I am currently preoccupied with the Two Body Problem and I was wondering whether the barycenter, or center of mass, is a static point or if it is moving, and how to calculate its position.

  • $\begingroup$ It might help if you specify moving with respect to what? $\endgroup$ – enumaris Dec 11 '18 at 17:24
  • $\begingroup$ In respect to the inertial frame of the initial position $\endgroup$ – Ian Ronk Dec 11 '18 at 18:29

In the absence of external forces, the barycenter of any mechanical system will move in a straight line with a uniform velocity, and if the system starts off at rest (i.e. the total momentum is zero) then the barycenter will remain static for all time.

If the barycenter does move, then it will satisfy the newtonian equation of motion \begin{align} \frac{\mathrm d}{\mathrm dt}\mathbf R(t) = \frac1m \mathbf P(t) \\ \frac{\mathrm d}{\mathrm dt}\mathbf P(t) = \mathbf F(t), \end{align} with $\mathbf P(t)$ the total momentum of the system and $\mathbf F(t)$ the total external force acting on the system.

  • $\begingroup$ Does it move into one direction and then turn and move the same way back or how does this work when one orbit around this point is completed? $\endgroup$ – Ian Ronk Dec 11 '18 at 18:28
  • $\begingroup$ Do you mean with total momentum is zero that the point masses in the systems don't have a velocity or is it also when they do have a velocity to create an orbit around this point? $\endgroup$ – Ian Ronk Dec 11 '18 at 18:31
  • $\begingroup$ Emilio is basically saying the Barycenter doesn't move (in the sense that you are thinking). Assuming no external forces, there is an inertial frame in which the Barycenter is static. Thus, in any other inertial frame, the Barycenter is moving at a constant velocity in a straight line. $\endgroup$ – enumaris Dec 11 '18 at 18:45

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