Delaunay Variable Problem Solution

 First obtain the perturbation term R in terms of Legendre functions:

R=  k ²  m ₃    [ 1/  r ₃ +  ½  r ²/ r ₃ ³ - 3/2  r ²/ r ₃ ³  cos ² S ] 

If we take  m ₃ =  mass of Jupiter,  m ₂ = mass  of Earth, and a ₃ =  semi-major axis of Jupiter we can calculate the first order perturbations in L, G,  ℓ and g using the reference Hamiltonian:

H   =  

 -  m ² / 2 L ²  -  k ²  m ₃    [1/  r ₃ +  ½  r ²/ r ₃ ³ - 3/2  r ²/ r ₃ ³  cos ² S ] 

We thereby obtain a functional Hamiltonian:

H  (L, G,  ℓ,  g, m , k ² , m ₃  ,  a ₃  ,  t)  

And can write out the differential equations to solve the problem.  One such equation would be:

dL/ dt =   - ¶ H   / ¶ ℓ         

  Integration yielding:

L  -  L _o =   ò ^t  _t o      F (ℓ)   dt    

Where F (ℓ)  =  F(L, G,  ℓ ,  g, constants, t) 

We  then substitute for each of the variables: L, G etc. Earth and Jupiter values, and also:    

 ℓ   -  ℓ  _o  ,   g   -   g  _o  ,   etc.  leaving everything else constant and taking the specific integral in each case.  Do this for L, G,  ℓ and g

Using the  mass values for Jupiter and Earth  expressed in terms  of solar (m ☉)

 

m ₃ =  1/ (1047.355 m ☉ )

m ₂ =  1/( 32930 m ☉ )   

   a ₃ =  5.2 AU  

 

On computation using the preceding, we get an error in the reference Hamiltonian:  

d H   =  k ²  m ₃    10   ^-2

For an error magnitude e  »   0.012