For illustrations of the meaning of many variables, see "Howell - Figures for Lucas Universal Force.odt", including :
Figure "General reference frames"
Figure "Chapter 4 reference frames"
The lists below include BOTH Lucas' notation (as required to understand his book) and mine (to follow my derivations).
Latin symbols
A amplitude of neutral dipole vibrations (A1, A2 (Figure 7-2,8-1))
A or A2 is a 2D angle (radians),
A1 1D angle (+,-)
A2 2D angle (radians)
A3 3D solid angle (steridians) -> Lucas symbol Ω, omega (5-16)
Aθ=AO 2D angle in O(theta) direction. This is the angle rooted at the [observer, particle,POI_root], that is measured from [a POI_start, or the direction of a vector (typically Vonv(particle))] to [a POI_end, or vector R_end ]. Note that the plane of intersection of [Vonv(particle),POIo or POIp (POIs FIXED in RFo or RFp)] is CONSTANT - ie there is no rotation or complication as the particle moves. The O(theta) direction is like a latitude.
AO is shorthand notation for fast typing, converted later.
Aφ=AP 2D angle in P(phi) direction. This is the angle IN A PLANE (often a plane that is perpendicular to Vonv(particle)), that is measured from [a POI_start, or the direction of a vector (typically the y-axis where the z-axis is aligned with Vonv(particle))] to [a POI_end, or vector R_end]. Apply the right hand rule with thumb in the direction of the vector to get positive and negative directions. The P(phi) direction is like a longitude (as suggested by Lucas's integration limits for φ from 0 to π p95h0.9 Equation (7-16)).
AP is shorthand notation for fast typing, converted later.
AQ is the angle between Rocv(observer to particle) the directed vector from observer to particle, and Vonv(particle), the constant velocity of the particle for Chapter 4. Note that the plane of intersection of [R,V] is CONSTANT - ie there is no rotation or complication as the particle moves. Note that this angle isn't used in Chapter 4, as it is assumed that the observer is located on the line L(Vocv(particle)) that is ANCHORED on the particle, such that AQ(Rocv(observer to particle), Vonv(particle)) = 0. Note that the plane of intersection of [r,v] is CONSTANT - ie there is no rotation or complication as the particle moves. This, along with ro(t=0) from the observer to the particle, firmly "links" the observer and particle frames of reference.
a acceleration = d2[d^2t: rv] - 2nd derivative of displacement vector
ag or as acceleration wrt center of galaxy,
au or a acceleration wrt center of universe
B magnetic field, B0 - from static charges , Bi - induced
NOTE : It would help clarity if the [electric, magnetic fields] were designated [E,B]_[0,V,S] (for [static, induced from velocity across field, induced from changing field]. In Chapter 4, ALL B are induced! (i.e. no magnetic materials)
<<< 29Mar2018 - 3 types of [E,B]!! >>>
b = β b = v/c (default scalar)
c speed of light in a vacuum (default scalar, taken as dimensionless for Chapter 4 for dimensional analysis)
D electric displacement field D = ε0*E + P = ε0*εr*E
where P = polarization density
(see https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law)
d[dt : f(t)] derivative short-hand notation eg d(dt : f(t)) = d/dt(f(t) Default meaning is "total" derivative
dt[dt : f(t)] total" derivative eg dt(dt : f(x,v,a,t)) = d/dt f(x,v,a,t)
dp[dt : f(t)] "partial" derivative eg dp(dt : f(x,v,a,t)) = ∂/∂t f(x,v,a,t)
d^3r =dx*dy*dz
E electric field, E0 static or primary, Ei - induced, Ei′(r′,t′) - new, different from Ei but not defined (see Lucas p68,69)
NOTE : It would help clarity if the [electric, magnetic fields] were designated [E,B]_[0,V,S] (for [static, induced from velocity across field, induced from changing field strength]. In Chapter 4, ALL B are induced! (i.e. no magnetic materials)
<<< 29Mar2018 - 3 types of [E,B]!! >>>
I assume that Ei′(r′,t′) = E0(r′,t′) + Ei(r′,t′) = total E, but this contradicts statements p68h0.3 where Ei is induced only (doesn't include static)
e charge of proton,
-e charge of electron (7-13)
F force, F_G gravity, F_I inertial, F(2+,1+) etc (7-11, 7-25)
Fu Howell's notation specific to derivations [of, from] Lucas's "Universal force"
Fu_G Lucas's "Universal force" expression for gravity
f frequency (Fiigure 7-2,8-1)
G Newton's universal gravitational constant
H is the magnetic H-field (in ampere per metre, Am−1) https://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_law
H is the magnetic H field (also called "auxiliary magnetic field", "magnetic field intensity", or just "magnetic field")
H = B/μ0 - M
h Planc's constant (Barry Setterfield : h*c = constant) (2-6)
J is the total current density (in ampere per square metre, Am−2) https://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_law
J displacement current J_D = ∂/∂t(D(r,t)) (4-9)
J includes magnetization current density[13] as well as conduction and polarization current densities.
J the current density contribution actually due to movement of charges, both free and bound
J = Jf+ Jd + Jm
Jd displacement current
Jf enclosed conduction current or free current density
Jm magnetization current density?
