Demonstration
One potential, two integrations.
Fitted integration
A fitted force law is not exactly central, so it turns the conserved vector, and not exactly inverse square, so the conic opens. The faint line is the encoded column.
Encoded integration
The kick is exactly along the line to the centre, so nothing in the update can torque the angular momentum. The conic closes and the plane holds.
Angular momentum is carried by the construction here rather than learned, so conservation is a property of the update itself.
How to read this figure
The road the body has taken, brightest where it is now.
The angular momentum. Its length fixes the conic, its direction fixes the plane.
The plane the orbit began in, so a tip out of it is visible.
How far the vector has left its start. The dashed line is where it began.
Angular momentum is carried by the construction here rather than learned, so conservation is a property of the update itself. It is a vector, so both its length and its direction are measured. Both columns start from the same condition, in the same potential, at the same step, under the same integrator, and differ by one term. The fitted column loses nine per cent of the momentum and tips the plane twenty degrees; the encoded column holds both at double precision, and holds them at every step size the figure runs.