Application of the Parker-Sochacki Method to Celestial Mechanics

Joseph W. Rudmin

*Physics Dept., James Madison
University, Harrisonburg, VA 22807*

**Abstract: **

We
present a tutorial demonstrating the theory and usage of the Parker-Sochacki
method of numerically solving systems of differential equations. Solutions are demonstrated for the case of
projectile motion in air, and for the classical Newtonian N-body problem with
mutual gravitational attraction.

**I. Introduction **

Physics is the
mathematical study of the interactions of matter and energy in the observable
universe. The key word in this description is “mathematical”. It is mathematics which gives physics the
analytical and predictive power which so distinguishes it from the other fields
of human knowledge. Mathematics is a
process of creating symbols, and rules for manipulating the symbols, in ways
which abstract, formalize, and enhance human logic. Without mathematics, physics would be nothing but lore,
experience, and stamp-collecting.

Historically, physics and
mathematics have been synergistically entwined. Physics has added to mathematics the subjects of geometry,
trigonometry, vectors, calculus, and distribution theory. Mathematics has supplied physics with such
tools as algebra, probability, complex mathematics, Boolean algebra, group
theory, and most importantly, the concept of abstract quantities such as
energy, entropy, angular momentum, and fields, and their rules of behavior—the
laws of physics. Over the centuries, a
major impetus for studying mathematics has come from the benefits it confers
through physics, and its daughters the engineering fields. Conversely, for the educated layman, perhaps
the very best reason for studying physics is that it makes a person
mathematically competent, and the mathematics thus learned is much more widely
applicable in life than in just physics.

Once in a while,
mathematicians create a new tool which is so powerful and so widely applicable,
that to slow its dissemination might significantly retard development across
the whole field of physics and engineering.
It is the belief of this author that the Parker-Sochacki method of
solving differential equations is such a tool.
For this reason, I have sought to publish it in a broad-spectrum journal
such as American Journal of Physics, rather than in a journal read by a smaller
subset of scientists and engineers.

The
Parker-Sochacki method is an extension of the Picard iteration, which in turn
is an algorithm for solving simultaneous differential equations. It is perhaps more easily shown than
described. It has been said that when
you are holding a hammer, everything looks like a nail. Similarly, the Parker-Sochacki method can be
summarized by the principle “When you have a Picard iteration, everything looks
like a polynomial. Or at least it
should.” The method has been formally
published elsewhere, [1], but I will present it more informally here, and will
apply it to two examples, one simple and one complicated.

Consider the
case of an object of mass m falling through air. The air friction force is assumed to be of the form DACs^{2 }where
D is the air density, A is the cross-sectional area, C is a drag coefficient,
and s is the speed of the object. Let x
be the horizontal position, y be the vertical position, u be the horizontal
velocity component, and v be the vertical velocity component. To simplify, let B = DAC/m. Then the equations of motion can be written

_{} (1)_{}

_{}
(2)

_{}

_{} and (3)

_{}
(4)

With a suitable choice of units, g for accelerations
and sqrt(g/B) for velocities, we can replace g and B with 1. Let’s try solving these using the Picard
iteration. Assume x and u can be
expressed as a truncated Maclaurin series in time t:

_{} (5)

and

_{} (6)

Substituting (5) and (6)
into (1) permits us to recover the next higher term in the x series, which
yields

_{} (8)

Continuing this process
constitutes the Picard iteration, which consists of expressing each right-hand
member of equations (1) through (4) as a power series of order n in t, and then
using the equations to increment the number of terms for the series
representing each left-hand member.
This was published by Picard in 1928 [2], but has been since regarded as
an impractical formalism, because it soon runs into practical difficulties, as
we shall see.

_{} (9)

where s is the speed of the
projectile. Similarly, (4) can be

written

_{}
(10)

To express the right member
of (9) as a Maclaurin series, we first need to express s as a power series in
time. However, to do so, we first need
to work out algebraic expressions for the coefficients, and after the first two
or three terms, these become so monstrously complicated that it cannot
practically be done. This type of
difficulty halted widespread application of the Picard iteration for the past
sixty years. Now, however, Ed Parker
and Jim Sochacki of the James Madison University Mathematics department have
succeeded in bypassing this barrier with some creative insight. The solution is this. Since the usual method of expanding the
square root s fails to give the desired polynomial expansion in time, simply
treat s as another variable to be expressed as a power series, whose
coefficients are also to be discovered through the Picard iteration.