J charge density (Jackson1999 p554h0.7 Equation (11.127)
L(v) = λ(v) velocity-dependent proportionality between Ei and E0 (4-31) (default scalar)
L(POI) is a line running through the POI that is parallel to Vonv(particle)
L(particle) = L(POI) for POI = particle
L(observer) = L(POI) for POI = observer
Lx(vo) is a line that is colinear with rx(POI), but note that it is the SAME for ALL POI!!
l path or line (4-2)
M magnetization field - how strongly magnetized a region is
Mg mass of spiral galaxy hosting a neutral electric dipole (10-6)
Mu mass of the universe (10-6)
magnetic field - The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter (symbol: Am−1 or A/m) in the SI. B is measured in teslas (symbol:T) and newtons per meter per ampere (symbol: Nm−1A−1 or N/(mA)) in the SI.
B is most commonly defined in terms of the Lorentz force it exerts on moving electric charges.
m mass (see also M - of galaxy, Mu of universe (10-6) )
n quantum level or number (2-7)
n unit basis vector, eg Cartesian coordinates x = n1*x1 + n2*x2 + n3*x3
n1,n2,n3 (note : often i j k are used)
n_hat unit vector normal to a surface (4-2, 4-3)
PI= Shorthand notation for fast typing, converted later. see constant π
POI (POIo,POIp) = "Point Of Interest" is a specific point, and is typically where estimates of [B,E,F, etc] are required. Normally, the POI is STATIONARY in EITHER :
RFp - particle reference frame (eg POIp,POIo(t))
RFo- observer reference frame (eg POIo,POIp(t))
Note that a POI cannot be simultaneously be fixed in BOTH RFo and RFp at a specific time t, EXCEPT when BOTH vo & vp = zero.
POIo(t) = POIp fixed in particle RFp, but seen in the observer Rfo at time t (eg for R,A,E,B,etc)
POIp(t) = POIo fixed in observer RFo, but seen in the particle RFp at time t(eg for R,A,E,B,etc)
See Section ""Points of Interest" (POI)s" for more details.
Q is the angle between r (directed vector from observer to particle) and v (the constant velocity of the particle for Chapter 4). Note that the plane of intersection of [r,v] is CONSTANT - ie there is no rotation or complication as the particle moves.
q charge (4-3, 4-6, 4-14)
R displacement (distance, direction) between two masses
Re radius of Earth (position on surface of the Earth (8-15)
Rg displacement (distance, direction) to center of galaxy
Ruc for a universe with spherical symmetry, Ruc is the distance to the center of the universe from the vibrating neutral electric dipole (8-13)_(10-6)
R(POIstart,POIend) is used to denote [scalar distances, vectors, unit vectors, displacement vectors, coordinate displacement vectors] depending on the qualifiers of the symbol (eg [s,v,h,c]). If both POI are present, POIstart indicates the start point, POIend the endpoint. If only one POI is present, then it denotes the endpoint, whereas POIstart is the coordinate origin - but this is also emphasized by using the qualifier "c", giving "redundancy" to reduce errors .
Examples :
using the particle frame of reference (RFp) : eg [Rpcv,Rpds,Rpnh,Rpd]
using the observer frame of reference (RFo) : eg [Rocv,Rods,Ronh,Ros,]
Special notations for r :
[R2,R1] often used by Lucas for polar ends of vibrating dipole etc, r21 = ?? (7-11), R to center of universe,
RDEpdh(Rpch(POIo,t,dt)) unit vector in direction of dp[dt : E0odv(POIo,t)] , that is, the differential change in static electric field due to particle movement relative to a POIo.
Rθ0och unit vector in the direction of θ=0 (theta=0). In Chapter 4 this is the direction of the particle velocity in both the particle and observer reference frames (RFp,RFo), ie Rθ0pch = Rθ0och.
Rθ0ocs(POIo) Rθ0pcs(POIp) etc
Rθ0ocs(POIp(t),t) Rθ0pcs(POIo(t),t) etc
scalar distance that is the Rθ0ch direction component of a coordinate displacement vector from the origin of the observer reference frame (RFo) to a point of interest (POI,t) (i.e. a distance from coordinate origin along the direction of the particle movement).
As another way to say it, Rθ0cs is the magnitude of a displacement vector that is perpendicular to L(POI) and that is anchored to the POI (Point Of Interest). See Figure ???
RθPI2och(POIp(t),t) RθPI2pch(POIo(t),t) etc
unit vector, orthogonal to Rθ0ch and intersecting POI, that is at an angle of ???Pc(POI)??? to RθPI20ch. Note that this applies to either [RFo,RFp], and that in Chapter 4 RθPI2och = RθPI2pch.
RθPI2cs(POI,t) scalar distance that is the RθPI2ch direction component of a coordinate displacement vector rpc. Note that this applies to either [RFo,RFp], and that in Chapter 4 RθPI2ocs = RθPI2pcs
rx(POI) = perpendicular distance from the particle center to the L(POI). In Chapter 4, given constant vov(t) = vov, and Chapter 4's (constant vov, aligned particle & observer reference frames (RFp & RFo)), rx(POI) is NOT a function of time nor Pph=Poh.
rxh is a unit vector in the direction of rx
S Poynting vector (5-15)
s position
s symbol qualifier - means scalar (eg magnitude of a vector)
t time
tx time at which POIo = POIp(tx), or POIp = POIo(tx)
U generalized potential (usually wrt Universal force or component) (5-2)
v velocity
vs velocity of circular orbits (10-10)
vtan speed (component of velocity) tanjent to curve
x Position vector in Cartesian coordinates (comparing to more conventional notations :
x1 = x,i; x2=y,j; x3=z,k
Z charge state of [atom, particle,...]
z red shift (7-29))