1. Equation of Motion for the Two-Body Problem
To get the equations of motion, write the Lagrangian for a two-body system:
is the reduced mass of the system,
is the distance between the two bodies,
is the angular velocity and
is the potential energy of the system.
The equations of motion are obtained by differentiating the Lagrangian with respect to the two generalized coordinates
where the constant
is the angular momentum.
equation (radial equation) is given by:
2. Solution to the Radial Equation
The solution to the radial equation gives
as a function of time
. In order to get the equation of the orbit in terms of the polar equation (i.e.
as a function of
), recast it in a slightly different form using the variable substitution
. Without going into the details of the transformation, the result is:
is a constant, equal to
is the force constant, equal to
for the gravitational potential
is the eccentricity of the curve.
3. Eccentricity and Energy
Eccentricity and energy are related by
Also, for bound orbits, the minimum and maximum radial distance are
Snapshot 1: circular orbit having eccentricity
and minimum energy; bound system
Snapshot 2: elliptical orbit having negative energy; bound system (Earth orbiting the Sun)
Snapshot 3: parabolic orbit having
and energy = 0; the system is just free
Snapshot 4: hyperbolic orbit having
and positive energy; system is unbound
 J. R. Taylor, Classical Mechanics
, Sausalito, CA: University Science Books, 2005.
 D. Morin, Introduction to Classical Mechanics: With Problems and Solutions
, New York: Cambridge University Press, 2008.