Lagrangian of two body problem with spin

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Discussion Overview

The discussion centers on the Lagrangian formulation of the two-body problem, specifically in the context of binary systems like the Earth-Moon system, with a focus on incorporating spin into the model. Participants explore the implications of including spin, the absence of spin in the general solution of the Kepler problem, and the potential effects of spin on the characteristics of motion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses interest in adding a spin term to the Lagrangian of a binary system, questioning its feasibility and potential effects on motion.
  • Another participant suggests that including spin is straightforward, but changes in dynamics depend on interactions with the spins.
  • A participant seeks clarification on how to model tidal effects within the Lagrangian framework and their implications for the system.
  • Concerns are raised about whether the inclusion of a spin term alters the equations of motion, with a participant noting that dynamics may remain trivial if only kinetic energy is considered.
  • Questions are posed regarding the specific effects of tidal forces on the Earth-Moon system, including changes in velocity and orbital characteristics.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the impact of including spin in the Lagrangian or the specific effects of tidal forces on the system. Multiple viewpoints regarding the role of spin and its interactions remain present.

Contextual Notes

There are unresolved questions about the nature of tidal effects and how they could be modeled within the Lagrangian framework. The discussion highlights the complexity of incorporating additional terms into the Lagrangian and their implications for the dynamics of the system.

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.
 
Last edited:
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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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