1. Celestial Mechanics VII
Lagrange’s planetary equations
Series development of the perturbing function
Secular perturbations, proper elements
The stability of the Solar System

2. Perturbed two-body motion
U, the central force function, equals minus the potential of a central point mass – this gives rise to Keplerian motion
The second term equals minus the potential of all perturbing forces under consideration – these are assumed to be conservative, so the system is Hamiltonian
Planetary perturbations on heliocentric motion
Perturbations caused by Earth’s oblateness in geocentric motion

3. Lagrange’s planetary equations
First-order differential equations for the orbital elements involving partial derivatives of the perturbing function
The orbital elements vary slowly, but the rate may vary quickly, since the perturbing function depends on position in the orbit
Let us see how they can be derived!

4. Perturbed vs unperturbed motion
Let xec be one of the cartesian, ecliptic coordinates
Introduce  as the orbital element vector
The xec velocity component can be written:

5. First osculation condition
The osculating elements are defined such that the velocity in the true, perturbed orbit is the same as the Keplerian velocity given by the osculating elements!
Hence we have the following condition on the time derivatives of the osculating elements:

6. Second osculation condition
Taking the second time derivative, we note that the Keplerian acceleration is the one given by the central force function
Hence we have another condition on the time derivatives of the osculating elements:

7. The Lagrangian brackets
We have six equations involving the time derivatives of the osculating elements and partial derivatives of the perturbing function w.r.t. the coordinates
Rearrange by multiplication and subtraction:
Introduce the so-called Lagrangian brackets:

8. Differential equations for 
Add the x, y and z equations together:
There are six such equations, featuring Lagrangian brackets (functions of orbital elements) as coefficients. One example is:

9. Properties of Lagrangian brackets
Total number of permutations: 36
But [a,a]=0 etc  30
But [a,e]=-[e,a] etc 15
• In addition, each Lagrangian bracket is explicitly independent of time

10. Evaluating the brackets
xec, yec and zec are composed of factors that depend on either (a,e,) or (i,,)

11. Angular elements (i,,)
Performing the derivatives of Rorbec(i,j) w.r.t. i and , we obtain the following expressions:

12. Geometry & time elements (a,e,)
Perform the derivatives of x and y w.r.t. a and e, using the near-perihelion approximations:
Then we obtain:

13. Results for mixed brackets

14. Using the bracket expressions
Example:
Full set of equations:

15. Solving the planetary equations
By successive elimination, we obtain Lagrange’s Planetary Equations:
- first-order differential equations for the orbital elements
- could easily be integrated, if we knew the explicit time dependence of the perturbing function
- but this would mean that we had already solved the problem!
We proceed by successive approximations, starting from a linear perturbation theory, computing the perturbing function from unperturbed orbits

16. Series developments
We express the coordinates, and thus the perturbing function, as trigonometric series in the angular elements (M,,) with coefficients that are non-linear expressions in the non-angular elements (a,e,i)
We truncate this expression at a certain order in the small quantities (e and i)
Resonance problem: M increases linearly with time t, so integration over t gives rise to terms with denominators of the form (kn+kpnp), where k and kp are integers. If n and np are nearly commensurable (close to resonance), some terms become extremely large

17. Action-angle variables
By truncating the series developments, we necessarily get an integrable system of equations
By transforming the variables, we can put this system on a canonical form, which means that the Hamiltonian (energy of the system) only depends on three of them (the action variables) but not on the other three (angle variables)
The integrals are then the action variables and the frequencies of the angle variables (constants of motion)
The semi-major axis a is such a constant in the linear theory, and M is an angle variable with constant frequency n

18. Proper Elements
The linear theory also identifies the coupled variations of (e,) and (i,) in terms of quasi-circular patterns in e.g. the (ecos,esin) plane
The offset of the center is the “forced eccentricity” and the radius of the circle is the “proper eccentricity”
The proper eccentricity and inclination are quasi-constants of motion over very long periods of time
Example from the long-term integration of (11798) Davidsson

19. Secular Resonances
The secular frequencies of the lines of apsides and nodes of the giant planets (due to their mutual perturbations) are characteristic frequencies of the entire Solar System
If an asteroid has the same frequency of variation of  or , the resonance leads to large variations of e and/or i
This is important for bringing main belt asteroids into planet-crossing orbits

20. Stability of the Solar System
Can one prove the stability of the planetary orbits on very long (quasi-infinite) time scales?
The constancy of the mean semi-major axis is not enough, since the eccentricity might increase until orbit crossing occurs
Poincaré showed more than 100 years ago that the series will generally diverge, and he made the first exploration of the properties of chaotic motion

21. Long-term stability, ctd.
In an integrable system, the phase space trajectories are locked to “invariant tori” surrounding periodic orbits (quasi-periodic motion)
The KAM (Kolmogorov-Arnol’d-Moser) theorem from the 1960’s proves that, if the perturbations applied to such a system are sufficiently small, the tori will generally survive even though chaos exists
However, in reality such smallness is not guaranteed
Thus, the Nekhoroshev theorem, stating that chaotic diffusion of trajectories is bounded over very long time scales, is very important
Finally, reality may be more complex!