_{} (13)

where _{} (14)

Then the time derivative of
s is

_{} (15)

Substituting in equations
(9) and (10) gives

(15b)

This is no help at all,
because the same difficulty as before arises due to s being in the denominator
of the last term. All we have achieved so far is to increase the number of
equations to be solved. Ed and Jim’s
creative inspiration is to repeat this exercise, which has just failed us!

Let r = 1/s. (16)

Then _{}. (17)

And now (15b) can be
rewritten as

_{} (18)

Now we see that the whole
mess has simplified beautifully.

Gathering the equations
together:

_{} (1)

_{} (2)

_{}
(9)

_{} (10)

_{}
(15b)

and

_{}
(17)

where _{} and _{}.
(18)

Now suppose we know the
expansions of each of the variables up through order n. Applying the Picard iteration to (1) and (2)
gives

_{} and _{}. (19)

For (9) a bit more work is
required. su is the product of two
expansions:

_{}. (20)

Multiplying these term by
term gives the result that the coefficient of the nth-order term for the
product is

_{}, or

_{} (21)

Then applying the Picard
iteration to (3) gives

_{} (22)

Similarly, _{}.
(23)

From (15b),

_{} (24)

where

_{}.
(25)

Now let _{}. (26)

Then

_{} (27)

Equations (19) and (22)
through (27) implement the Picard iteration.

Now look at the beauty of
what Ed Parker and Jim Sochacki have done.
First, every term in the expansions has been calculated simply and in
closed form. The number of operations
required is not only finite, but small.
Once a coefficient in the expansions is calculated, it is never changed
again. The only limit on its precision
is the digital accuracy to which it is first calculated. The calculations can even be done
analytically, displaying the exact algebraic expressions for terms of all
orders. An algebraic manipulator such
as Macsyma or Maple can generate and display these coefficients to any order
desired. All the required operations on
series have been reduced to just three:
integration of a series, and addition and multiplication of two
series. Of these, the first two are
trivial, and the third is not difficult.
Finally, the only arithmetic operations used are multiplications,
additions, and subtractions. The only
divisions required are the inverses of small integers, and these can be
calculated just once and stored in a table.
The serendipitous absence of divisions makes the method ideally suited
for high speed computation in computers.

As an aside, note that in
demonstrating the method, we have also solved for the motion of a projectile
with a quadratic drag force--itself an important problem which, to this
author's knowledge, has not been previously published. Note that it would not be very difficult to
extend this calculation to include, say, an exponential atmosphere, buoyancy, g
varying with height, coriolis forces, and wind forces.

The world of theoretical
physics is well-stocked with first-order approximations. Now all the higher order-terms have been
made available as well.

III. Chronology

Ed Parker and Jim Sochacki,
of the James Madison University Mathematics Department, discovered this
approach in the late 1980's when they were studying chaotic systems arising in
population dynamics. Having achieved a
series solution, but they wondered what series it was that they were getting. With some further effort, they discovered
that in the population dynamics problem, the solution they were getting was the
Maclaurin series. They then succeeded
in proving five theorems which are published in reference [1]. Summarizing the results of these theorems:

(1) The polynomial solution
produced by the Picard iteration is unique, and is therefore identical with the
Maclaurin series.

(2) In computing the term
n+1 of a Picard iteration, only the first n terms of the other series need to
be used.

(3) Defining a property
called "projectively polynomial", which is equivalent to a real
function having a polynomial generator, they show that this property is
preserved by addition, multiplication, and differentiation (using the
chain-rule).

(4) The Picard-generated
polynomial approximations to the solutions of the equations on any finite
interval can approach the solutions arbitrarily closely if the solutions are
analytic functions.

(5) The solutions reached by
the Picard iteration satisfy a Lipshitz condition on any locally analytic interval. Of these, probably the most important for
the practicing engineer or physicist is the first. It guarantees that the expansion produced in this process is not
just an approximation polynomial, but in fact is the Maclaurin series. It allows us to safely assume all the
powerful properties for the Maclaurin series, including the fact that if the
differential equation has a unique solution, and if the series converges as n
increases, it will converge to that solution.

