Can there be an energetically unbounded three-body orbit where escape is impossible due to conservation of angular momentum?

Solution 1:

The answer is no... conservation of angular momentum, by itself, can't be used to prove boundedness of a 3-body system with positive total energy (in the frame where the center of mass is stationary at the origin). For sufficiently large $t$, all escaping bodies (there must be at least 2) will have essentially fixed velocities ${\bf v}_i$ and linearly evolving positions ${\bf x}_i + t {\bf v}_i$. The total angular momentum is $\sum_i \left({\bf x}_i + t{\bf v}_i\right) \times m_i{\bf v}_i = \sum_i m_i{\bf x}_i \times{\bf v}_i$, also a constant. But note that the angular momentum can be changed to any value without changing the total energy, the total momentum, or the center of mass, by adding appropriate offsets to the ${\bf x}_i$. (Keeping the center of mass fixed imposes one vector constraint on these offsets; since at least two bodies are escaping, there is at least one vector degree of freedom remaining.)

In short, conservation of angular momentum doesn't help you because each "escape scenario" belongs to an equivalence class of scenarios (with the same total energy and momentum) that differ only in their angular momenta.