Some time back I read a paper by Carl Bender on numerical approximation using the structure of Mechanics. Now I find out that Celestial Mechanics is using the so-called symplectic structure of classical mechanics, to improve numerical calculations.
Both findings sound true to me. There are several ways to discretize a continuous problem. One may expect that it helps to impose extra conditions in the discrete version. Conservation laws come to mind, and as I write in this note, also the geometric structure of phase space.
I am just coming to grips with the concepts of geometry of phase-space. In real space a riemannian metric may be enough, but when the equations of motion have an intrinsic sign difference between the two sets of equations that describe the mechanical system, one should expect more goodies.
I am looking for some of those, and it seems that celestial mechanicists already found some.
In another interesting paper, Carl Bender writes:
"The purpose of a transform is to convert an apparently complicated problem into one that is obviously simple. In order to be useful a transform must have an inverse. One applies the transform to a difficult-looking problem, solves the resulting easy problem, and applies the inverse transform to obtain the solution to the original problem."
Nifty.
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