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 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.

4. Kepler Orbits  The right side of the orbit equation is constant. Equation is integrableEquation is integrable Integration constants: e,  0Integration constants: e,  0 e related to initial energye related to initial energy Phase angle corresponds to orientation.Phase angle corresponds to orientation.  The substitution can be reversed to get polar or Cartesian coordinates.

5. Conic Sections focus  r s  The orbit equation describes a conic section.   init orientation (set to 0) s is the directrix.  The constant e is the eccentricity. sets the shape e < 1 ellipse e =1 parabola e >1 hyperbola

6. Apsidal Position  Elliptical orbits have stable apses. Kepler’s first lawKepler’s first law Minimum and maximum values of rMinimum and maximum values of r Other orbits only have a minimumOther 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

7. Angular Momentum  The change in area between orbit and focus is dA/dt Related to angular velocityRelated to angular velocity  The change is constant due to constant angular momentum.  This is Kepler’s 2 nd law r dr

8. Period and Ellipse  The area for the whole ellipse relates to the period. semimajor axis: a=(r 1 +r 2 )/2.  This is Kepler’s 3 rd law. Relation holds for all orbits Constant depends on   r s r1r1 r2r2

9. 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