# Degrees of freedom in restricted circular coplanar three body problem

How many degrees of freedom does a mechanical system consisting of three bodies, the Sun, Jupiter and an asteroid, have in the restricted circular coplanar problem of the three bodies?

I know that if we consider the three bodies as material points, each will have three degrees of freedom, so the system will have 9. However, if the three bodies are forced to remain in the same orbital plane, it would mean that each to have 2 degrees of freedom? So the system will have 6 degrees of freedom in total?

• Thank you for for your answers, Messrs.! @DavidHammen, I would like to ask you, if you allow me, why the primary bodies, the Sun and Jupiter, do not have degrees of freedom? Jun 25, 2021 at 5:22
• @Augustin First look at the Newtonian two body problem. This problem addresses two point masses orbiting one another, with the only interaction between the two being Newtonian gravitation. From the perspective of a Newtonian inertial frame of reference, this two body system has but one degree of freedom. This is why the concept of Keplerian orbital elements work so nicely. Jun 25, 2021 at 11:19
• The only thing that is special with regard to Newtonian inertial frames is that fictitious accelerations vanish in such frames. Non inertial frames of reference are equally valid; one merely needs to account for those fictitious accelerations. In the Circular Restricted Three Body Problem (CR3BP), all (and I do mean all) analyses are done from the perspective of the "synodic frame". This is a rotating frame whose rotational axis is the same as the angular momentum vector of the two orbiting bodies that rotates at the same rate as the orbital rate of the two bodies. Jun 25, 2021 at 11:25
• @DescheleSchilder The restricted coplanar three body problem is a toy model, and it's totally valid to ask questions about that model without focusing on any imperfections or deviations from reality. You're saying "Well, the model's not totally realistic", and while that's true, it's not relevant here. As another example: black bodies are good models for some objects, and we can understand objects by treating them as black bodies, even though they're not. But it's unhelpful to sidestep a question by saying "Well, the model isn't perfect." Jun 25, 2021 at 15:20
• @DescheleSchilder Stop thinking about planets. The question asks about the CR3BP, which is an idealization using point masses and a universe comprising only two bodies with significant mass. (The third body has infinitesimal mass.) Because the orbits of many of the planets are nearly planar and are nearly circular, the CR3BP remains a very useful approximation. Jun 25, 2021 at 15:36

In the most general case, there are three (spatial) degrees of freedom for each body, for a total of 9 degrees of freedom.

The circular restricted three-body problem forces the two larger masses to be in perfectly circular orbits defined by their masses and the chosen orbital radii (with the third body having negligible mass and thus no influence on their orbits), so they have no degrees of freedom.

In the planar (or "coplanar") circular restricted three-body problem, there is no motion or moment in the $$z$$ direction allowed (where $$z$$ is perpendicular to the orbital plane), so there are only two degrees of freedom left: the $$x$$ and $$y$$ position of the third body.

(The circular restricted three-body problem defines the bodies to be point masses, so there are no extra degrees of freedom for things like rotation.)

• No degrees of freedom? That makes it pretty hard to move for one mass. You are making the mistake to confuse a state of motion with a boundary induced, material induced restriction of the degrees of freedom. There is no such boundary in this problem so all masses simply have three degrees of freedom. Jun 24, 2021 at 21:06
• Please re-read my answer: I said the restricted problem eliminated the degrees of freedom for the two massive objects; the planar restrictions means there are two degrees of freedom ($x$ and $y$) for the third object. And you are confusing a mathematical problem with reality. Jun 24, 2021 at 21:13
• But we are not discussing math here. Jun 24, 2021 at 21:17
• @DescheleSchilder The circular restricted three body problem is math. It is always solved in the synodic frame, a non-inertial reference frame that rotates with the orbit of the two larger masses about one another such that neither of the two larger masses is moving in that frame of reference. No motion = zero degrees of freedom. The test mass is an object with negligible mass; it does not perturb the larger masses orbits. The test body is the only body with has any degrees of freedom. With motion constrained to the larger bodies' orbital plane, there are only two degrees of freedom. Jun 25, 2021 at 2:19
• In other words, this answer is correct. Jun 25, 2021 at 2:19