Motion in a central potential

Consider a particle moving in a central potential.  In a central potential the energy E and the angular momentum M are conserved.  The motion is in a plane.  Choose this plane to be the x-y plane.  Then

.

.

Kepler’s second law:

.

The area swept out per unit time is constant for all central potentials.

Equations of motion involving r only:

yields

.

Lagrange’s equations yield

.

Equations for the orbit:

.

From

we obtain

,

,

.

From

we obtain

,

or

.

The Kepler problem

Let

.

The equation for the orbit yields

.

.

This is the equation of a conic section.  Here e is the eccentricity.

The ellipse and the circle are closed orbits.  For closed orbits we have

semi-major axis:

semi-minor axis:

From

we find the period

.

This is Kepler’s third law.