If f(r) = f(|r|) = f(r) we have Ñ_{r}f(r) = f'(r)r^{-1}r = f'(r)r^{~} and Ñ_{r}^{2}f(r) = r^{-1}f'(r) + f"(r).
This then is the Newtonian model. The 3D gravitational force field G(r) is characterised by it's
scalar "integral" f(r) = -Ñ_{r}^{-1} G(r) ;
a 0-potential scalar
field which satisfies Poissons equation everywhere and provides the acceleration on a test mass at r
as G_{N}(r)=-Ñ_{r}f(r) . We can in principle compute f(r_{0}) as the infinite volume integral
f(r_{0}) = gò_{Â3} m_{r}(r-r_{0})^{-½} |dr^{3}|
The scalar field m_{r} fully determines G_{N} so the "flow" of matter is relevant in determining the instantanous gravitational
potential only by its magnitude. It's direction effects only how G_{N} changes over time.
Gravity is such that the force exerted on a test particle is proportional to the mass of the test particle
so that the acceleration experieneced is independent of the mass, More generally we have a force field
F_{p} = Ñ_{p}f_{p} .
Effective Potential
Under a central force with classical 0-potential f(r)=f(r) ,
the 2-blade angular momentum L=mrÙr' and scalar energy E=½m(r')^{2} + V(r) are both conserved
and the radial energy equation
½m(r')^{2} + V(r) - ½m^{-1}r^{-2}L^{2} = E
for r³0 implies that the shape of the 1D scalar effective potential
f(r) - ½m^{-1}r^{-2}L^{2}
= f(r) + ½m^{-1}r^{-2}L^{2}
determine the
possible orbits for a given L=|L^{2}|^{½} and E combination.
If the particle is released from rest L=0 , but otherwise the effective potential contains a
term equivalent to that contributed by a repulsive inverse cube force
m^{-1}L^{2} r^{-3}
Orbital Stability Indicator
Suppose a particle of inertia m is subject to a time t-independant central force
¦(r,t)r^{~} = ¦(r)r^{~}.
If x=r-r_{0} denotes the deviation of r from cirular orbit value r_{0} then
mx^{..} » (r_{0}^{-1}3¦(r_{0})+¦'(r_{0}))x
to first order in r^{-1}x
where ^{..} denotes second time derivative. For gravitationl forces
¦(r) is proportionate to the inertia m.
mr^{..} = ¦(r)+L^{2}r^{-3} where
angular momentum L=mr^{2}q^{.} is
conserved.
Let x=r-r_{0}. We have
mx^{..} = ¦(r_{0}+x) + L^{2}(r_{0}+x)^{-3}
»
¦(r_{0})+x¦'(r_{0})+½x^{2}¦"(r_{0}) + 3!^{-1}¦"'(r_{0})x^{3} + ...
+ r_{0}^{-3}L^{2}(1-3r_{0}^{-1}x + 6r_{0}^{-2}x^{2} - 10r_{0}^{-3}x^{3} +...
For circular orbit at r_{0} we require L^{2} = -mr_{0}^{3}¦(r_{0}) ,
w = (-r_{0}^{-1}m^{-1}¦(r_{0}))^{½}
so
mx^{..}
» ¦(r_{0})+x¦'(r_{0})+½x^{2}¦"(r_{0}) + 3!^{-1}¦"'(r_{0})x^{3} + ...
- ¦(r_{0})(1-3r_{0}^{-1}x + 6r_{0}^{-2}x^{2} - 10r_{0}^{-3}x^{3} +...)
= x¦'(r_{0})+½x^{2}¦"(r_{0}) + 3!^{-1}¦"'(r_{0})x^{3} + ...
+ ¦(r_{0})(3r_{0}^{-1}x - 6r_{0}^{-2}x^{2} + 10r_{0}^{-3}x^{3} +...)