They also raised two
unanswered questions in their article.
First, how can one obtain a good estimate for the accuracy of the
solution? Second, they have shown that
the generators which are projectively polynomial are dense in the analytic
functions. Are they the set of analytic
functions? I suggest a third
question: What are the (or some of the)
differential equations for which the method fails?

When Ed and Jim first
discovered the method in the late 1980's they didn't yet realize how widely
applicable it was. At that time, I was
supervising a student--Timothy MacDevitt--in trying a new approach to celestial
mechanics. We decided to see if we
could improve on celestial mechanics calculations by extrapolating Hermite
interpolation polynomials of large order from previously calculated
points. Although I was aware that
Lagrange interpolation polynomials were subject to unstable oscillations, I was
optimistic in this case because we intended to extend the polynomials to two
higher derivatives. That is, we would
create a polynomial which at n different values of time, would fit the
position, velocity, and acceleration of the orbiting particle. The acceleration was to be calculated from
Newton's laws of motions. We would then
extrapolate this polynomial forward in time to get later positions. The project failed spectacularly. We found that we could create a polynomial
which would fit all three derivatives at n points in an orbit of radius one,
which between those points would oscillate to values of one million. Increasing the number and density of points
only made the oscillations worse, not better.
This taught me the following lesson:
There are many polynomial approximations which can satisfy a differential
equation on a finite number of points, but there is only ONE polynomial which
will approach the solution BETWEEN those points, and that is the truncated
Maclaurin series.

Tim graduated and moved on
to graduate school, and I turned to other research. In the summer of 1994, I was awarded the LaRose Fellowship by the
James Madison University Foundation. This enabled me to hire a student,
Geoffrey Williams, for a summer research project. This project was to install a CCD on the JMU observatory
telescope, with a goal of tracking asteroids.
To calculate the asteroid orbits, I decided to see if the method
developed by Parker and Sochacki could be applied to celestial mechanics. Ed said he would try it, and succeeded
beautifully, as the rest of this paper will show.

Before continuing, I again
want to say what the Parker-Sochacki method can do. Suppose you want to solve a set of n differential equations with
initial conditions, such as

x'= F(x,y,z,t)

y'= G(x,y,z,t)

z'= H(x,y,z,t).

Try to write the right-hand
members in such a way that if x,y, and z are polynomials in t, then F,G, and H
also give polynomials in t. To do this
will require replacing non-polynomial functions with new polynomial
approximations, thus increasing the number of variables needing solution. If you succeed, then the Picard iteration is
guaranteed to generate the Maclaurin series.
The question arises "Are there some systems of differential
equations for which you cannot fulfill the required conditions?" Ed and Jim say that they do not know the
answer to that question, but they have applied the method to roughly 100
different systems, and have not yet found a system for which it fails.

IV. Celestial Mechanics

A. Parker-Sochacki Solution for the Classical N-body Problem

We now turn to the problem
of high-precision computation of the coordinates and velocities of N particles
orbiting under mutual gravitation, neglecting relativistic effects. This problem has not been previously solved
exactly, and perturbation theories and methods of averaging have provided only
incomplete and approximate solutions. [3]
Our subject in this case is the solar system. First, we note that the center of mass of the three-particle
system consisting of the sun, Jupiter, and Saturn lies outside the surface of
the sun. Thus during the Jovian year,
the sun moves around a region exceeding its diameter. Therefore, the model of the system in which the sun is fixed and
the planets move in ellipses, is clearly no more accurate than about one part
in ten thousand per Jovian year. If we want to compute the orbits within one
part in a billion per year, then we need to use better computational
methods. At this level of precision,
perturbation theory also fails, because the orbital elements need to be
expressed as polynomials, and so many terms need to be carried in the
computation that, given the complexity of the functions, there is no advantage
in using elliptic orbits over using Cartesian coordinates.

It is fair to ask what the
reasons are for requiring this level of precision. I suggest three. The first is tracking asteroids. In this problem the most interesting cases
are the non-elliptic orbits--those in which the particle undergoes a deflection
by a larger body, for it is just these collisions which may shift orbits from
safe to earth-threatening. Also, if an
object does appear to be headed near the earth, it is a great advantage to be
able to predict its trajectory more precisely.

