Dynamic Theory Gives New insights

Williams Dynamic Theory shows that 
         F[x,v] =  ((Sqrt[1 - ((V^( 2))/(\[ConstantC]^2))]) ) F[x]

All forces must be functions of velocity.
There is a limiting velocity, (Subscript[v, 0])
(Subscript[v, 0]) = \[ConstantC], the speed of light demonstrated experimentally for ElectroMagnetism

Upon reaching that velocity, all forces, including Gravity, drop to zero.
There is no further acceleration.
Mass does not increase as v -> \[ConstantC], rather force decreases.

Einstein' s Postulate, \[ConstantC] is constant, is Required by Generalized Thermodynamics

Williams’ Dynamic Theory predicts non-singular potentials: (1/r) (e^(- λ / r))

Compare the Classic Maxwell Potential from an Electron or Proton with 
Williams’ Non-Singular Potential, derived using his Dynamic Theory

Framed[
Plot[{(1/r),(( (E^(-λ/r)))/r)/.λ->0.2},{r,0,6},PlotRange->{0,3},PlotStyle->{Dashed,Thick},AxesLabel->{"r","Potential"},BaseStyle->{FontFamily->"Arial",24},Ticks->False,ImageSize-> 72 8,Filling->0,Epilog->{Blue,Text["(1/r)",{1,1.4}],Brown,Text["((E^(-λ/r))/r)",{0.8,.4}]}],
FrameStyle->Blue,FrameMargins-> 12,RoundingRadius->12,ImageMargins->4]

The Electric Field is the Derivative of Electric Potential:

Clear[er,erλ]
Labeled[
Framed[
Style[
Grid[
{{"Classical", "Dynamic Theory"},
{Style[ er[r_]=D[(1/r),r],36],Style[erλ[r_,λ_]=D[(( (E^(-λ/r)))/r),r],36]}},
Dividers->{{False,True},{True,True}}],
24,Bold],
FrameStyle->Blue,FrameMargins-> 12,RoundingRadius->12,ImageMargins->4],
Style["Non-Singular Electric Field
      of Small Particles",{28,Bold,Blue,FontFamily-> "Helve","Cener"}]
]

Framed[
Plot[{er[r],erλ [r,0.2]},{r,0,3},PlotRange->{-25,25},
PlotStyle->{{Dashed,Thick,Black},{Thick,Blue}},
AxesLabel->{"r","E-Field"},
BaseStyle->{FontFamily->"Arial",24},
Ticks->{{{0.2, "λ"},{1,"5 λ"},{2,"10 λ"}},False},
ImageSize-> 72 8,
Epilog->{Text["Dynamic Theory",{.2,15},{-1,0}],Text["Standard Model",{.3,-15},{-1,0}]}],
FrameStyle->Blue,FrameMargins-> 12,RoundingRadius->12,ImageMargins->4]

E-Field for Proton and Electron

Force on a particle is the product of its Charge and the imposed Electric Field.
The force of attraction between a proton and an electron is indefinitely high for 0 separation in the classic model.  
However, the force between particles is finite and holds some surprises in the Dynamic Theory model.

Remember:

The proton feels a force from the electron’s E-field, 
		Fp[r] = qp Ee = ((- e^2)/(4 π ϵo r^2))(1-((λe)/r)) (e^(- λe / r))

     The electron feels a force from the proton’s E-field, 
     		Fe[r] = qe Ep =  ((+ e^2)/(4 π ϵo r^2))(1-((λp)/r)) (e^(- λp / r))

DT, P255

Parameter Values Based on Experiments

Based on Measured Data from particle collisions, Williams estimated values for 

        λe =  10^-18 meter = 0.001 fermi
     
     λp =  10^-15 meter = 1 fermi
     
These values control the shape of the associated E - field.

