<p>The condition $$\sum_{i=1}^{3}m_iq_i'=0$$
of <a href="http://iopscience.iop.org/1538-3881/145/3/63/pdf/1538-3881_145_3_63.pdf" rel="nofollow">this paper</a> generalizes to
$$\sum_{i=1}^{n}m_iq_i'=0,$$
meaning the <a href="http://en.wikipedia.org/wiki/Kinetic_energy" rel="nofollow">barycenter</a> should be at the origin of the frame.</p>

<p>Equation (1) generalizes to
$$H=\frac{1}{2}\sum\frac{|p_i'|^2}{m_i}-\sum_{i=1}^n\sum_{j=1, j\neq i}^n\frac{m_im_j}{|q_i'-q_j'|}.$$
The first part is the total <a href="http://en.wikipedia.org/wiki/Kinetic_energy" rel="nofollow">kinetic energy</a> of the system relative to the barycenter, written in terms of <a href="http://en.wikipedia.org/wiki/Momentum" rel="nofollow">momenta</a>: $$E=\frac{1}{2}mv^2=\frac{1}{2}m\left(\frac{p}{m}\right)^2=\frac{1}{2}\frac{p^2}{m}.$$
The second part is the sum of all <a href="http://en.wikipedia.org/wiki/Potential_energy" rel="nofollow">self-potential energy</a> between each pair of (spherical) objects, with Newton's gravitational constant $G$ set to 1.</p>

<p>See also <a href="http://en.wikipedia.org/wiki/N-body_problem" rel="nofollow">n-body problem on Wikipedia</a>.</p>

<p>The 3-dimensional 3-body (or n-body) problem is like the planar version. Just take 3-dimensional vectors for $p_i'$ and $q_i'$.</p>

CC BY-SA 3.0
2014-04-29T20:12:18.443Z
2014-04-29T21:51:38.193Z
2014-04-29T21:51:38.193Z
Gerald
<p>Maybe move your planar restricted problem to a 3D orbital restricted one.
(i.e. first two bodies in fixed orbit then solve for third body off the plane)</p>

CC BY-SA 3.0
2014-12-19T18:11:15.367Z
2014-12-19T18:16:57.110Z
2014-12-19T18:16:57.110Z
Neomada
<p>i try to extend my analysis from planar restricted three body problem to the general one. </p>

<p>Is there anyone that can help me to find out the Hamiltonian for this prolem in cartesian coordinates? </p>

<p>I need the hamiltonian of 3D case somethink like eq.(1) in this <a href="http://iopscience.iop.org/1538-3881/145/3/63/pdf/1538-3881_145_3_63.pdf" rel="nofollow">article</a> or in this one <a href="http://www.aanda.org/articles/aa/pdf/2009/02/aa10843-08.pdf" rel="nofollow">paper</a> again eq.1 . Because my symplectic integrator can be used only if I have the fixed cartesian coordinates.</p>

<p>Thanks</p>

CC BY-SA 3.0
2014-04-29T18:41:20.933Z
This should probably be migrated over to Physics
Rory Alsop
CC BY-SA 3.0
2014-04-29T19:00:24.357Z
Could you please [edit] your question to include more information on what exactly you have problems with? The way you currently specified it, it's too broad and rather unclear what exactly it even pertains to (tags you used help a bit, but not too much). If I get you correctly, you're after Hamiltonian form of the three-body problem? If that's the case then you might find [this article](http://www.scholarpedia.org/article/Three_body_problem) useful.
TildalWave
CC BY-SA 3.0
2014-04-29T19:15:16.233Z
Yes, something like that but, in the wiki example the Hamiltonian is for the planar case. I need the hamiltonian of 3D case somethink like eq.(1) in this article [link](http://iopscience.iop.org/1538-3881/145/3/63/pdf/1538-3881_145_3_63.pdf) or in this one [link](http://www.aanda.org/articles/aa/pdf/2009/02/aa10843-08.pdf) again eq.1
Panichi Pattumeros PapaCastoro
<solar-system><stellar-dynamics>
Hamiltonian of general three body problem
Panichi Pattumeros PapaCastoro
# Hamiltonian of general three body problem

(Asked by: username_0 on 2014-04-29)

i try to extend my analysis from planar restricted three body problem to the general one.

Is there anyone that can help me to find out the Hamiltonian for this prolem in cartesian coordinates?

I need the hamiltonian of 3D case somethink like eq.(1) in this [article](http://iopscience.iop.org/1538-3881/145/3/63/pdf/1538-3881_145_3_63.pdf) or in this one [paper](http://www.aanda.org/articles/aa/pdf/2009/02/aa10843-08.pdf) again eq.1 . Because my symplectic integrator can be used only if I have the fixed cartesian coordinates.

Thanks

**Tags:** solar-system, stellar-dynamics

> **Comments:**
> **username_1**: This should probably be migrated over to Physics  
> **username_2**: Could you please [edit] your question to include more information on what exactly you have problems with? The way you currently specified it, it's too broad and rather unclear what exactly it even pertains to (tags you used help a bit, but not too much). If I get you correctly, you're after Hamiltonian form of the three-body problem? If that's the case then you might find [this article](http://www.scholarpedia.org/article/Three_body_problem) useful.  
> **username_0**: Yes, something like that but, in the wiki example the Hamiltonian is for the planar case. I need the hamiltonian of 3D case somethink like eq.(1) in this article [link](http://iopscience.iop.org/1538-3881/145/3/63/pdf/1538-3881_145_3_63.pdf) or in this one [link](http://www.aanda.org/articles/aa/pdf/2009/02/aa10843-08.pdf) again eq.1  

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## 2 answers

### Answer by username_3 on 2014-04-29. Score: 3

The condition $$\sum\_{i=1}^{3}m\_iq\_i'=0$$
of [this paper](http://iopscience.iop.org/1538-3881/145/3/63/pdf/1538-3881_145_3_63.pdf) generalizes to
$$\sum\_{i=1}^{n}m\_iq\_i'=0,$$
meaning the [barycenter](http://en.wikipedia.org/wiki/Kinetic_energy) should be at the origin of the frame.

Equation (1) generalizes to
$$H=\frac{1}{2}\sum\frac{|p\_i'|^2}{m\_i}-\sum\_{i=1}^n\sum\_{j=1, j\neq i}^n\frac{m\_im\_j}{|q\_i'-q\_j'|}.$$
The first part is the total [kinetic energy](http://en.wikipedia.org/wiki/Kinetic_energy) of the system relative to the barycenter, written in terms of [momenta](http://en.wikipedia.org/wiki/Momentum): $$E=\frac{1}{2}mv^2=\frac{1}{2}m\left(\frac{p}{m}\right)^2=\frac{1}{2}\frac{p^2}{m}.$$
The second part is the sum of all [self-potential energy](http://en.wikipedia.org/wiki/Potential_energy) between each pair of (spherical) objects, with Newton's gravitational constant $G$ set to 1.

See also [n-body problem on Wikipedia](http://en.wikipedia.org/wiki/N-body_problem).

The 3-dimensional 3-body (or n-body) problem is like the planar version. Just take 3-dimensional vectors for $p\_i'$ and $q\_i'$.


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### Answer by username_4 on 2014-12-19. Score: 0

Maybe move your planar restricted problem to a 3D orbital restricted one.
(i.e. first two bodies in fixed orbit then solve for third body off the plane)


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