The most common set of orbital elements used to parameterize Keplerian orbits consists of the major radius, ; the mean longitude at epoch,
; the eccentricity, ;
the inclination (relative to some reference plane), ; the longitude of the perihelion,
; and the longitude of the ascending node,
. (See Section 4.12.) The mean orbital angular velocity is
[see Equation (4.117)].
Consider how a particular Lagrange bracket transforms under
a rotation of the coordinate system , , about the axis (if we look along the axis).
We can write

(G.41) 
where

(G.42) 
Let the new coordinate system be
. A rotation about the
axis though an angle
brings the ascending node to the axis. See Figure 4.6.
The relation between the old and new coordinates is (see Section A.6)
The partial derivatives with respect to can be written


(G.46) 


(G.47) 


(G.48) 


(G.49) 
where


(G.50) 


(G.51) 


(G.52) 


(G.53) 
Let , , , and be the equivalent quantities
obtained by replacing by in the preceding equations.
It thus follows that
Hence,

(G.56) 
Now,
Similarly,
Let

(G.59) 
Because and
, it follows that
However,

(G.61) 
because the lefthand side is the component of the angular momentum per unit mass parallel to
the axis. Of course, this axis is inclined at an angle
to the axis, which is parallel to the angular momentum vector.
Thus, we obtain

(G.62) 
Consider a rotation of the coordinate system about the axis. Let the
new coordinate system be , , . A rotation through an
angle brings the orbit into the  plane. See Figure 4.6.
Let

(G.63) 
By analogy
with the previous analysis,

(G.64) 
However, and are both zero, because the orbit lies
in the  plane. Hence,

(G.65) 
Consider, finally, a rotation of the coordinate system about the axis. Let the
final coordinate system be , , . A rotation through an angle
brings the perihelion to the axis. See Figure 4.6.
Let

(G.66) 
By analogy with the previous analysis,

(G.67) 
However,

(G.68) 
so, from Equations (G.62) and (G.65),

(G.69) 
It thus remains to calculate
.
The coordinates
and
—where represents radial distance from the Sun, and
is the true anomaly—are functions of the major radius, ,
the eccentricity, , and the mean anomaly,
.
Because the Lagrange brackets
are independent of time, it is sufficient to evaluate them at
; that is, at the perihelion point. It is easily
demonstrated from Equations (4.86) and (4.87) that


(G.70) 


(G.71) 


(G.72) 


(G.73) 
at small . Hence, at
,


(G.74) 


(G.75) 


(G.76) 


(G.77) 


(G.78) 


(G.79) 
because
. All other partial derivatives are zero.
Because the orbit in the , , coordinate system only
depends on the elements , , and
, we
can write
Substitution of the values of the derivatives evaluated at
into this expression yields
and

(G.84) 
where
.
Hence, from Equation (G.69), we obtain

(G.85) 