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

This question evolved from a discussion below this answer which explains (among other things) that the total energy of a system offers insight as to the possibility of one (or all) members "escaping".

The total energy would be the sum of the kinetic and potential energies

$$E = \sum_{i=1}^{3}\frac{1}{2}m_i v_i^2 - \sum_{i=1}^{3} \sum_{j>i}^{3} \frac{m_i m_j}{r_{ij}}.$$

Can there be some three body orbit that is energetically unbounded ($$E>0$$) but where it is still impossible for any of the objects to escape due to do conservation of angular momentum?

Possibly helpful: Equations of motion for the n-body problem

notes:

1. I'm not asking if there exist orbits that are closed and periodic where escape is impossible for that reason.
2. I haven't written an expression for angular momentum because there is flexibility about which point it is calculated.

Batominovski's Clarification on the Bounty (as noted by Angela Pretorius in a comment). The energy should be measured with respect to the center-of-mass frame of the system. That is, the condition $$\sum_{i=1}^3m_iv_i=0$$ is enforced.

Based on comments here and my suspicion I've corrected $$i \ne j$$ to $$i > j$$ for the potential energy term to avoid double-counting.

• Regarding the link above, I think your post is fine here, although I believe that you might get a quicker answer if you post in the physics forum. May 6, 2020 at 6:26
• My suggestion is to post the question at Mathoverflow. May 6, 2020 at 19:16
• It seems unlikely, because as one mass gets very far away, you can give it a large angular momentum by giving it a tiny tangential velocity, which wouldn't significantly affect the other conservation laws. May 6, 2020 at 20:12
• @uhoh: Cross-posting is OK, as long as you state clearly in MO that this is a cross-post and give a link to MSE question. It happened to me in the past: I asked a question first on MSE, then on MO; it was well-received but it turned out to be a duplicate of an earlier MO question, so it was closed for that reason. May 8, 2020 at 4:44
• I should point out that the kinetic energy of a system can be made really high by changing the frame of reference. There are energetically unbounded systems which don't escape from each other, but I guess that you are really asking whether there are energetically unbounded systems which don't escape from a point which has zero velocity in the chosen frame of reference. May 8, 2020 at 6:06

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.)
• I am not quite sure how this post answers the question. Could you please elaborate? This post seems to prove only that, when there are at least two escaping bodies, then the total angular momentum can be changed arbitrarily without changing the total energy $E$. The question is whether there exists a bounded system with total energy $E>0$ that is bounded because the conservation of angular momentum. May 13, 2020 at 16:47