1. Equation of Motion for the Two-Body Problem

To get the equations of motion, write the Lagrangian for a two-body system:

,

where

μ 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

and

. The

equation gives:

,

where the constant

is the angular momentum.

The

equation (radial equation) is given by:

,

(

equation).

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:

,

where

is a constant, equal to

;

is the force constant, equal to

for the gravitational potential

; and

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

and

.

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

[1] J. R. Taylor,

*Classical Mechanics*, Sausalito, CA: University Science Books, 2005.

[2] D. Morin,

*Introduction to Classical Mechanics: With Problems and Solutions*, New York: Cambridge University Press, 2008.