Force Plot for Proton and Electron At Atomic Distances

Minimize[(1-λp/r) E^(-λp/r)/.λp->10^(-15),r]//N;

Column[{
LogLinearPlot[{-(1-λe/r) E^(-λe/r)/.λe->10^-18,(1-λp/r) E^(-λp/r)/.λp->10^-15},{r,10^-19,10^-11},
PlotStyle->{{Thick,Blue},{Thick,Purple}},
Frame->True,FrameLabel->{"r in meters","Normalized   Force"},
PlotLabel ->Row[{Style["⊖   Fe → ",Purple], "    Attraction Forces    ",Style[" ← Fp   ",Blue],Style["⊕",48,Blue]}],
BaseStyle->{FontFamily->"Arial",18,Bold},
ImageSize-> 72 9,
AxesOrigin->{Log[5 (10^(-16))],0},
Epilog-> {
Text[Style["Fe",Purple],{Log[10^-12], 0.9}],
Text[Style["Fp",Blue],{Log[10^-12], -0.9}],
Text[Style["rN = 0.5 λe",Black],{Log[(10^(-15))],0.2},{1,0}],
Text[Style["rH",Black],{Log[9 (10^(-12))],0.1}],
{Thick,Black,Line[{{Log[9 (10^(-12))],-0.05},{Log[9 (10^(-12))],0.05}}],Line[{{Log[5 (10^(-16))],-0.05},{Log[5 (10^(-16))],0.05}}]},
Text[Style["λp = 10^-18",Blue],{Log[0.7 (10^(-18))],-0.2},{-1.,0.}],
Thick,Blue,Line[{{Log[10 (10^(-19))],-0.05},{Log[10 (10^(-19))],0.05}}],
Text[Style["λe = 10^-15",Purple],{Log[0.7 (10^(-15))],-0.2},{-1,0}],
Thick,Purple,Line[{{Log[10(10^(-16))],-0.05},{Log[10 (10^(-16))],0.05}}]
}],

fp[lr_]=-(1-λe/r) E^(-λe/r)/.{λe->10^-18,r->10^lr};
fe[lr_]=(1-λp/r) E^(-λp/r)/.{λp->10^-15,r->10^lr};
Manipulate[
{Graphics[
{Thick,Blue,Arrow[{{2,0},{2+fp[lr],0}}],
Purple,Arrow[{{0,0},{fe[lr],0}}]},PlotRegion->{{-.1,-.1},{2.1,.1}},
AspectRatio->1/8,ImageSize->9 72
]},
{{lr,-15.5,"Log10[r]"},-19,-11,Appearance->"Open",AspectRatio->{3,1}}
]
}](*,
FrameStyle->Blue,ImageMargins->4,FrameMargins->12,RoundingRadius-12]*)

{e-, p+} Mutually Attract for r = molecular size, but NOT for atomic or nucleon size

Newton’s 3rd Law (equal & opposite forces) holds approximately for distances > 10^-11 meter
Newtons 3rd Law does not apply for distances < 10^ -12 meter
For r near 0.5 λp = 5 × (10^(-16)) meter: electron is repelled by the proton & proton is attracted to electron
For r near 0.5 λe = 5 × (10^(-19)) meter: proton is repelled by the electron & electron experiences no force from proton

Non-Relativistic Neutron Model with Bohr Circular Orbits

Find r, re, rp and Vp in terms of Ve by solving simultaneous nonlinear equations:

(Ve/re)  == (Vp/rp) 
r == rp - re 
mp((Vp/rp)) ==( ((-e^2)/(4 π ϵo r^2)))(1-((λe)/r)) (e^( - λe / r))
me((Ve/re)) ==( ((-e^2)/(4 π ϵo r^2)))(1-((λp)/r)) (e^( - λp / r))

There is a surprizing meta-stable geometry.
The electron and the proton can orbit a common point.
Their center of mass also orbits.
e, p, c.o.m. all have same angular velocity

The Neutron from Non-Relativistic Dynamic Theory

For r ~ 5 (10^(-16)) meter, p+ attracted to e- and e- repelled by p+

statNeu=Graphics[{
{Purple,Dashed,Circle[{0,0},0.2,{(1.1)π,2.9 π}]},{Blue,Dashed,Circle[{0,0},1,{π/15,(2) π}]},Arrow[{{0,0},{0,0.2}}],Arrow[{{0,0},{.707,.707}}],
Text["re",{-.04,.08}],Text["rp",{.38,.32}],Text[Style["e-",Purple],{-.2,-0.03}],Text[Style["p+",Blue],{1,0.1}],
Text["Ve",{-.2,0.26}],Text["Vp",{1,-0.55}],
Arrow[{{-.2,.2},{-0.2,0}}],Arrow[{{1,-0.5},{1,0}}]},
BaseStyle->{16,FontFamily->"Arial",Bold},
ImageSize->5 72];