= x(¦'(r_{0}) + 3r_{0}^{-1}¦(r_{0}))
+ x^{2}( ½¦"(r_{0}) - 6r_{0}^{-2})
+ x^{3}(3!^{-1}¦"'(r_{0}) + 10r_{0}^{-3})
+ _{O}(x^{4})
Thus if
orbital stability indicator
O_{SI}(r_{0}) º -3r_{0}^{-1}¦(r_{0}) - ¦'(r_{0})
= 3r_{0}^{-1}f'(r_{0}) + f"(r_{0})
is positive we have oscillations about the
circular orbit of frequency m^{-½} O_{SI}(r_{0})^{½} and so period
2pm^{½} O_{SI}(r_{0})^{-½} , a proportion
(-3-r_{0}¦'(r_{0})¦(r_{0})^{-1})^{-½}
of the orbit period 2p(-r_{0}^{-1}m^{-1}¦(r_{0}))^{-½}, while if O_{SI}(r_{0}) is negative we have hyperbolic deviation.
We will see below that O_{SI}(r) = 2r^{-1}E'(r) where E(r)=f(r)+½rf'(r) is the total energy
of a circular orbit of radius r.
If ¦'(r_{0}) + 3r_{0}^{-1}¦(r_{0}) vanishes, we have harmonic deviation equation
mx"
» x^{2}( ½¦"(r_{0}) - 6r_{0}^{-2})
= -P_{SI}(r_{0})x^{2}
where
P_{SI}(r_{0}) º
r_{0}^{-2}(6+½r_{0}^{2}f"'(r_{0}))
= 6r_{0}^{-2} + ½f"'(r_{0})
.
This is analytically problematic since while x" = ax^{2} has a particular known solution
x(t)=6a^{-1}(t+6a^{-1}x(0)^{-1})^{-2} this is not general enough to match a given x'(0) .
The normalised orbilital stability indicator of a circular orbit of radius r_{0}
under a central force ¦(r) £ 0 is the value
O_{SI}(r_{0}) º (3+r_{0}¦(r_{0})^{-1}¦'(r_{0}))^{½}
, where ^{½} denotes that
for negative values we take the negative square root of the absolute value.
In eliminating the magnitude of ¦ by multiplying O_{SI}(r) by |r^{-1}¦(r_{0})|^{½},
we aquire an infinite O_{SI}(r) when ¦(r)=0. The sign of ¦(r) has also been eliminated so O_{SI}(r) of a repuslive
central force is the O_{SI} of the attactive force -¦(r). For nonzero ¦(r), O_{SI}(r) is the
number of orbits about 0 per peturbation oscillation.
O_{SI}(r) provides a measure of deviation from shell condition p(t)^{2} = p(0)^{2}
under acceleration p"(t) = -f'(|p(t)|)p(t)^{~} where |p(t)| º |p(t)^{2}|^{½}, assuming
p(0)¿p'(0) is small. Large O_{SI} suggests rapid oscillations across the shell with integer values suggesting petal forms.
O_{SI}<1 suggesting slowly corrected slightly noncircular orbits.
"Power law" attractive force f(r)=-ar^{k} has O_{SI}(r) = (3+k)^{½}
so stable circular orbits require k>-3 . Inverse square forces have unit O_{SI}(r), so the period of the deviation
coincides with that of the orbit, stretching it into an ellipse for example.
An integer O_{SI}(r) indicate a single repeated orbit
while rational O_{SI}(r) pq^{-1} implies that after q orbits each of which contain p deviation cycles the trajectory will repeat itself.
An r^{-1} orbit takes ½^{½} orbits to complete an oscillation and, as this is irrational,
the trajectory in theory never repeats. However, as the oscillatory nature of deviations is only correct to first order in small deviations, such
multi-orbit analysis is profoundly suspect.
High O_{SI}(r) indicate rapid oscillations about the circular orbit while high negative O_{SI} indicate rapid hyperbolic deviations,
with the caviat that ¦(r) changing sign gives infinite O_{SI} when ¦(r)=0 corresponding to deviations from the straight line
zero forces trajectory.
Zero O_{SI} indicates paraboloid deviations from u with q correspending to paraboloid deviation from u_{0} with q
while O_{SI} j as for ¦(r)=-ar^{j2-3} corresponds to oscillating j times faster than the orbit.
