## Thursday, November 12, 2020

### Solution of Delaunay Variable Problem

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 =  mass of Jupiter,  m = mass  of Earth, and a =  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 general functional Hamiltonian:

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

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

dL/ dt =   -  H   /

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:

-    ,   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 =  1/ (1047.355 m ☉ )

m =  1/( 32930 m ☉ )

3 =  5.2 AU

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

H   =  k 2  m 3    10    -2

For an error magnitude e  »   0.012