The dynamics of the solar system

Historically, the first
description of the orbits of the Solar System's planets
was empirical, given by the three Kepler's laws. These
empirical laws were demonstrated by Newton, as a simple
2body application of the theory of universal
gravitation. But, in reality, the
Solar System is a nbody system and not a 2body one,
since the orbit of each planet is not only determined by
the Sun's attraction. Every planet is also infleuenced by
the other planets. So, considering the Solar System a
2body system is a very rough estimation of the reality. This problem can be overridden in two ways:  first it is possible to use computers to simulate to a certain extent a nbody problem;  a second solution is to consider that the planets' orbits can be described by Kepler's laws, applying to the orbits some corrections due to the fact that planets interact gravitationally. 
The solar
system described by Kepler's Laws: a simple 2body problem
Kepler's
laws are three empirical laws that describe the orbits of the
planets of the Solar System as simple ellipses, with a not
constant speed and a period proportional to the characteristics
of the orbit .
Newton was the first to demonstrate
mathematically these laws, applying the law of universal
gravitation to a very simple, approximated 2body problem, where
each planet is only attracted by the Sun, and where some
approximations are made.
These approximations are quite simple: first of all, since
the Sun is extremely more massive than the other objects of the
solar system, in a first approximation the motion of the planets
can be studied only considering the interaction between the Sun
and every single planet. Furthermore, in a first approximation it
is possible to ignore the interactions between planets .
To formulate
this simple demonstration, Newton invented the infinitesimal
calculus.
In the reality, planets do interact
between them, and these interactions cannot be ignored. The
motion of the planets is therefore much more complicated than the
Kepler's laws can describe.
Planetary perturbations: correcting
Kepler
A good description of the Solar System can
be given considering the gravitational interactions between the
planets as simple corrections of orbits determined applying
Kepler's laws. In fact, even if the interactions between planets
cannot be completely ignored, they are sufficiently light to be
considered as mere perturbations.
These perturbations are essentially of two
kinds:
Short period perturbations: these are small
variations around the mean Keplerian orbit of the planet (called
here the main body). These perturbations don't have effects on
the orbit of the main body over long periods. In fact the
perturbing planets move all around the main body, in different
directions. Therefore, these perturbations have
characteristic periods comparable to the orbital periods of the
planets: if the effect of these perturbations is summed over
longer periods the overall effect is absent. The typical
period of this kind of perturbation goes form months to years.
Secular (or long period) perturbations:
due to the action of the surrounding planets, in addiction to its
keplerian orbit, the main body will follow a second very
slow motion that can be seen as the sum of two distinct motions
of the Keplerian orbit:
 a rotation of the line where the major axis of the orbit lies
on the orbital plane (animation on the left);
 a precession of the orbital plane around a fixed direction
(animation on the right).
This secular perturbation can be seen as a very slow
periodical variations of the values of the characteristics of the Keplerian orbit:
the values of eccentricity, inclination and major semiaxes
fluctuate periodically around their mean values.
The typical period of these perturbations is of 10.000 years.