To maximise O_{SI}(r) º r^{-1}3f'(r) + f"(r)
we consider O_{SI}'(r) = -3r^{-2}f'(r) + 3r^{-1}f"(r) + f"'(r)
which is zero when
f"'(r) = 3r^{-2}(f'(r) - rf"(r)) whereupon
P_{SI}(r) =
r^{-2}(6+(3/2)(f'(r) - rf"(r)))
= (3/2)r^{-2}(4 + f'(r) - rf"(r))
.
P_{SI}'(r) = -12r^{-3} + ½f^{(4)}(r)
vanishes everywhere for f(r) = 6r(ar)^{↓} + P_{4}(r)
= 6(a^{↓}r + 6r(r)^{↓} + P_{4}(r)
for arbitary a and quartic polynomial P_{4}() with
f'(r) = 6(ar)^{↓} + 6 + P_{4}'(r).
f"(r) = 6r^{-1} + P_{4}"(r).
f"'(r) = -6r^{-2} + P_{4}"'(r).
f"(r)+(N-1)r^{-1}f'(r)-(N-1)
O_{SI}(r_{0})=
3r_{0}^{-1}f'(r_{0}) + f"(r_{0})
= 18r_{0}^{-1}(ar_{0})^{↓} + 18_roinv + 3P_{4}"(r)
+ 6r^{-1} + P_{4}"(r).
= 18r_{0}^{-1}(ar_{0})^{↓} + 24_roinv + 4P_{4}"(r)
.
ò_{0}^{r0} dr 6r(ar)^{↓}
= 6 ò_{0}^{ar0} dx a^{-1}x (x^{↓})
= 6a^{-1} [
x^{2}(½(x)^{↓}-¼)
]_{0}^{ar0}
= 6ar_{0}^{2}(½(ar_{0})^{↓}-¼) .
However, this is less intersting than P_{SI}'(r)=0 coinciding with
O_{SI}'(r)=0 and P_{SI}(r)=0 .
Stability Indicators of a Klein Gordan 0-Potential
If Ñ_{x}^{2}f(r) = af(r) then
f"(r) =
±af(r) - (N-1)r^{-1}f'(r)
according as x^{2} is ±,
and so
O_{SI}(r) = (4-N)r^{-1}f'(r) ± af(r)
and for N=4 we have O_{SI}(r) = ±af(r) .
O_{SI}'(r)
= (±a - (4-N)r^{-2})f'(r) + (4-N)r^{-1} f"(r)
= (±a - (4-N)r^{-2})f'(r) + (4-N)r^{-1}
(±af(r) - (N-1)r^{-1}f'(r))
= (±a - (4-N)r^{-2} - (4-N)(N-1)r^{-2})f'(r)
±a(4-N)r^{-1}f(r)
= (±a - N(4-N)r^{-2})f'(r)
±a(4-N)r^{-1}f(r) .
Thus O_{SI}(-r) is extreme when
= f'(r) =
-/+a(4-N)r^{-1}(±a - N(4-N)r^{-2})^{-1}
f(r) .
Further,
f"'(r) = ±af'(r) - (N-1)r^{-1}
(±af(r) - (N-1)r^{-1}f'(r))
+ (N-1)r^{-2}f'(r)
=
(±a + N(N-1)r^{-2})f'(r) -/+a(N-1)r^{-1}f(r)
according as x^{2} is ±; and so
P_{SI}(r) = r^{-2}(6 + ½r^{2}f"'(r))
=
O_{SI}'(r) =
-3r^{-2}f'(r) + 3r^{-1}f"(r) + f"'(r)
=
-3r^{-2}f'(r) + 3r^{-1}(
±af(r)-(N-1)r^{-1}f'(r)
)
+ (±a + N(N-1)r^{-2})f'(r) -/+a(N-1)r^{-1}f(r)
Circular Orbits
-f'(r)r^{~} = -mw^{2}r for inertial mass m>0 yields w = ±|m^{-1}r^{-1}f'(r)|^{½} Þ wr = ±m^{-½} |rf'(r)|^{½} , and so circular orbits under central force ¦(r)r^{~} = -f'(r)r^{~} have
The potential f(r)=-lr^{b} with l>0 has f'(r)=br^{-1}f(r) and hence squared angular momentum mlbr^{b+2} and kinetic energy |bf(r)|. For b<0 the toal orbit energy is thus (b+1)f(r) and so in a Coloumb potential with b=-1 all circular orbits have total energy zero, while if b=-2 all circular orbits have angular momentum (2ml)_sqdrtcnj and positve total energy -f(r).