Secondly, there may still be
one or more undiscovered gravity sources in the solar system. The anomalies of Neptune's orbit, which led
to the discovery of Pluto, lost their explanation when the discovery of Charon
revealed Pluto's small mass. According
to the Astronomical Almanac, a satisfactory ephemeris for Uranus for the 1980's
could be computed only by excluding observations made before 1900. More precise computational methods may
permit higher resolution estimates of the anomalous forces in the system. [4]

Finally, with the
proliferation of computers, it is now possible for amateur and professional
astronomers to generate their own ephemerides, rather than relying on
approximation formulas and tables.
Better algorithms will facilitate this.

Taking the solar system as a
model for demonstrating the calculation technique, we will assume N_{p}
planets with masses Mj = 1, ... N_{p},
Cartesian coordinates x_{i,j}
, i = 1,2,3, and velocity components v_{i,j}. Planet one is the sun and planet ten is
Pluto. Following the Astronomical
Almanac, let M_{0} be the mass
of the sun, G be Newton's Gravitational Constant, and T be one earth year. There is a defined constant called the
Gaussian Gravitational Constant, k, which determines the length of the solar
day as used in astronomy.

_{} or _{}

days.

This is in turn is used to
define the Astronomical Unit, A, which is approximately the radius of the
earth's orbit around the sun.:

_{}

Effectively, you can think
of A as an historical unit:

A = 1.32712440 x 10^{20} m (28)

and 2p/k as the number of days in the orbital
period of an object of negligible mass orbiting a much greater mass at that
distance. In this calculation, it is
assumed that

D = 1 day = 86400 seconds,
and

GM_{0} = 1.32712440x10^{20} m^{2}/s^{3}

In this case, the natural
unit of time is 1/k = T/2p.

In these units, the
equations of motion can be written

_{}, where i = 1,2,3, and j = 1,..,N_{p} (29)

_{}
(30)

where s_{jk} is the separation between particles k and
j:

_{}. (31)

In (30), m_{k}
is the mass of the kth planet divided by the mass of the sun. The term s_{jk}^{3} in the denominator of (30) makes the
integrals unsolvable. Therefore,
following Parker and Sochacki, we replace these factors with a polynomial
approximation: Let this polynomial be

_{}. (32)

For convenience, define _{} for all k. We now need an equation which gives _{} as a function of
time. From the chain rule,

_{}
(33)

From (31), _{} (34)

The sum in the right side of
(34) has the units of action divided by mass, so we will denote it by _{}. Then (33) and (34)
can be combined to give

_{} (35)

We now have a closed set of
differential equations to use in the Picard iteration, at the price of having
increased the number of unknowns. For a
solar system of 10 planets we initially needed to calculate 30 position
coordinates and 30 velocity components, for a total of 60 unknowns. To this we have added 55 inverse
separations, for a total of 115 unknown variables. This is a substantial increase, but it is a small price to pay
for the benefits of the Picard iteration.

As in the previous example,
we can now derive the expressions for calculating the coefficients of the terms
in the Taylor series. We assume that we
know the coefficients for terms up to order m-1, and want to find the
coefficients for terms of order m.

Let _{} (36)

and define coefficients _{} _{}, and _{}

for the velocity,
separation, and action similarly. From
(29) we get

_{} (37)

From (30),

_{} (38)

For the four-factor product
in (35), it is easier to simplify it by breaking the multiplication into
smaller steps. Define the coefficients
of the square and cube for the inverse separation as follows:

_{} (39)

and _{}.
(40)

From the definition of
A , and the expression (21) for a

jk

coefficient of the product of two series, we get

_{} (41)

Then from (35), _{}. (42}

Equations (36) through (42)
constitute the Picard Iteration. It can
be implemented with less than 50 lines of code in Basic, Fortran, or C, as
shown in the following example, written in Power Basic [5].

Table I: Basic Source Code for Solving the N-body
Problem

‘Note: variables bdginning with I through L are
integers.