dynNeu=Manipulate[
Graphics[{
{Point[{1.16,1.16}],Point[{-1.16,-1.16}]},
{Dashed,Circle[{0,0},0.2],Circle[{0,0},1]},
{Purple,Disk[-0.2{Cos[θ+π],Sin[θ+π]},0.05],Blue,Disk[1{Cos[θ],Sin[θ]},0.15]}
}],
{{θ,7.π/4},0,2 π,Appearance->{"Open","Labeled"}}];

Row[{statNeu,dynNeu} ]

Not to Scale

Dynamic Theory Non-Relativistic Neutron in 3D

dynNeu=Manipulate[
Graphics3D[{
{Point[{1.16,1.16,.16}],Point[{-1.16,-1.16,-.16}]},
{PointSize->Medium,
Rotate[Point@Table[{Cos[u],Sin[u],0},{u,0.,2 Pi,2 Pi/30}],{{1,0,0},{0,1,0}}],
Rotate[Point@Table[0.2{Cos[u],Sin[u],0},{u,0.,2 Pi,2 Pi/15}],{{1,0,0},{0,1,0}}]
},
{Purple,Sphere[0.2{-Cos[θ+π],-Sin[θ+π],0},0.05],Blue,Sphere[{Cos[θ],Sin[θ],0},0.15]},Lighting->"Neutral"
}, AspectRatio -> Automatic,AxesOrigin->{0,0,0},Axes->True,Ticks->False,ImageSize->72 9],
{{θ,π/4.},0,2 π,Appearance->{"Open","Labeled"}}]

Not to Scale

Electron attracts Proton (central force)
Proton repels Electron (central force)
Orbital Motion Balances Forces (radial force)
Meta-Stable ... Half Life ~ 600 seconds

Non - Relativistic Deuteron Model

deutPlot=Graphics[{{Purple,Disk[{0,0},0.02],Text["e-",{0,-0.06}]},
{Dashed,Circle[{0,0},1,{π/15, π}],Circle[{0,0},1,{π(1+1/15),2 π}]},
Arrow[{{0,0},{.707,.707}}],
Text["r",{.38,.32}],
{Blue,Disk[{1,0.1},0.1],Disk[{-1,-0.1},0.1],White,Text["p+",{-1,-0.1}],Text["p+",{1,0.1}]},
Text["Vp",{1.,-0.57}],Text["Vp",{-1,0.57}],
Arrow[{{-1,.5},{-1,0}}],Arrow[{{1,-0.5},{1,0}}]},
ImageSize-> 72 7,BaseStyle->{24,FontFamily->"Arial",Bold}]

Not to Scale

Protons repel each other and near λp, also repel the electron.
Both protons are attracted by the electric field of the electron.
Electron repels protons that approach too closely.

Low Activation Energy for Deuterium Fusion to He-4 in a Magnetic Field

Graphics3D[{{
Texture[deutPlot],
Polygon[{{-1,-1,-1},{1,-1,-1},{1,1,-1},{-1,1,-1}},
VertexTextureCoordinates->{{0,0},{0,1},{1,1},{1,0}}
]},
{Texture[deutPlot],Polygon[{{-1,-1,1},{1,-1,1},{1,1,1},{-1,1,1}},
VertexTextureCoordinates->{{0,0},{1,0},{1,1},{0,1}}]}},
Lighting->"Neutral",Boxed->False,ImageSize-> 9 72]

Use Magnetic Field to orient deuterons as shown.
Electrons and Protons Repel Each Other.
Electrons Attract Protons.
α has 4 protons in Plane and 2 electrons above and below plane.

