1. Kepler

2. Inverse Square Force  Force can be derived from a potential.  < 0 for attractive force  Choose constant of integration so V (  ) = 0. m2m2 r1r1 F 2 int r2r2 R m1m1 F 1 int r = r 1 – r 2

3. Kepler Orbits  Right side of the orbit equation is constant. Equation is integrable.Equation is integrable. Integration constants: e,  0Integration constants: e,  0  Equation describes a conic section.   init orientation (often 0)   init orientation (often 0) e sets the shape: e 1 hyperbola. e sets the shape: e 1 hyperbola. s is the directrix. s is the directrix. focus  r s

4. Kepler Lagrangian  The lagrangian can be expressed in polar coordinates.  L is independent of time. The total energy is a constant of the motion.The total energy is a constant of the motion. Orbit is symmetrical about an apse.Orbit is symmetrical about an apse. constant

5. Apsidal Position  Elliptical orbits have stable apses. Kepler’s first lawKepler’s first law Minimum and maximum values of r.Minimum and maximum values of r. Other orbits only have a minimum.Other orbits only have a minimum.  The energy is related to e: Set r = r 2, no velocitySet r = r 2, no velocity  r s r1r1 r2r2

6. Angular Momentum  Change in area between orbit and focus is dA/dt It is constantIt is constant  Kepler’s 2 nd law  Area for the whole ellipse relates to the period. semimajor axis: a=(r 1 +r 2 )/2.  Kepler’s 3 rd law r dr

7. Effective Potential  Treat problem as a one dimension only. Just radial r term.Just radial r term.  Minimum in potential implies bounded orbits. For  > 0, no minimumFor  > 0, no minimum For E > 0, unboundedFor E > 0, unbounded V eff 0 r 0 r unbounded possibly bounded