PolyGen: ' Generate the polynomials.

for m = 1 to nt

mm1 = m-1

um = 1./m

for j = 1 to Np

for i = 1 to 3

xx(i,j,m) = vv(i,j,mm1)*um

a = 0

for k = 1 to Np

b = 0

for L = 0 to mm1

mm1mL= mm1 - L

b = b + (xx(i,k,L) -xx(i,j,L))*u3(j,k,mm1mL)

next L 'Note u3(j,j,m) = 0

a = a + b*amass(k)*um

next k

vv(i,j,m) = a

next i

jm1 = j-1

for k = 1 to jm1

a = 0

for L = 0 to mm1

mm1mL = mm1-L

a = a - u3(j,k,L)*aa(j,k,mm1mL)

next L

u1(j,k,m) = a*um :
u1(k,j,m) = a*um

a = 0

for L = 0 to m

mmL = m - L

a = a + u1(j,k,L)*u1(j,k,mmL)

next L

u2(j,k,m) = a : u2(k,j,m) = a

a = 0 : b = 0

for L = 0 to m

mmL = m - L

b = b + u2(j,k,L)*u1(j,k,mmL)

for i = 1 to 3

a = a + (xx(i,j,L) - xx(i,k,L))*(vv(i,j,mmL) - vv(i,k,mmL))

next i

next L

aa(j,k,m) = a : aa(k,j,m) = a

u3(j,k,m) = b :
u3(k,j,m) = b

next k

aa(j,j,m) = 0 : u1(j,j,m) = 0

u2(j,j,m) = 0 : u3(j,j,m) = 0

next j

next m

return

Laurence G. Taff, in his excellent text* Celestial Mechanics, A Computational Guide for the Practitioner*,
writes the Newtonian equations of motion for N bodies orbiting under mutual
gravitation, and then comments, "No compelling evidence exists that a
successful numerical solution of Eq. 12.1 has even been carried out. Moreover, much evidence to the contrary does
exist." The preceding 47 lines of
code demonstrate that Taff's statement is no longer true. What is stunning is the simplicity of the
solution.

B. Tests of the Algorithm

A computer program was written for a PC-type computer in
compiled Basic, [5], using extended-precision floating point arithmetic
(18-digit accuracy). Three tests of the
algorithm were run. The first test was
to check the behavior of a two-particle system. The result was the expected elliptic orbits.

The second test was to use solar system data taken from page E3
of the 1991 and 1992 editions of the Astronomical Almanac [6]. These tables give the position and velocity,
relative to the sun, for each of the planets in the solar system, at two times
separated by 200 days. In this test,
polynomial approximations were generated using the Parker-Sochacki method for
the energy and angular momentum of the solar system. The Taylor series coefficients for the center of mass-position,
momentum, angular momentum, and energy were displayed. The results are shown below in Table 2.

Table 2. Taylor Series Coefficients for Coordinates
and Momentum of the Center of Mass

Table 2. Taylor Series Coefficients for Coordinates
and

Momentum of Center of Mass
of the Solar System

m |
x |
y |
z |
px |
py |
pz |

0 |
9.95E-19 |
3.01E-22 |
4.00E-19 |
5.01E-20 |
-3.55E-19 |
3.65E-20 |

1 |
1.01E-22 |
3.64E-21 |
5.01E-20 |
1.02E-21 |
3.65E-20 |
8.29E-22 |

2 |
1.82E-21 |
5.54E-22 |
5.12E-22 |
3.18E-22 |
4.14E-22 |
1.07E-22 |

3 |
3.20E-23 |
5.56E-22 |
1.11E-24 |
2.93E-22 |
1.07E-22 |
5.94E-23 |

4 |
1.39E-22 |
4.45E-22 |
7.32E-23 |
6.34E-21 |
1.49E-23 |
3.27E-22 |

5 |
1.68E-22 |
2.12E-20 |
1.78E-21 |
3.08E-21 |
1.62E-22 |
4.10E-21 |

6 |
3.53E-21 |
4.32E-21 |
1.31E-21 |
4.62E-21 |
1.19E-21 |
9.22E-21 |

7 |
3.61E-21 |
1.15E-21 |
2.16E-21 |
2.91E-20 |
1.37E-21 |
3.56E-20 |

8 |
1.44E-22 |
7.12E-19 |
3.63E-21 |
5.10E-20 |
4.44E-21 |
3.39E-20 |

9 |
7.56E-20 |
3.35E-19 |
5.75E-21 |
4.55E-19 |
3.08E-21 |
7.51E-19 |

10 |
4.86E-20 |
7.65E-19 |
4.98E-20 |
5.50E-18 |
6.01E-20 |
2.41E-19 |

Taylor Series Coefficients
for the components of the total angular momentum, L_{x}, L_{y},
L_{z} and energy E.

