<p>In the most general case, there are three (spatial) degrees of freedom for each body, for a total of 9 degrees of freedom.</p>
<p>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 <em>no</em> degrees of freedom.</p>
<p>In the <em>planar</em> (or &quot;coplanar&quot;) circular restricted three-body problem, there is no motion or moment in the <span class="math-container">$z$</span> direction allowed (where <span class="math-container">$z$</span> is perpendicular to the orbital plane), so there are only <em>two</em> degrees of freedom left: the <span class="math-container">$x$</span> and <span class="math-container">$y$</span> position of the third body.</p>
<p>(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.)</p>

CC BY-SA 4.0
2021-06-24T21:06:21.153Z
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.
Deschele Schilder
CC BY-SA 4.0
2021-06-24T21:13:26.820Z
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.
Peter Erwin
CC BY-SA 4.0
2021-06-24T21:17:51.653Z
But we are not discussing math here.
Deschele Schilder
CC BY-SA 4.0
2021-06-25T02:19:22.310Z
@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.
David Hammen
CC BY-SA 4.0
2021-06-25T02:19:38.073Z
In other words, this answer is correct.
David Hammen
CC BY-SA 4.0
2021-06-25T08:45:44.350Z
@DavidHammen Im not saying its not correct. It depends. The small mass can move in every direction. Tthis means it has three degrees. The sma holds for the other two masses. What prevents them from moving in other directions?
Deschele Schilder
CC BY-SA 4.0
2021-06-25T08:51:55.167Z
On top of that the bodies have a rotational degree of freedom too(the real ones). You can make it a math problem but this is an astronomy site. If there were people living on the bodies they would be very disappointed to find they couldnt move in space! I guess its reoutation that talks here (and highly sophisticated use of language covering up the obvious).
Deschele Schilder
CC BY-SA 4.0
2021-06-25T11:07:33.587Z
@DescheleSchilder The Circular Restricted Three Body Problem (CR3BP for short) is a mathematical exercise. You can google "CR3BP" and find lots and lots of articles about this concept. In this problem, the objects are point masses, so there are no rotational degrees of freedom. The two larger masses are frozen in the synodic frame. This is the frame in which the five Lagrange points (aka Lagrangian points, aka libration points) appear. While this is a mathematical simplification, those Lagrange points are real, even in the Elliptical Restricted Three Body Problem (ER3BP for short).
David Hammen
CC BY-SA 4.0
2021-06-25T11:14:41.047Z
@DavidHammen If the motion is in a mathematical 2d plane (without a third one) then its obvious that there are two degrees for each mass and not zero for two and two for one as is stated in the first answer. The two circing heavy points still have two degrees of freedom. How else can they move on a circle around each other? The problem might be well documented but that  doesnt take away the fact that its not physical.
Deschele Schilder
CC BY-SA 4.0
2021-06-25T11:43:35.147Z
@DescheleSchilder From the perspective of a Newtonian inertial frame, there is but one degree of freedom in the Newtonian two body problem. Kepler deduced this without explaining why. Newton later explained why this was the case. In the CR3BP, that singular degree of freedom can easily be eliminated.
David Hammen
CC BY-SA 4.0
2021-06-25T11:50:15.860Z
@DavidHammen A mass falling straight to earth falls in one directio. But depending on where you lett the mass fall it has three independent possibilities to fall. So three degrees of freedom.
Deschele Schilder
CC BY-SA 4.0
2021-06-25T15:21:40.643Z
Just as a particle moving on a straight line executes just one of its degrees of freedom so are the masses in the plane. You can use any two coordinates to desribe their motion. xy  xz  yz. Anyhow, it's time to move on. Im done.
Deschele Schilder
CC BY-SA 4.0
2021-06-24T20:09:32.387Z
2021-06-24T20:09:32.387Z
Peter Erwin
<p>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?</p>
<p>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?</p>