Dynamic Theory Model for Non-Relativistic He-4 Nucleus

Manipulate[

Graphics3D[
{
{PointSize->Tiny,Point[{1.21,1.21,.21}],Point[{-1.21,-1.21,-.21}]},,
{Purple,Opacity->0.5,Sphere[{0,0,0.2},0.05],Sphere[{0,0,-0.2},0.05]},
{Blue,Opacity->0.5,Sphere[{Cos[u],Sin[u],0},0.2],Sphere[{Cos[u+π/2],Sin[u+π/2],0},0.2],Sphere[{Cos[u+π],Sin[u+π],0},0.2],Sphere[{Cos[u+3π/2],Sin[u+3π/2],0},0.2]},
{Dashed,Line[{{0,0,-0.15},{0,0,0.15}}],Line[{0.8{Cos[u],Sin[u],0},0.8{Cos[u+π],Sin[u+π],0}}],Line[{0.8{Cos[u+π/2],Sin[u+π/2],0},0.8{Cos[u+3π/2],Sin[u+3π/2],0}}]}
},
Boxed->False,Lighting->"Neutral",AxesOrigin->{0,0,0},Ticks->False,ImageSize->7 72],

{u,0,2 Pi,Appearance->{"Open",Large}}]

Not to Scale

Effects of Applying Restrictions to the Full Set of 5-D Equations
with: 3 (Space) + 1 (Time) + 1 (MassDensity) = 5 variables

One applies restrictions to the general equations to find established equations of physics
     One set of restrictions leads to Newton’s laws
     Another similar set leads to classic Maxwell Equations for Electromagnetism
     A slightly less restrictive set leads to Special Relativity
     A similar set leads to Relativistic Maxwell Equations for Electromagnetism
     A different set leads to Schrödinger equation for Quantum Mechanics with new insights into what is waving in the wave functions (it is Weyl’s guage function)
     A different set of restrictions yields the equations for General Relativity with new insights about the connection between electric, magnetic and gravitational forces ... 
     These are equations that Mathematica is now capable of solving.

Exmples of Predictions by Dynamic Theory

- Quantization is required in isolated, isentropic systems not during irreversible processes.
- Non-Singular Potential with λ ∝ mass exhibits behavior of strong and weak forces.
- Strong force between like particles, (Subscript[λ, 1]) = (Subscript[λ, 2]) 
- Weak force between unlike particles,  (Subscript[λ, 1]) ≠ (Subscript[λ, 2])
 -Non-Singular Potential for all forces ==> no singularities, no renormalization needed. 
- All Forces propagate at speed < \[ConstantC].
- Gravity and Mass Density are functions of time.
- Speed Limit for force eliminates singularities in mass.
- Time dependence of Gravity & speed limit explains spiral arms without dark matter.
- Proton is 2 positrons in orbit around an electron.
- Neutron is proton in metastable orbit around an electron. 
- Deuterium nucleus is 2 Protons in orbit around two electrons.
- Alpha Particle is 4 Protons in orbit around 2 electrons.
- Cosmic Red Shift has two components:  distance and gravitational gradient.

- Simultaneous red shift measurements in orbit & on earth could revise Quasar Distances.
- Gravitational Acceleration in a bore hole on earth differs from Newton’s Model.

- Induction between E and B Fields is confirmed with new insights.

- Induction between G and B Fields is predicted, consistent with available experimental evidence.

- Mass Estimates for atoms with Z < 16 fit measured values better than standard model.

- Deuterium Fusion at lower T and P than standard model if in strong magnetic field.
- Restricting the 5-D 1st and 2nd Laws of Thermodynamics to fewer variables reproduces proven equations for Newton, Quantum Mechanics, General and  Special Relativity, Quantum electrodynamics, Maxwell Equations, and provides new insights into each.
- Relevatistic solutions needed for all atomic and nuclear orbital models.

Need Solutions for Relativistic Particle Orbits
Positron + Electron inside Proton
Electron + Proton inside Neutron
Electrons + Protons inside Deuterons, Alpha and larger nuclei
Mathematica Tensor Calculus Functionality Now Seems Capabile Of That
I, working alone, am not yet capable.  Any Volunteers?

see: Dynamic Theory--A New View of Space-Time-Matter by Pharis Williams  (2011)

Learn more about Williams’ Dynamic Theory & Equations Needing Solutions

For easy reading :  
Physics - Against The Odds 
by Pharis E. Williams (Feb 10, 2009)

For Details of Mathematics  :  
The Dynamic Theory - A New View of Space-Time-Matter... 
by Pharis E Williams (Mar 23, 2011)