3. Two-Body Central-Force Problems Hasbun Ch 8 Taylor Ch 8 Marion & Thornton Ch 8

4. Central Force Examples Gravitational Force Conservative force, so the force can be derived from a potential energy, U(r 1, r 2 ) Coulomb Force Also a conservative force, described by the potential energy function: O r1r1 r = r 1 - r 2 r2r2

5. White Boards Write the Lagrangian for a two-body central-force system. The Problem: find the possible motions of these two bodies

6. Center of Mass and Relative Coordinates What generalized coordinates should we use to solve the two-body central force problem? Relative position, r Position of center of mass, R, where O r1r1 CM r2r2 R The total momentum of the system is: B/c total momentum is constant, CM frame is an inertial reference frame.

8. White Boards Write r 1 and r 2 in terms of R and r. Write the kinetic energy T in terms of R and r. Write the Lagrangian L in terms of R and r.

9. White Boards Write the equations of motion from the Lagrangian (one equation for R, one equation for r)

10. Center of Mass Frame If we choose the center-of-mass frame as our reference frame (we can do this b/c it’s an intertial frame), then The center of mass part of L is zero, so the Lagrangian becomes: Note this is the Lagrangian for a single particle moving in a central potential!

11. Conservation of Angular Momentum Angular momentum of the two particles is conserved. Rewrite this in the COM frame Because the direction of angular momentum is constant, the motion remains in a fixed plane, which we can take to be the x-y plane. Always a 2D problem in COM frame

12. White Boards The easiest coordinates for this 2-D problem are polar coordinates. Remember that the velocity in polar coordinates is given by Write the Lagrangian in polar coordinates, and find the equations of motion.

13. Equations of Motion Angular Equation Radial Equation Centrifugal Force

14. White Board Use the phi equation to eliminate phi-dot from the radial equation

15. White Board Use the phi equation to eliminate phi-dot from the radial equation

16. Bound & Unbound Orbits At r min, the energy is equal to U eff

17. Matlab Problem Write down the actual and effective potential energies for a comet (or planet) moving in the gravitational field of the sun. Plot the 3 potential energies involved (U, Ucf, Ueff) and use the graph of Ueff vs r to describe the motion as r goes from large to small values. UseG=1m1=m2=l=1

18. Matlab Problem