The Coulomb Potential f(r)=-lr^{-1} with f'(r)=lr^{-2} thus has
w = ±|m^{-1}lr^{-3}|^{½} ;
period
2p|m^{½}l^{-½} r^{3/2} ;
speed ±|m^{-1}lr^{-1}|^{½} ;
angular momentum ± m^{½}r^{½}
;
classical kinetic energy
½lr^{-1} ; and
radial acceleration -m^{-1}lr^{-2}.
O_{SI}(r) = r^{-1}3f'(r) + f"(r)
= -lr^{-3}
while
O_{SI}(r) = 1.
More generally a small test particle launched from r_{0}e_{1} with velocity r_{0}w_{0}e_{2} + s_{0}e_{1}
will seek a circular orbit having
mr_{0}^{2}w_{0} = m^{-½}r^{3/2}f'(r)^{½}
and
f(r_{0}) + ½m(r_{0}^{2}w_{0}^{2}+s_{0}^{2}) =
f(r) + ½rf'(r) .
Quantised Orbits
There are two natural ways to particularise orbits under a central oscillatory potential f(r).
The maxima of O_{SI}(r) provide the stablest orbits, while orbits at r satisfying
|rf'(r)| = mv^{2} provide the orbits having a given speed v.
In particular, v=1 yeilds the lightspeed orbits with inertial mass m
regarded as a coulping constant rather than a mass-energy.
Ring Potential
The ring potential f^{oR}_{z}(r) º
2R ò_{0}^{p} dq f((z^{2}+r^{2}+R^{2}-2rR cosq)^{½})
is the potential at r of a circle of radius R lieing in a plane containing
0 and r.
Setting s º (z^{2}+r^{2}+R^{2}-2rR cos(q))
we have ds = 2rR sinq dq.
cos(q) = (2rR)^{-1}(s-z^{2}-r^{2}-R^{2}) so
for q Î [0,p] we have
sin(q)
= (2rR)^{-1} (4r^{2}R^{2}
- (s - z^{2}-r^{2}-R^{2})^{2})^{½}
= (2rR)^{-1}
(2s(z^{2}+r^{2}+R^{2}) -s^{2}
+ 4r^{2}R^{2}
- (z^{2}+r^{2}+R^{2})^{2})^{½}
= (2rR)^{-1} Q(s)^{½}
where
Q(s) º
-(s^{2} - 2s(z^{2}+r^{2}+R^{2})
+ (z^{2}+ r^{2}+R^{2})^{2}
- 4r^{2}R^{2} )
= -s^{2} + 2s(z^{2}+r^{2}+R^{2}) - z^{4}
- 2z^{2}(r^{2}+R^{2}) - (r^{2}-R^{2})^{2}
=?= -(s-z^{2}-(r-R)^{2})(s-z^{2}-(r+R)^{2})
is a quadratic in s with roots z^{2} + (r±R)^{2}.
Thus
f^{oR}_{}(r)
= 2r^{-1} ò_{z2+(r-R)2}^{z2+(r+R)2} ds f(s^{½}) ( sinq)^{-1}
= 4R ò_{(r-R)2}^{ (r+R)2} ds Q(s)^{-½} f(s^{½}) .
For z=0 , Q(s) = -(s^{2} - 2s| (r^{2}+R^{2}) + (r^{2}-R^{2})^{2} ) ,
Morse Potential
Typically used for r representing the seperation between two covalently bonded atoms,
f(r) = f_{¥}(1-(-(½kf_{¥}^{-1})^{½}(r-r_{0}))^{↑})^{2}
for dissassociation energy f_{¥}>0, equilibrium or zero force distance r_{0},
and bond force constant
k>0.
