I've read a little bit about Delaunay variables, but I can't understand what they are good for. Do they make calculations easier? What is the advantage of using them? Where can I read a bit more about it? There's not so much in the books.

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    $\begingroup$ Have you studied Hamiltonian mechanics? Most books about solar system dynamics cover them. Yes, they're simplifying equations because, as wiki says, they are action-angle coordinates. $\endgroup$ Jan 21, 2020 at 1:15

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Long story short: Delaunay variables are something which is called action-angle variables, which makes them extremely useful in perturbation theory. Let me first talk about what action-angle variables in general and then mention a few facts about Delaunay variables themselves.

Let us consider a Hamiltonian system of $N$ degrees of freedom with generic canonical coordinates $p_i,q^j, i,j=1,...,N$ and a general integrable Hamiltonian $H(p_i,q^j)$. Action-angle coordinates are a special type of canonical coordinates $J_\beta(p_i,q^i), \psi^\alpha(p_i,q^i),\, \alpha,\beta = 1,...,N$ such that the Hamiltonian is expressed only as a function of the actions $J_\beta$, $H =H(J_\beta)$. Consequently, the equations of motion are $$\dot{\psi^\alpha} = \frac{\partial H}{\partial J_\alpha} \equiv \Omega^\alpha(J_\beta)$$ $$\dot{J_\beta} = -\frac{\partial H}{\partial \psi^\alpha} = 0$$ In other words, we can easily write the explicit (and general) solution of the equations of motion as $J_\beta(t) = J_{0\beta}, \,\psi^\alpha(t) = \psi^\alpha_0 + \Omega^\alpha(J_{0\beta})t$. Of course, the issue is that it is usually quite difficult to find the transformation $J_\alpha(p_i,q^i), \psi^\alpha(p_i,q^i)$ from the generic canonical coordinates $p_i,q^i$ you start with. In fact, it is typically about as hard as finding the explicit solution to the equations of motion in the original system of coordinates.

So why make the transformation? The answer is that it becomes extremely useful once the original system is perturbed such that the new Hamiltonian is $H' = H(J_\beta) + \epsilon K(J_\beta,\psi^\alpha)$ with $\epsilon \ll 1$. There is then a straightforward algorithm to transform to new, slightly deformed coordinates $\psi^{\alpha'}, J'_\beta$ such that $H' = H(J'_\beta) + \epsilon \bar{K}(J'_\beta)$, where the new addition to the Hamiltonian is obtained by averaging over the angle coordinates and subsituting the new actions: $$\bar{K}(J'_\beta) = \langle K(J_\beta,\psi^\alpha)\rangle_{\psi}\Big|_{J_\beta \to J'_\beta}$$ In other words, action-angle coordinates provide you not only with the exact solution but also a way to describe the exact solution of any close system (up to special points called resonances...). For more details see e.g. Arnol'd, Kozlov & Neishtadt (English ed. 2006) - Mathematical Aspects of Classical and Celestial Mechanics.

The Delaunay variables are action-angle variables for the Newtonian two-body problem. Even more, they are "the" action-angle variables, since any other set of action-angle variables is related to another one by a discrete rotation.

The three Delaunay action variables are usually denoted $L,G,H$, and the respective angles as $l = M, g=\omega, h = \Omega$. $G$ and $H$ have the meaning of total and azimuthal angular momentum respectively, and $L$ has the meaning of a "total orbital action" including not only the angular momentum but also the "oomph" (also known as action) of the radial motion. The angles $\omega, \Omega$ are argument of periapsis and longitude of ascending node and in the unperturbed Keplerian problem they are degenerate (they have a zero associated frequency $\Omega^\alpha$). The final angle $M$ is just the mean anomally containing the only real dynamics in the Kepler problem. The nice thing about Delaunay variables is that one just needs to solve the Kepler equation to make the transform to the action-angle coordinates. However, when passing to perturbation theory, I recommend to make a simple discrete transform to a modified set of coordinates because Delaunay variables are degenerate near circular and/or equatorial orbits.

For the construction and properties of Delaunay variables, see the excellent textbook of Morbidelli (2002) - Modern Celestial Mechanics (this is also a good resource for perturbation theory mentioned earlier).


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