4. 1) Perturbations in Orbital Elements
As we have said several times already the actual orbit does
not deviate much fromtl:le Keplerian plane motion because the disturbing potential R is much smaller than the potential of the central
motion in a central field formulated by the general
is not linear (as
is in the classical mechanics).
of the non-linear terms results in the orbital period to
be slightly longer than 2Vmaking thus the perigee to adva
To evaluate the coefficients X
(t) we use the orbital elements
describing the lntermediate orbit and the value of J 2 determined from the
1-st approximation of the equations of motion (see 4.3).
practice, the evaluation of t
oe(t) "'"' 1 a.
ar ( t) ,.
+ (n-2p+q)M + m(; - e)
21/! - ch/J)
Alternative Deriy,s.tion of Lagrangian Equations of Motion
We first define the Lagrangian potential Las
( q ' q ) .I
Then we define the action functionS, known sometimes as Hamilton 1 s
principal function, as
It describes the ac
where we have substituted the orbital parameters for
Review of Classical Mechanics
Let us begin with the convention that throughout the course we
shall be using subscript notation for vectors and matrices.
letters will denote indices running from 1 to 3; Greek letters will be
and we may recall having seen them in section 1.3 already.
We shall see
later that these euqations of motion can be further generalized and then
converted into the canonic equations we are looking for.
When describing prob
Hence 1 defining a new potential
called Lagrangian potential,or just Lagrangian 1 we can rewrite the
equations above in a
form (for U conservative)
This is the new form of our generalized equations of motion of which the
while the second term is nothing but the total energy E, which we know
is constant for a motion in conservative field
But, f,r;om our earlier explanations, we know that the potential
energy of the central field is given by
Since the described coordinate system is not inertial with
respect to fixed stars, its acceleration (with respect to an inertial
system) can be observed as a change of geometry of the potential.
This appears as non-conservative part of the disturbing p