m L_{x} L_{y} L_{z} Energy

0 9.288E-05
-1.379E-03 3.255E-03 -1.123E-04

1 -3.438E-22 2.723E-21
-5.302E-21 -4.765E-22

2 -2.729E-23
1.969E-22 7.079E-23 -1.800E-21

3 -2.484E-22 1.147E-22
-4.129E-22 -2.541E-21

4 7.794E-22
-1.508E-21 1.116E-21 3.494E-21

5 2.836E-21
-8.924E-24 3.044E-21 1.016E-20

6 -1.456E-21
7.495E-22 2.415E-20 2.033E-19

7 -4.533E-20 2.402E-20
-6.175E-20 -3.930E-19

8 -5.526E-20 -5.399E-20 -1.891E-19 -1.084E-18

9 2.662E-19 5.654E-20 -3.556E-19 -4.554E-18

10 1.024E-18 -6.606E-19
1.262E-18 1.214E-17

Examining Table 2, we see
that the position and coordinates of the center of mass remain zero, within the
digital accuracy of the computer. In
the columns showing the angular momentum coefficients, we note that the initial
values of angular momentum are mostly in the y and z directions. The y component is substantial since the z
axis points in the direction of the earth's axis, which is not perpendicular to
the plane of the ecliptic. The m=1
terms are about 10^{-18} of the m=0 terms, and are non-zero due to
round-off error. As higher order-terms
are calculated, the round-off error propagates and grows until by term 10, the
angular momentum coefficient is about 10^{-15} of the m=0 term, and the
energy is about 10^{-13} of the m=0 value.

As a third test of the
algorithm, the program was used to propagate the solar system between the two
dates given in the table shown in the 1992 Astronomical Almanac [6]. This table, described as "low
precision", gives the velocity and position coordinates of the planets at
two dates 200 days apart. The largest
inconsistency in this table appears to be for the position of Venus, with an
inconsistency of about 2 x 10^{-6 }AU or 300 km. That is, the Parker-Sochacki algorithm was
used to propagate a solar system from the first date to the second, and the
positions and velocities from the Almanac table and from our computer results
were compared for the second date. When
our code ran at very high precision, its highly self-consistent results
disagreed with the Almanac's coordinates for Venus by about 2 x 10^{-6}
AU. We decided to experiment with the
polynomial degree and step size to give an ephemeris of about this
precision. The most inaccurate
resulting coordinates were found to be in the position of Mercury. Therefore, we sought the combination of
polynomial degree and step size (200 days / # of steps) which would give the
shortest computation time, and a precision of 300 km or better. The computer used was a PC with an 133 MHz
80586, roughly equivalent to a 100 Mhz Pentium. We also repeated this experiment for a precision of 10^{-7}
Au or 15 km. The results are shown
below in Table 3. The running times
were found to vary by a factor of roughly 2, perhaps due to pipelining in the
microprocessor. The fastest times are
shown.

Table 3. 200-Day Computation times for a 100Mhz
Pentium, as a Function of Polynomial Degree and Step Size for Two Different Precisions

300 km Precision 15 km Precision

Poly'l Min # Step
Comp Poly'l Min #
Step Comp

Degree Steps Size
Time Degree Steps
Size Time

(Days)
(Sec) (Days) (Sec)

5 244 0.8
9 12 38
5.3 14

6 107 1.9
8 14 37
5.4 11

7 83 2.4
8 15 35
5.7 9

8 62 3.2
6 16 34
5.9 10

9 44 4.5
7 18 30
6.7 14

10 38 5.3
7

11 33 6.1
8

12 33 6.1
7

13 33 6.1
10

The point of this table is
to see that high levels of precision can be obtained in short computation
times, and that the most rapid computation is generally obtained by using a
higher-order polynomial, than is conventionally used in other methods.

In 1889, a prize for the
best mathematical paper answering one of four questions, was offered in honor
of the sixtieth anniversary of the King of Sweden. One of the questions, posed by Weierstrasse, was this.

