Why isn't a jagged orbit ever observed in the two-body problem?

Question

The two-body problem deals with two planets revolving around a common center relative to one another. Why doesn't the model ever exhibit a jagged orbit and it is always smoothly elliptical? Is there something about the math actively enforcing the smooth orbit?

Answer 1

As you are asking a question about inertial motion and gravitational forces, this question might have been better suited to Physics SE.

It is not only the two-body problem which has smooth trajectories. n-body problems also result in smooth (although far more complex) trajectories.

In special cases gravitational trajectories can change direction abruptly. For example, the 1D motion of a smaller body oscillating in a tunnel through the centre of a larger body. However, this trajectory is the limiting case of an ellipse with eccentricity $e \to 1$, and the velocities change smoothly through zero at the extremes so there is never an infinite acceleration. Another example is the 'sling-shot' motion of a light point-like particle passing very close to a much heavier moving particle. On a 'grand' scale this looks like a discontinuous change of speed and direction but on a sufficiently fine scale the trajectory is smooth and the acceleration remains finite (although possibly very large).

The motion of objects results from the interaction between a Force Law $f(r)$ and Inertial Law $f(r)=m\ddot r$. The gravitational and electrostatic force laws $f(r)=G\frac{Mm}{r^2}$ and $f(r)=k\frac{Qq}{r^2}$ are long-range and vary smoothly and continuously with separation $r$ between objects. The inertial law (Newton's 2nd Law) which links force and motion is linear; finite accelerations are guaranteed because mass $m$ is constant and finite. Because force $f(r)$ changes smoothly then so does acceleration $\ddot r$.

By contrast, contact forces are discontinuous (for practical purposes) which is why the trajectories of billiard balls are not smooth (when viewed macroscopically). And it is conceivable (in a different universe) that mass could vary discontinuously with force, resulting in infinite accelerations and non-smooth trajectories.