Lagrange planetary equations
Now,
with
.
Hence,


(G.89) 


(G.90) 


(G.91) 


(G.92) 


(G.93) 


(G.94) 
with all other partial derivatives zero. Thus, from Equation (G.85), the
only nonzero Lagrange brackets are


(G.95) 


(G.96) 


(G.97) 


(G.98) 


(G.99) 


(G.100) 
Hence, Equations (G.32) yield


(G.101) 


(G.102) 


(G.103) 


(G.104) 


(G.105) 


(G.106) 
Finally, Equations (G.95)–(G.106) can be rearranged to give
Equations (G.107)–(G.112), which specify the time evolution
of the osculating orbital elements of our planet under the action of the disturbing function, are known collectively as the Lagrange
planetary equations (Brouwer and Clemence 1961).
In fact, the orbital element
always appears in the disturbing function in the
combination
. This combination is known as the mean longitude, and is
denote
. It follows that
The integral appearing in the previous equation is problematic.
Fortunately, it can easily be eliminated by replacing the variable
by
. In this case, the Lagrange planetary equations become
where
is taken at
constant , and
at constant
(Brouwer and Clemence 1961).