CC BY-SA 4.0
2021-06-25T05:22:38.417Z
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?
Augustin
CC BY-SA 4.0
2021-06-25T05:23:34.113Z
Can we also compare the motion of the asteroid with that of a mathematical pendulum in which the asteroid is "suspended" by the rod formed by the Sun and Jupiter?
Augustin
CC BY-SA 4.0
2021-06-25T11:01:12.253Z
Good question in your first comment!
Deschele Schilder
CC BY-SA 4.0
2021-06-25T11:19:48.327Z
@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.
David Hammen
CC BY-SA 4.0
2021-06-25T11:25:53.387Z
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.
David Hammen
CC BY-SA 4.0
2021-06-25T11:27:11.370Z
So in the synodic frame, the two larger bodies do not move in the case of circular orbits. In other words, the two larger bodies have zero degrees of freedom in this frame.
David Hammen
CC BY-SA 4.0
2021-06-25T14:41:18.120Z
@DavidHammen Jupiter the sun and an asteroid are not moving in such a frame though. They are moving through plain 3d Eucliden space with no material boundaries. Jupiter can like all planets move out of the plane.
Deschele Schilder
CC BY-SA 4.0
2021-06-25T14:58:35.613Z
@DescheleSchilder Please read up on the circular restricted three body problem. I'm done.
David Hammen
CC BY-SA 4.0
2021-06-25T15:03:26.280Z
@DavidHammen Me too. But you still havent answered the question if planets cant move out of the plane.
Deschele Schilder
CC BY-SA 4.0
2021-06-25T15:20:17.277Z
@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."
HDE 226868
CC BY-SA 4.0
2021-06-25T15:24:14.623Z
@HDE226868 You are comparing oranges with apples. Can a planet move out of its plane of motion? It seems a question that is hard to anwer...I have asked it five times already.
Deschele Schilder
CC BY-SA 4.0
2021-06-25T15:31:23.920Z
@HDE226868 Im not saying the model is not perfect. Its a perfect model but a 2d space doesnt exist in the real world.
Deschele Schilder
CC BY-SA 4.0
2021-06-25T15:36:45.333Z
@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.
David Hammen
CC BY-SA 4.0
2021-06-25T15:40:00.037Z
@DescheleSchilder It can't move out of the plane in this model by the definition of the problem's restrictions, and that's all that matters.
HDE 226868
CC BY-SA 4.0
2021-06-25T15:46:29.717Z
@HDE226868 But you can use any pair of coordi nates  xy xz or yz you are free to do. If motion in one ditectio was restricted you couldnt do that. Anyhoe its not that important :) Ï choose the phydical approach and you the math (and dont  tell me there is one approach only...:)
Deschele Schilder
CC BY-SA 4.0
2021-06-25T15:48:01.853Z
@DescheleSchilder In the two body point mass model, the Newtonian gravitational force is instantaneous and is purely radial. This inherently means that two body problem orbits are strictly planar under Newtonian gravity. Adding a third body of negligible (infinitesimal) mass does not change this. Why are you being so difficult?
David Hammen
CC BY-SA 4.0
2021-06-25T15:55:19.853Z
@ DavvidHammen Tbe thing is that they still have the liberty  to move in the two anģular ditections too. What prohibits this freedom?
Deschele Schilder
CC BY-SA 4.0
2021-06-25T16:15:40.060Z
he façt that they move along two degrees only (or less) doesnt take that liberty away. I wouldnyt be able to escape from prison!
Deschele Schilder
<orbit><solar-system><astrophysics><orbital-mechanics><celestial-mechanics>
Degrees of freedom in restricted circular coplanar three body problem
Augustin
# Degrees of freedom in restricted circular coplanar three body problem

(Asked by: username_0 on 2021-06-24)

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?

