[stephencanon]

The orbital example where BDF loses momentum is really about the difference between a second-order method (BDF2) and a fourth-order method (RK), rather than explicit vs implicit (but: no method with order > 2 can be A-stable; since the whole point of implict methods is to achieve stability, the higher order BDF formulas are relatively niche).

There are whole families of _symplectic_ integrators that conserve physical quantities and are much more suitable for this sort of problem than either option discussed. Even a low-order symplectic method will conserve momentum on an example like this.

[Certhas]

Obviously^1. But it illustrates the broader point of the article, even if for the concrete problem even better choices are available.

1) if you have studied these things in depth. Which many/most users of solver packages have not.

[_alternator_]

The fascinating thing is that discrete symplectic integrators typically can only conserve one of the physical quantities exactly, eg angular momentum but not energy in orbital mechanics.

[srean]

I have always wanted to know if there is any theorem that says one cannot preserve all of the standard invariants.

For example, we know for mappings that we cannot preserve angles, distances and area simultaneously.

[ekelsen]

leapfrog!