Yellow shows potential f(r);
green is force f'(r); while red is the OSI (3+f'(r)^{-1}rf"(r))^{½} with imiginary values focred negative real.
For rÎ[0,r_{0}] we have f(r)£f_{0}=f(0)= f_{¥}(1-((½kf_{¥}^{-1})^{½}r_{0})^{↑})^{2} with f(r)³f_{¥} for r£ r_{0} - (2f_{¥}k^{-1})^{½}(2)^{↓}. f(r) ® f_{¥} from below as r®¥ and to ¥ as r®-¥ though we might typically impose r³0. We have a global minimum at r=r_{0} with f(r_{0})=0 , f'(r_{0})=0, and f"(r_{0})=k. |
The comnplex Morse potential
f(r) = b^{2}(-2(r-r_{0}))^{↑} - b(1+2D)(-(r-r_{0}))^{↑}
= (b(-(r-r_{0}))^{↑} - (½+D))^{2}
- (½+D)^{2}
gives real energy eigenvalues E_{n}=-(n-D)^{2}
for integer n<D. [Saaidi]
Born Mayer Potential
The Born-Mayer potential is the exponentiion f(r) = f_{0} (-ar)^{↑} with
with a>0. Since f^{(k)} = (-a)^{k} f(r) we have
O_{SI}(r) = a(a+3r^{-1}) f(r) ;
O_{SI}(r) = (3+ar)^{½}
; and P_{SI}(r) = 6r^{-2} - ½a^{3} f_{0}(-ar)^{↑}
zero when
r^{-2}(ar)^{↑} = 12^{-1}a^{3}f_{0}^{-1} .
Lennard-Jones Potential
The Lennard-Jones potential is a convenient fiction used to approximate potentials that strongly repel close up but
attract at a distance.
f(r) = 2f_{r0}(r_{0}^{-1}r)^{-6}(1 - ½(r^{-1}r)^{-6})
where f_{r0}<0 .
This is +¥ at r=0, 0 at at 2^{6-1}r_{0} » 1.12246r_{0} and ®0 as r®¥.
Since
f'(r) = 12f_{r0}r^{-1}(r_{0}^{-1}r)^{-6}(1 - (r^{-1}r)^{-6}) ,
f(r) is minimised at r=r_{0} by f(r_{0}) = f_{r0}.
Tangential occlusion potential
Constructing an inverse square particle-mediated field is unrealistic because the r^{-1}
factor representing an inverse square "dispersion" of a constant signal fails for low r .
If the absorbing particle is not attracting the
carriers but merely absorbing those that reach a small perimeter distance h ,
then we would expect the proportion of carriers absorbed to
be in proportion to the area content of the spherical cap defined by the cone
tangential to the h radius sphere centered on the absorber.
A sphere of radius h at distance r
from 0 subtends a cone of demiangle q=
sin^{-1}(r^{-1}h) =
sin^{-1}(u) where uºhr^{-1} ,
which occludes an area
2p(1- cos(q))d^{2} =
2p(1-(1-u^{2})^{½})d^{2}
of the 4pd^{2} surface area of a sphere of radius d at 0, rather than
an area proportionate to r^{-2}.
Attractive central force
¦(r) = (1-(hr^{-1})^{2})^{½} - 1 = (1-u^{2})^{½} - 1 for r³h (ie. u £1) has ¦'(r) = h^{2}r^{-3}(1-u^{2})^{-½} = h^{-1}u^{3}(1-u^{2})^{-½} = h^{2}r^{-3}(1+¦(r))^{-1} . For r>>h we have ¦(r) » -½(hr^{-1})^{2} - 1/8(hr^{-1})^{4} - K_{3}(hr^{-1})^{6} - K_{4}(hr^{-1})^{8} - ... where K_{k} º 1*3*5*7*...*(_2k-3) / (2*4*6*8*...*(2k)) = 2^{2-2k}(2k-3)! / ((k-2)! k! ) . For large r (so small u=hr^{-1}), the inverse square "Coulomb" force will be massively dominant. We find that circular orbits are stable only for u^{2}< 3/4 (ie. r>(3^{-½})2h) and that the 3 + r¦'(r)¦(r)^{-1} indicater tends to 1 from below as r®¥ . |
For a screened force
F(r) = (-lr)^{↑} ¦(r) we have
Z(F)(r) = F(r) + 3^{-1}rF'(r)
= (-lr)^{↑} ( Z(¦)(r) - 3^{-1}lr¦(r) ) .
