Periods and Motion of Orbits, CM and Relative Motions

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SUMMARY

The discussion focuses on calculating the orbital period of a mass m1 in a circular orbit around a fixed mass m2 using Newtonian mechanics and Lagrangian mechanics. The derived period for part (a) is τ=2πr^(3/2)/√(GM2). In part (b), the challenge arises when applying the Lagrangian approach to a system where both masses orbit their common center of mass (CM), leading to confusion regarding the correct interpretation of the system dynamics. The participant expresses uncertainty about whether to treat the two bodies as orbiting around the CM or as the CM itself orbiting an arbitrary origin.

PREREQUISITES
  • Understanding of Newtonian mechanics, specifically gravitational orbits.
  • Familiarity with Lagrangian mechanics and the concept of the Lagrangian function.
  • Knowledge of center of mass (CM) calculations and reduced mass (μ).
  • Basic grasp of orbital dynamics and the implications of mass separation in two-body problems.
NEXT STEPS
  • Study the derivation of the orbital period using Newton's laws in two-body systems.
  • Learn about the application of Lagrangian mechanics in orbital dynamics, focusing on systems with variable mass distributions.
  • Explore the concept of reduced mass and its significance in simplifying two-body problems.
  • Investigate the implications of the center of mass in gravitational interactions and orbital mechanics.
USEFUL FOR

Students and educators in physics, particularly those studying classical mechanics, orbital dynamics, and Lagrangian mechanics. This discussion is beneficial for anyone seeking to deepen their understanding of orbital motion and the mathematical frameworks used to analyze it.

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Homework Statement


(a) Using elementary Newtonian mechanics find the period of a mass m1 in a circular orbit of radius r around a fixed mass m2 (solved, but placing for context).
(b) Using the separation into CM and relative motions, find the corresponding period for the case that m2 is not fixed and the masses circle each other a constant distance r apart. Discuss the limit of this results if m2 →∞.

Homework Equations


R = Centre of mass = m1r1+m2r2/m1+m2... let m1+m2 be M.
r = r1-r2
μ = m1m2/M = reduced mass
U(r) = Gm1m2/r

Lagrangian can be written as ½M d/dt(R)^2 + ½μ d/dt(r)^2 + U(r)
L=Lcm + Lref, where Lcm is the Lagrangian of the centre of mass and Lref is the Lagrangian of reduced mass and r.

The Attempt at a Solution


I solve part a and found a solution of τ=2πr^(3/2)/√GM2, which I am pretty confident with. I decided to solve the Lagrangian for the second part to see if I got a solution that would end up being similar to this solution (this is what I am expecting). But I found that v= GMt/r^2 + v0 was the equation for radial velocity, and thus the period comes out nothing like the period that I found using Newtonian mechanics.

I am wondering if I am interpreting the question wrong. Or, perhaps, I don't even use the Lagrangian at all (I have chosen this method because I am fresh with the derivation for Lagrangian orbits, but I see that the radius is constant and I'm not sure if this means that I should just be doing vector analysis).

Any help would be welcomed (it has helped just typing this out).
 
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I would appreciate some clarification with the conceptual side of the problem. Am I setting up a problem where these two bodies are orbiting about the CM, or am I treating these two bodies as if they ARE the CM and the CM is orbiting about some arbitrary origin?
 

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