Tricky interesting physics problem about magnetism

Question

Can someone please help in finding a solution? I've never solved a problem like this one and it seems interesting but I can't get my head around it.

The problem goes like this:

The interacting electric particles move in a constant magnetic field with induction 𝐵. The masses and charges of the particles are denoted as 𝑚1, 𝑞1 and 𝑚2, 𝑞2 respectively. Show that the dynamics problem of the system can be decomposed into two independent problems, one describing the motion of the system's center of mass and the other describing the motion of the particles relative to each other, only when 𝑞1𝑚2 = 𝑞2𝑚1. Write down the separate equations by introducing the total system charge 𝑄 = 𝑞1 + 𝑞2 and the reduced charge q ̃=q1q2/Q.

Answer 1

Assume that the particles have small enough speeds relative to B that radiative and relativistic effects are negligible. Then the two-particle Lagrangian is

L=1/2 m₁v₁² + 1/2 m₂v₂² + q₁v₁(A+A₂) + q₂v₂(A+A₁) - q₁Φ₂ - q₂Φ₁

Further assume that magnetic interactions between them are also negligible, and the only electric interactions are electrostatic. Then L reduces to

L=1/2 m₁v₁² + 1/2 m₂v₂² + (q₁v₁ + q₂v₂)A - kq₁q₂/r

Introduce center-of-mass coordinates,

MR = m₁r₁+m₂r₂, r=r₁-r₂,

solve for r₁ and r₂, let V=dR/dt, v=dr/dt, μ=m₁m₂/M, q=q₁q₂/Q,

then L becomes

L=1/2 MV² + 1/2 μv² + QVA + (q₁m₂-q₂m₁)vA/M - kQq/r.

If q₁m₂=q₂m₁, the fourth term disappears, and the motion decouples into center-of-mass motion determined by B=curl(A), and the well-known effective one-body problem (Kepler problem) for r.