Now -3^{-1}lr¦(r) is positive
so exponentially damping ¦ makes circular orbits unstable
whenever
-1 + (1-u^{2})^{-½}(1-(2/3)u^{2}) - 3^{-1}lr((1-u^{2})^{½}-1) ³ 0
Û -1 + (1-u^{2})^{-½}(1-(2/3)u^{2}) + 3^{-1}lr ³ 3^{-1}lr(1-u^{2})^{½}
Û -1 + 3^{-1}lr ³ (3^{-1}lr - 1 + (2/3)u^{2})(1-u^{2})^{½}
Comparing this with indicater (1-3^{-1})r^{k} for ¦(r)=r^{k} we see
that for large r, circular orbits are only just stable;
verging on the the instability associated with inverse cube orbits.
The apsidal angle p(3+r¦'(r)¦(r)^{-1})^{-½}
= p(3 +
d^{2}r^{-2}(1-u^{2})^{-½}
(-1 + (1-u^{2})^{½})^{-1} )^{-½}
=
p(3 +
u^{2}( (1-u^{2})^{½} (-1 + (1-u^{2})^{½}) )^{-1}
)
^{-½}
=
p(3 + u^{2}( 1-u^{2}-(1-u^{2})^{½} )^{-1}
)
^{-½}
When u^{2}=½ this is
p(3 + ½( ½ - (½)^{½} )^{-1}
)
^{-½}
=p(3 + (1-2^{½})^{-1} )^{-½}
» 1.30p .
It is a rational multiple of p only when
3 + u^{2}( 1-u^{2}-(1-u^{2})^{½} )^{-1} is a squared rational number,
which occurs
when 1-u^{2}-(1-u^{2})^{½} is a squared rational,
.
We will here refer to V(r) = ò_{r}^{¥} dr ¦(r)
= r¦(r) + h sin^{-1}(hr^{-1})
= r((1-(hr^{-1})^{2})^{½}-1) + h sin^{-1}(hr^{-1})
= h(u^{-1}((1-u^{2})^{½}-1) + sin^{-1}(u))
as the tangential occlusion potential.
[ Proof :
ò_{r}^{¥} dr ¦(r) = ò_{u}^{0} (-r^{2}h^{-1} du) ¦(r)
= h ò_{u}^{0} du -u^{-2}¦(r)
= h ò_{0}^{u} du ((1-u^{2})^{½} - 1)u^{-2}
= h
[
-(1-u^{2})^{½}u^{-1} - sin^{-1}(u)
+ u^{-1}
]_{0,u}
= h(u^{-1}(1-(1-u^{2})^{½}) - sin^{-1}(u))
= -r¦(r) - h sin^{-1}(hr^{-1})
.]
(¶/¶r)^{2}V + 2r^{-1}¶V/¶r
= -r^{-1}(1-u^{2})^{-½}¦(r)^{2}
[ Proof : ¶V/¶r = -¦(r) and
(¶/¶r)^{2}V + 2r^{-1}¶V/¶r =
= -¦'(r) + 2r^{-1}¦(r)
=
-h^{2}r^{-3}(1-u^{2})^{-½} +
2r^{-1}((1-u^{2})^{½}-1)
= r^{-1}(1-u^{2})^{-½}(-u^{2} + 2(1-u^{2}) - 2(1-u^{2})^{½})
= r^{-1}(1-u^{2})^{-½}(2-u^{2} - 2(1-u^{2})^{½})
= r^{-1}(1-u^{2})^{-½}(1 - (1-u^{2})^{½})^{2}
= -r^{-1}(1-u^{2})^{-½}¦(r)^{2}
.]
