Lagrangian of two body problem with spin

AI Thread Summary
The discussion focuses on the complexities of modeling binary systems, specifically the Earth-Moon system, using Lagrangian mechanics while considering spin effects. It highlights that while incorporating spin into the Lagrangian is straightforward, the dynamics only change significantly when there is an interaction between the spins, such as tidal forces. The participants express curiosity about how these tidal interactions affect orbital characteristics and whether the inclusion of spin alters the equations of motion. They seek clarification on how to model tidal effects in a simplified manner and the resultant changes in the system's dynamics. Overall, the conversation emphasizes the challenges of integrating spin and tidal interactions into the analysis of binary systems.
Vrbic
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I know how to solve "typical" Kepler problem but I'm interested in a global view to "binary" systems. For example Earth - Moon. If I set lagrangian of system as ##L=\frac{1}{2}(m_1\dot{r}_1^2 + m_2\dot{r}_2^2)-V(|r_2-r_1|)## there isn't included a spin.
My questions are:
1) If it is solved as two body (I guess, two points) problem. Is possible to put there a term describing a spin?
2) Why the spin isn't in general solution of Kepler problem?
3) Whether it is possible. How? What term could describe the spins? ##L_s=\frac{1}{2}(J_1\dot{\phi}_1^2 + J_2\dot{\phi}_2^2)##? ; J - moment inertia
4) Anyway, if or not would be possible to create lagrangian with spin. Does it change some characteristic of motion? (shape of orbit, period, etc.)?

Thank you for your replies.
 
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Spin is easy to include, but the dynamics only change if there is an interaction with one or both spins.

Including the interaction (for example to account for tides) is the hard part.
 
Dr. Courtney said:
Spin is easy to include, but the dynamics only change if there is an interaction with one or both spins.

Including the interaction (for example to account for tides) is the hard part.
Thank you very much for you reply.
So if I understand in good way, it is answer to my question 1),2) and 4). Could you please comment the third?

I totally hope, it change something. I know, changes are arising from tidal effects. But why it doesn't change anything? Lagrangian is changed and I believe that extra term is not a total derivative of some function...or? Because this is a only one case, which I know, when the equations of motion (Lagrange eq.), are not changed.

If I may, I have last question: How could the term, I mean the easiest one (not real), which describes for example tidal effect (total toy model)?
 
Vrbic said:
If I may, I have last question: How could the term, I mean the easiest one (not real), which describes for example tidal effect (total toy model)?
When I'm thinking about tidal force, honestly I have to say, I don't know results of it. What do tidal forces cause on system Earth-Moon? Which way? Changes in velocity of orbiting, spinning or...? I'm not asking for exact mechanisms, but their results on system.
 
Your term for spin is right, but if there is only a kinetic energy term in a variable (no potential energy), the dynamics are trivial.
 
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