**Tags:** orbit, solar-system, astrophysics, orbital-mechanics, celestial-mechanics

> **Comments:**
> **username_0**: Thank you for for your answers, Messrs.! @username_2, I would like to ask you, if you allow me, why the primary bodies, the Sun and Jupiter, do not have degrees of freedom?  
> **username_0**: Can we also compare the motion of the asteroid with that of a mathematical pendulum in which the asteroid is "suspended" by the rod formed by the Sun and Jupiter?  
> **username_1**: Good question in your first comment!  
> **username_2**: @username_0 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.  
> **username_2**: 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.  
> **username_2**: So in the synodic frame, the two larger bodies do not move in the case of circular orbits. In other words, the two larger bodies have zero degrees of freedom in this frame.  
> **username_1**: @username_2 Jupiter the sun and an asteroid are not moving in such a frame though. They are moving through plain 3d Eucliden space with no material boundaries. Jupiter can like all planets move out of the plane.  
> **username_2**: @username_1 Please read up on the circular restricted three body problem. I'm done.  
> **username_1**: @username_2 Me too. But you still havent answered the question if planets cant move out of the plane.  
> **username_3**: @username_1 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."  
> **username_1**: @username_3 You are comparing oranges with apples. Can a planet move out of its plane of motion? It seems a question that is hard to anwer...I have asked it five times already.  
> **username_1**: @username_3 Im not saying the model is not perfect. Its a perfect model but a 2d space doesnt exist in the real world.  
> **username_2**: @username_1 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.  
> **username_3**: @username_1 It can't move out of the plane in this model by the definition of the problem's restrictions, and that's all that matters.  
> **username_1**: @username_3 But you can use any pair of coordi nates  xy xz or yz you are free to do. If motion in one ditectio was restricted you couldnt do that. Anyhoe its not that important :) Ï choose the phydical approach and you the math (and dont  tell me there is one approach only...:)  
> **username_2**: @username_1 In the two body point mass model, the Newtonian gravitational force is instantaneous and is purely radial. This inherently means that two body problem orbits are strictly planar under Newtonian gravity. Adding a third body of negligible (infinitesimal) mass does not change this. Why are you being so difficult?  
> **username_1**: @ DavvidHammen Tbe thing is that they still have the liberty  to move in the two anģular ditections too. What prohibits this freedom?  
> **username_1**: he façt that they move along two degrees only (or less) doesnt take that liberty away. I wouldnyt be able to escape from prison!  

---

## 1 answers

### Answer by username_4 on 2021-06-24. Score: 8 | Accepted answer

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.)

> **Comments:**
> **username_1**: 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.  
> **username_4**: 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.  
> **username_1**: But we are not discussing math here.  
> **username_2**: @username_1 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.  
> **username_2**: In other words, this answer is correct.  
> **username_1**: @username_2 Im not saying its not correct. It depends. The small mass can move in every direction. Tthis means it has three degrees. The sma holds for the other two masses. What prevents them from moving in other directions?  
> **username_1**: On top of that the bodies have a rotational degree of freedom too(the real ones). You can make it a math problem but this is an astronomy site. If there were people living on the bodies they would be very disappointed to find they couldnt move in space! I guess its reoutation that talks here (and highly sophisticated use of language covering up the obvious).  
> **username_2**: @username_1 The Circular Restricted Three Body Problem (CR3BP for short) is a mathematical exercise. You can google "CR3BP" and find lots and lots of articles about this concept. In this problem, the objects are point masses, so there are no rotational degrees of freedom. The two larger masses are frozen in the synodic frame. This is the frame in which the five Lagrange points (aka Lagrangian points, aka libration points) appear. While this is a mathematical simplification, those Lagrange points are real, even in the Elliptical Restricted Three Body Problem (ER3BP for short).  
> **username_1**: @username_2 If the motion is in a mathematical 2d plane (without a third one) then its obvious that there are two degrees for each mass and not zero for two and two for one as is stated in the first answer. The two circing heavy points still have two degrees of freedom. How else can they move on a circle around each other? The problem might be well documented but that  doesnt take away the fact that its not physical.  
> **username_2**: @username_1 From the perspective of a Newtonian inertial frame, there is but one degree of freedom in the Newtonian two body problem. Kepler deduced this without explaining why. Newton later explained why this was the case. In the CR3BP, that singular degree of freedom can easily be eliminated.  
> **username_1**: @username_2 A mass falling straight to earth falls in one directio. But depending on where you lett the mass fall it has three independent possibilities to fall. So three degrees of freedom.  
> **username_1**: Just as a particle moving on a straight line executes just one of its degrees of freedom so are the masses in the plane. You can use any two coordinates to desribe their motion. xy  xz  yz. Anyhow, it's time to move on. Im done.  

---