"For a system of
arbitrarily many mass points that attract each other according to Newton's
laws, assuming that no two points ever collide, give the coordinates of the
individual points for all time as the sum of a uniformly convergent series
whose terms are made up of known functions....
This problem, whose solution would considerably extend our understanding
of the solar system, would seem capable of being solved using analytical
methods presently at our disposal...
Unfortunately, we know nothing about [the deceased Dirichlet's]
method... We can nevertheless suppose, almost with certainty, that this method
was based not on long and complicated calculations, but on the development of a
fundamental and simple idea that one could reasonably hope to recover through
persevering and penetrating research...".
[7]

The prize was won by
Poincare for the development of phase-space mechanics. It seems possible that the lost method of
solving differential equations, which Dirichlet took with him to his grave, was
the Parker-Sochacki method. Had this
method been entered in the 1889 contest, it would have won the prize.

V. Conclusions

Looking ahead, there are
several directions, both in the fields of celestial mechanics, and in the area
of general computation, which appear promising. For celestial mechanics, these might include improved planetary
ephemerides, searching for an explanation for the anomalies in the orbits of
Neptune and Uranus, proliferation of desk-top software to assist astronomers,
and precision computation of the orbits of asteroids.

The method needs to be
extended to include lowest order relativistic effects for Mercury, and to
include the effects of the larger moons on their host planets. The relativistic effects on Mercury can
probably be simulated by a quadrupole (or oblateness) term in the sun's field. The planet-moon systems can be handled by
first finding the orbits of the planets in the solar system, treating each
planet-moon system as a point, and then going back and recalculating the
positions of the moons and their host planets as a two-body (earth), or
five-body (Jupiter), system with the sun and other planets as a background
field. This should prove feasible for
projections of a few centuries into the future.

The Parker-Sochacki
algorithm can also be used to check various methods of averaging, such as the
simplectic method and other statistical methods. If implemented with parallel processors, it could even be used
for direct high-precision orbit computation over periods of several tens-of
millions of years, for a system of ten particles.

What are the intractable
problems? Comets appear to be
unsolvable, because of the unpredictable forces caused by vapor emissions. Chaos is also still present--an immeasurably
small change in the velocity or position of an asteroid may cause it to pass on
the opposite side of a planet centuries later.
The effects of ocean tides on the moon's position over eons of time
would also seem difficult if not impossible, since this is affected by
glaciation as well.

In the area of general
computation, the Parker-Sochacki method is clearly a fertile ground for
parallel computation. In the celestial
mechanics problem, the mth coefficient for all 115 unknowns could have been
computed in parallel. Widespread
adoption of the method could provide a substantial motivation for the
development of parallel processing hardware.

It is hard to overstate the
importance of the Parker-Sochacki method.
It has solved the problem of celestial mechanics, which has occupied
many of the greatest minds of mathematics for over two centuries, as far as it
every will or can be solved. But the
method has much wider application. It
may be the greatest advance in the solution of differential equations since the
development of orthogonal functions. Coupled with the modern computer, it may
have more impact on the solution of dynamical systems than any other method in
the history of mathematics.

VI. Acknowledgments

In addition, of course, to Ed Parker and Jim Sochacki, I would
like to acknowledge Laurence Taff for his honest exposition of the challenges
of celestial mechanics, P. Kenneth Seidlemann for encouragement and
consultation, and Geoffrey Williams for assistance in developing the computer
codes. Other members of the JMU physics
department including ??? read the
manuscript and provided valuable criticism.
This work was supported by the James Madison University Foundation and
the Robert LaRose Fellowship program.

References

[1] G. Edgar Parker and
James S. Sochacki, "Implementing the Picard Iteration", Neural,
Parallel, and Scientific Computations 4 (1996), 97-112.

[2] E. Picard, "Traite
D'Analyse", Volume 3, Gauthier-Villars (1922-28).

[3] Laurence G. Taff,
Celestial Mechanics, A Computational Guide for the Practitioner, John Wiley and
Sons, New York, 1985.

[4] The Astronomical Almanac
for the Year 1991, U.S. Government Printing Office, p L1.

[5] Available from Power
Basic, 316 Mid Valley Center, Carmel, CA 93923, email info@powerbasic.com, URL
www.powerbasic.com. This is an
excellent technical programming language for DOS-based personal computers.

[6] Astron. Alm., pE3.

[7] Newton's Clock, Chaos in
the Solar System, Ivars Peterson, W. H. Freeman and Company, New York, 1993.

[8] J. Wisdom and M. Holman,
Astron. J. 102, 1528 (1992).