Setting V(r) = (1-u^{2})^{a} gives
¶V/¶r
= a(1-u^{2})^{(a-1)}2h^{2}r^{-3}
so r^{2} ¶V/¶r = 2h^{2}a(1-u^{2})^{(a-1)}r^{-1}
whence
Ñ^{2} V(r) = r^{-2} (¶/¶r)(r^{2} ¶V/¶r)
=
2r^{-2}h^{2}a (
(a-1)(1-u^{2})^{(a-2)}2h^{2}r^{-4}
- (1-u^{2})^{(a-1)}r^{-2})
= 2r^{-4}h^{2}a (1-u^{2})^{(a-2)}(
(a-1)2u^{2} - (1-u^{2}) )
= 2r^{-4}h^{2}a (1-u^{2})^{(a-2)}(
(2a-1)u^{2} - 1 )
Yukawa potential
f(r)=r^{b}(ar^{g}) has
f'(r) = (gar^{g-1}+br^{-1})f(r) ;
f"(r)
= (g(g-1)ar^{g-2}-br^{-2})f(r)
+ (gar^{g-1}+br^{-1})f'(r)
= r^{-2}(g(g-1)ar^{g}-b
+ (gar^{g}+b)^{2}
)f(r)
= r^{-2}(g^{2}a^{2}r^{2g}
+ g(g-1+2b)ar^{g}
+ b(b-1))f(r)
Thus
f"(r)+(N-1)r^{-1}f'(r)
= (g(g-1)ar^{g-2}-br^{-2})f(r)
+ ((gar^{g-1}+br^{-1}
+ (N-1)r^{-1})f'(r)
= r^{-2}(g(g-1)ar^{g}-b
+ (gar^{g}+b+N-1)
(gar^{g}+b))f(r)
= r^{-2}(g(g-1)ar^{g}-b
+ g^{2}a^{2}r^{2g} +
gar^{g}(2b+g+N-1) + b(b+N-1)f(r)
= r^{-2}(
g^{2}a^{2}r^{2g} +
gar^{g}(g+2b+N-2)
+ b(b+N-2))f(r)
Setting b=2-N, g=N-2 gives f"(r)+(N-1)r^{-1}f'(r) = (N-2)^{2}a^{2}r^{2(N-3)}f(r), thus for N=3 we have central f(r)=r^{-1}(ar)^{↑} solving Klien Gordan equation Ñ_{p}^{2}f =a^{2}f.
Taking a=-m<0, b=-1, g=1, d=0 gives the
real scalar
Yakuwa potential
or
screened-Coulomb potential
±(-mr)^{↑} r^{-1}
with Ñ_{x}^{2} (-mr)^{↑} r^{-1}
= m^{2} (-mr)^{↑} r^{-1} .
This can be loosely regarded as the potential for a putative inverse square force (with negative potential generating an attractive force) mediated by decaying carriers, although it actually only approximates ò_{R}^{¥} dr r^{-2}(-mr)^{↑} = (-mS)^{↑}R^{-1} + m^{-1}E(-m^{-1}R) where E(s) º ò_{-¥}^{s} dt t^{-1}(t)^{↑} is the integral exponential function, small for large negative s. [ Proof : Setting s=-mr: -m ò_{-mR}^{-¥} ds s^{-2}(s)^{↑} = m(- [(s)^{↑}s^{-1}]_{-¥}^{-mR} + ò_{-¥}^{ -mS} ds s^{-1}(s)^{↑} = (-mS)^{↑}R^{-1} + mEi(-m^{-1}R) .] |
To explain the strong nucleonic (proton and neutron) forces,
observed to act at a range of about R=10^{-15} m, Yakuwa postulated massive mediator particles
of lifetime Dt = c^{-1}R. Uncertainty principle
DEDt ³ h
then implies DEc^{-1}R ³ h with uncertainty minimised
by DE = chR^{-1} » 200 MeV.
