Lagrangian of two body problem with spin

In summary, the conversation discusses the inclusion of spin in the "typical" Kepler problem and the possibility of including it in the Lagrangian. It is mentioned that spin is easy to include, but the dynamics only change if there is an interaction with one or both spins, such as tidal effects. It is also noted that including this interaction is the difficult part. The conversation ends with a question about the effects of tidal forces on the Earth-Moon system.
  • #1
Vrbic
407
18
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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  • #2
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.
 
  • #3
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)?
 
  • #4
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.
 
  • #5
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.
 

1. What is the Lagrangian of the two body problem with spin?

The Lagrangian of the two body problem with spin is a mathematical function that describes the dynamics of two bodies with spin interacting with each other through a potential energy. It takes into account the positions, velocities, and spins of both bodies, as well as the potential energy between them.

2. How is the Lagrangian derived for the two body problem with spin?

The Lagrangian for the two body problem with spin is derived using the Lagrangian formalism, which is a method for describing the dynamics of a system in terms of its generalized coordinates and their time derivatives. The potential energy between the two bodies and the kinetic energy of their spins are included in the Lagrangian function.

3. What is the significance of the two body problem with spin in physics?

The two body problem with spin is significant in physics because it allows for the study of the dynamics of two interacting bodies with intrinsic angular momentum (spin). This is important in understanding the behavior of particles at the quantum level, as well as in the study of celestial bodies such as planets and stars.

4. How does the Lagrangian of the two body problem with spin differ from the Lagrangian of the two body problem without spin?

The Lagrangian of the two body problem with spin includes additional terms for the kinetic energy of the spins and their interaction with each other. In the Lagrangian of the two body problem without spin, these terms are not present.

5. Can the Lagrangian of the two body problem with spin be solved analytically?

In most cases, the Lagrangian of the two body problem with spin cannot be solved analytically and requires numerical methods for finding solutions. However, there are some special cases where analytical solutions can be found, such as when the two bodies have equal masses and their spins are aligned or anti-aligned.

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