1. Central Forces

2. Two-Body System  Center of mass R  Equal external force on both bodies.  Add to get the CM motion  Subtract for relative motion m2m2 r1r1 F 2 int r2r2 R m1m1 F 1 int F 1 ext F 2 ext r = r 1 – r 2

3. Reduced Mass  Internal forces are equal and opposite.  Express in terms of a reduced mass .  less than either m 1, m 2  less than either m 1, m 2  approximately equals the smaller mass when the other is large.  approximately equals the smaller mass when the other is large. for

4.  Central motion takes place in a plane. Force, velocity, and radius are coplanar.Force, velocity, and radius are coplanar.  Orbital angular momentum is constant.  If the central force is time-independent, the orbit is symmetrical about an apse. Apse is where velocity is perpendicular to radiusApse is where velocity is perpendicular to radius Central Motion Use J to avoid confusion with Lagrangian L

5. Central Force Equations  Use spherical coordinates. Makes r obvious from central force.Makes r obvious from central force. Generalized forces Q  = Q  = 0.Generalized forces Q  = Q  = 0. Central force need not be from a potential.Central force need not be from a potential.  Kinetic energy expression

6. Coordinate Reduction  T doesn’t depend on  directly.  Constant angular momentum about the polar axis. Constrain the motion to a planeConstrain the motion to a plane Include the polar axis in the planeInclude the polar axis in the plane  Two coordinates r, . constant

7. Angle Equation  T doesn’t depend on  directly.  Also represents constant angular momentum. A constant of the motionA constant of the motion  Change the time derivative to an angle derivative. constant

8. Orbit Equation Let u = 1/r

9. Central Potential  Central force can derive from a potential.  Rewrite as differential equation with angular momentum.  Equivalent Lagrangian next