Historically first the muon, then the pion, were regarded as filling this role,
necessitating a high coupling constant of about 14, making peturbation theories
neglecting higher order couplings inapplicable for the strong nucleonic forces.
Trigonometric Potential
Taking a = ia gives f(r)=r^{-1}(ari)^{↑} with
Ñ_{x}^{2}f = -a^{2}f .
f_{1}(r) = r^{-1} sin(ar) = a Sin(Q)
f'(r) = -r^{-2} sin(ar) + ar^{-1} cos(ar) = ar^{-1}( cos(Q)= Sin(Q)) f"(r) = (2r^{-3}-a^{2}r^{-1}) sin(ar) - 2ar^{-2} cos(ar) = ar^{-2}(-2 cos(ar) + (2(ar)^{-1} - ar) sin(ar)) O_{SI}(r) = ar^{-2}( cos(ar) - R^{-}_{a,N} sin(ar)) = ar^{-2} ((ar)^{2} + (ar)^{-2} + 3)^{½} sin(ar- cot^{-1}(R^{-}_{a,N})) where R^{-}_{a,N}=ar+(ar)^{-1}. For small ar, f_{1}(r) = a - 3!^{-1}a^{3}r^{2} + _{O}(r^{-1}(ar)^{5}) is approximately harmonic. | |
O_{SI}(r) is extremal when
Q - tan^{-1}(3^{-1}Q(3+Q^{2})) = kp
which is a condition of Q independant of a.
[ Proof : O_{SI}'(r) = 2ar^{-3}( - cos(ar) + (ar + (ar)^{-1}) sin(ar) ) - ar^{-2}( +a sin(ar) + (a - a^{-1}r^{-2}) sin(ar) + a(ar + (ar)^{-1}) cos(ar) ) = ar^{-3}( -2 cos(ar) + 2(ar + (ar)^{-1}) sin(ar) - ( +ar sin(ar) + (ar - (ar)^{-1}) sin(ar) + ar(ar + (ar)^{-1}) cos(ar) ) = ar^{-3}( (-2-ar(ar+(ar)^{-1})) cos(ar) + (2(ar + (ar)^{-1}) - ar - (ar - (ar)^{-1}) ) sin(ar) ) = ar^{-3}( -(3+(ar)^{2}) cos(ar) + 3(ar)^{-1} sin(ar) ) = a^{4}Q^{-3}( -(3+Q^{2}) cos(Q) + 3Q^{-1} sin(Q) ) = a^{4}Q^{-3} ( (3+Q^{2})^{2} + 9Q^{-2})^{½} sin(Q - tan^{-1}(3^{-1}Q(3+Q^{2})) .] | |
Define W_{k} to solve x - tan^{-1}(1/3Q(3+Q^{2})) = kp . [spctime1] |
E(r) = ½(a cos(ar)- Sin(ar))
Logarithmic Potential
f(r) = r^{g}(ar^{b})^{↓} + m
has f'(r)
= r^{g-1}(g(ar^{b})^{↓}+b) ;
f"(r)
= r^{g-2}((g-1)g(ar^{b})^{↓}+(2g-1)b)
so that
f"(r)+(N-1)r^{-1}f'(r).
= r^{g-2}((g+N-2)g(ar^{b})^{↓}+(2g+N-2)b)
Setting g=2 gives
f(r)
= r^{2}(ar^{b})^{↓} + ½N^{-1}(N+2)b
= r^{2}(a^{↓} + br^{↓}) + ½N^{-1}(N+2)b
with f'(r) =
= r(2(ar^{b})^{↓}+b)
solving
Ñ_{x}^{2} f(r) = ±2Nf(r) for any a,b according as x^{2} is ±.
O_{SI}(r) = (4-N)r^{-1}f'(r) ± 2Nf(r)
= (4-N)(2(ar^{b})^{↓}+b) ±
(2Nr^{2}(ar^{b})^{↓} + N(N+2)b)
= 2((4-N) ± Nr^{2})(ar^{b})^{↓}.
Next : 1-Potentials