Show That F is Continuous on −∞ ˆž

You're quite right about multipole not being worth it for stock KSP; not only are we dealing with low N but the system is very flat and the rotational aspect completely dominates phase space. On that note, 2D/4D clustering using volumetric 3D space partitioning (octrees) is just silly. If we define a system of say 5 gas giants in the 0.5 < M_Jup < 6 range and put more than an Earth-mass of moons in orbit around each, plus trojans and spartans at the L-points and add a considerable sphere of scattered objects, well then FMM might be good (one can dream, right)

We'll have asteroids to deal with in 0.24, so that's not completely out of scope.

I believe there is a way to describe all these "leapfrog in higher orders" algorithms in terms of partitioned RK schemes,

"leapfrog in higher orders" is pretty much what SPRKs are about, though a stricter generalisation of leapfrog is an FSAL SPRK. They give some really nice explanations for the idea behind the SPRKs in this book though, rather than expressing them all as a bunch of coefficients. I think I'll add it to the recommended reading section, it looks pretty accessible.

I've seen Yoshida referenced everywhere in the litterature, so I guess I better read up on Lie algebras and Baker–Campbell–Hausdorff in order to understand his work ....

You'll have a hard time doing that, since papers about symplectic integrators assume you know what this is all about and general treatments of the Baker-Campbell-Hausdorff formula fail to mention the relation to Hamiltonian mechanics. Here's a rough outline, you should be able to fill in the blanks from Wikipedia.

-- Notations:

M is the phase space, in this case â„Â^3NÃâ€"(â„Â^3N)^*;

{ . , . } : C^{∞}(M)² → C^{∞}(M) is the Poisson bracket;

{f, . } := g ↦ {f, g}, i.e., {f, . } g = {f, g}, so that {f, . } is a linear endomorphism of the space C^{∞}(M) of smooth functions of the phase space.

We want the solution z to z' = -{H, z} (Hamilton's equations, where z = (q, p) are the generalised coordinates).

It is of course given by the exponential of the linear operator applied to the initial conditions, z = exp(-t {H, . }) z ₀, so that z(Ä) = exp(-{Ä H, . }) z ₀, where Ä is the timestep.

We take smooth functions f(q, p) and g(q, p) in C^{∞}(M) (H, z = id, L, p, the Lagrangian ℒ, T, V, are examples of such functions) and look at the commutator of the linear operators {f, . } and {g, . }:

          
[{f, . }, {g, . }] = {f, . } {g, . } - {g, . } {f, . }          
          = {f, {g, . }} - {g, {f, . }}          
          = {f, {g, . }} + {g, { . , f}}      -- As the Poisson bracket is antisymmetric,
          = - { . , {f, g}}                   -- As the Poisson bracket satisfies the Jacobi identity,
          = {{f, g}, . }                      -- Antisymmetry.
        

It follows that the commutator on the operators of the form {f, . } works essentially like the Poisson bracket, so it will satisfy the Jacobi identity too. It is bilinear and alternating, and the {f, . }s form a vector space, so it is a Lie algebra.

Note that we actually started with the Lie algebra (C^{∞}(M), { . , . }), whose the Lie bracket was the Poisson bracket (a Poisson algebra, mathematicians have creative naming conventions), what we just did was taking the adjoint representation (see the relevant Wikipedia article), thus getting a subalgebra of End(C^{∞}(M)). The usual notation for the adjoint representation is ad f := {f, . }; I'll use this notation from now on, the brackets are getting rather cumbersome.

Back to the problem at hand, finding exp(Ä ad H). We have H = T + V (or some other partition of the Hamiltonian). We want to know how to write exp(Ä ad(T + V)) = exp(Ä ad T + Ä ad V) as a function of exp(Ä ad T) and exp(Ä ad V), the separate evolutions. ad T and ad V don't commute, so it's not just the product. The Baker-Campbell-Hausdorff formula tells you just that: it says

          
log(exp(Ä ad T) exp(Ä ad V)) = Ä ad T + Ä ad V + 1/2 [Ä ad T, Ä ad V] + 1/12 [Ä ad T, [Ä ad T, Ä ad V]] - 1/12 [Ä ad V, [Ä ad T, Ä ad V]] + ...
        

I think this should enable you to read Yoshida's works.

I would recommend starting with Yoshida's Symplectic Integrators for Hamiltonian Systems: Basic Theory [Yoshida 1992] in order to see the general definitions and ideas, his 1990 paper Construction of higher order symplectic integrators focuses only on proving that such integrators exist, with no concern for efficiency, and only talks about even-order integrators. See the discussion with Wisdom at the end of [Yoshida 1992].

The main idea is the same as for Runge-Kutta (see my introduction linked in the OP) except instead of trying to match terms in the Taylor expansion of the solution, you try to match terms in the Baker-Campbell-Hausdorff formula: the convergence rate is for the Hamiltonian here. Of course you do that by composing symplectic maps so that you don't have an energy drift.

I think you'd be surprised at how many companies enforce strict C89 lexical standards in 2014, even if you're coding something like C# in a fully utf-8 environment. I've even seen source control systems that reject check-ins if you don't use strictly alphanumeric US ASCII extended by only precisely a set of characters from the host language (operators etc). And this in Europe! Inconceivable!

We'll always have Mordac.

there should be hats on those f's!

Which f's?

Unrelated: while investigating formalisations of physical quantities and reference frames, I found this fascinating post by Terence Tao. It turns out that deep down formalising them works similarly. The post is pretty enlightening: it give a clue as to how to formalise the fact an average of temperatures makes sense, a difference of temperatures makes sense, a sum of temperature differences makes sense, but a sum of temperatures doesn't (or similarly, why a barycenter makes sense while a sum of positions doesn't).

Edited March 5, 2014 by eggrobin
signs.

browningdecien.blogspot.com

Source: https://forum.kerbalspaceprogram.com/index.php?/topic/162200-wip181-191-1101-1110%E2%80%932-1122%E2%80%933-principia%E2%80%94version-hilbert-released-2022-07-28%E2%80%94n-body-and-extended-body-gravitation/page/7/

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