Orbital equations of motion = theory of nonlinear ordinary differential equations
The model works.
Let's see what the experts in the field have to say.
“Kolmogorov (1954) made the momentous discovery that catastrophes in planetary systems
are all too likely when he found that the minor terms in Newcomb’s series used to
solve the n-body problem for planetary motion in Newtonian mechanics were, in
fact, easily subject to random, unpredictable perturbations which could quickly lead
to turbulence; cf. Arnol’d (1997). In the meantime, the buzzword ‘chaos’ is applied
to such turbulence – also called ‘catastrophe,’ to the dismay of Shapley and many
other astronomers sharing Newton’s dream of an eternally stable solar system . . .
Subsequently, in the later fifties, Siegel & Moser (1971) and Moser (1973) further
extended mathematical knowledge of divergent [orbits of planets], Newcomb
series.
That ‘deterministic’ systems, like those obeying Newtonian mechanics, should
be subject to unpredictable catastrophes came as a real shock.
Next came pioneering computer simulations of planetary – system evolution
by Hills (1969). Wisdom (1981), finding even more advanced numerical methods
and using faster computers, achieved the breakthrough of repeatedly observing
chaos (or catastrophes) in computer-simulated planetary orbits, splendidly
confirming Kolmogorov and putting the dream of Newton in an absolutely stable
and orderly cosmos to rest in a way most astronomers could not have previously
imagined. All experts now agree that even apparently stable [planetary] orbits will
occasionally experience unpredictable chaotic disturbances.”
E. Köhler, Induction and Deduction in Science
“these [stability] models neglect solar mass loss and the effect of passing
stars . . . [whose perturbations] should shorten the lifetimes of the systems [change to instability]
by several orders of magnitude.”
J. J. Lissaur, Q. N. C. Lin, “Diversity of Planetary Systems and Unsolved Problems,” From Extrasolar
Planets to Cosmology: V. L. T. Opening Symposium
“Instead of using full equations of motion, Laskar focused on a special
formulation that spotlights gradual but cumulative changes in an orbit’s shape
[eccentricity] and orientation [inclination]. He worked with equations that smooth
out the recurring wiggles and wobbles in planetary orbits leaving only long term
trends . . .
“By applying a similar strategy to celestial curves [eccentricities], Laskar could
isolate these [non-gravitational] parts of a planet’s motion that correspond to lasting
changes in key characteristics of its orbit.”
Ivars Peterson, Newton’s Clock
“This differential system is a close approximation to the real solar system, and in
particular, the inner solar system . . . but the exact meaning of ‘close’ is still difficult to evaluate.”
J. Laskar
When one employs non-gravitational theory instead of “full equations of motion” to prove
stability, one has removed the solution of the problem from reality. But even believing in these
non-gravitational equations, Laskar cannot tell what a “close approximation” is. It is a theoretical
construct that comes out of non-gravitational math.
C. Ginenthal
Repeatedly the eccentricities and inclination of the positions of the orbits of the planets
in the solar system are relied on to determine whether or not there was stability in the past or
stability in the future. Because this tool, that is not based strictly on gravitational theory using
only masses and their separations, is the basis for all reckoning of stability, it is essentially a great
fudge factor that has been swallowed hook, line and sinker by astronomers, never noticing the
distinction between their heuristic mathematics and the underlying forces that should have been
employed in the first place.
C. Ginenthal
“Efforts to settle the question of the solar system’s stability face a serious,
perhaps insurmountable obstacle. As Scott Tremain has remarked: ‘In some sense,
you end up having to deal with probabilities. You can never rule anything out
completely. Even if a [planetary] system is well behaved, there’s always a small
chance of its wandering by some narrow path to just about any configuration.’ In
other words, with a mathematical model that automatically incorporates chaotic
behavior, there’s no way to prove, with absolute certainty, that something can’t ever
happen.”
Ivars Peterson, Newton’s Clock
“From a physical point of view, this model [of the solar system] is obviously
hard to accept, but cannot escape these conclusions, if one idealizes the problem
mathematically . . . In fact, the [mathematical] idealization goes further: We are not
talking about the motions of planets under realistic forces, but of the n-body
problem taking into account only Newton’s force laws and referring to mass points
with some smallness restriction on masses.”
Jurgen Moser (one of the giants in the field, proved the exceedingly difficult KAM theory)
J. Moser, “Stability in Celestial Mechanics,” The Stability of the Solar System and of Small Stellar
Systems Symposium
“The word ‘chaotic’ summarizes many fundamental concepts characterizing
a dynamical system such as complex predictability and stability. But above
all, it acts as a warming of the difficulties which are likely to arise when trying to
obtain a reliable picture of its past and future evolution. As an example, a
commonly accepted definition states that a system is ‘unstable’ if the trajectories of
two points that initially are arbitrarily close . . . diverge quickly in time. This has
strong implications, as small uncertainties in initial conditions . . . might [also] be
consistent with completely different future trajectories: The conclusion is that we
can exactly reproduce the motion of a chaotic system only if WE KNOW, WITH
ABSOLUTE PRECISION, THE INITIAL CONDITIONS – A STATEMENT
THAT, IN PRACTICE, CAN NEVER BE TRUE."
Alessandra Celletti, Ettore Perozzi, Celestial Mechanics: The Waltz of the Planets
Sussman and Wisdom's 1992 integration of the entire solar system displayed a disturbing dependence on the timestep of the integration (measurement of the Lyapunov time).
Thus, different researchers who draw their initial conditions from the same ephemeris at different times can find vastly different Lyapunov timescales.
Wayne Hayes, UC Irvine