The motion of a particle under the action of a conservative force that always points toward a fixed point (i.e. a conservative central force) is governed by two conservation principles. Conservation of energy is implied because the force is conservative; conservation of angular momentum holds because the torque of the force with respect to the fixed point is zero. The following formulas use polar coordinates; see [1].

Conservation of angular momentum gives:

, (1)

where

is the constant angular momentum of the particle. Conservation of energy gives:

, (2)

where

is the constant energy of the particle,

is the kinetic energy, and

is the potential energy, which is a function of only the distance

. The conservative force acting on the particle is

.

Equation (2) gives:

. (3)

From equation (3) we can obtain

, which, used in equation (1), gives:

. (4)

The constants

and

are evaluated from the initial position and velocity of the particle.

These equations can be used to calculate the motion of an artificial satellite of mass

in orbit around Earth. If the only force acting on the satellite is Earth's gravitational force, we have:

,

where

,

is the mass of Earth, and

.

It is well known that in this case the orbit of the satellite is a conic section. The orbit is elliptical, parabolic, or hyperbolic according to whether the energy

is negative, zero, or positive, respectively. Parabolic orbits are highly unlikely, because the energy

must be exactly zero. In the case of an elliptical or hyperbolic orbit, the equation of the orbit is:

, (5)

where

is the semimajor axis of the orbit and

is the eccentricity of the orbit. Then

(6)

.

In this Demonstration the satellite is originally moving in a circular orbit of radius

. In this case the gravitational force of Earth acts as a centripetal force and the speed of the satellite

is constant with

. A force

constant in magnitude and directed along the line that connects the centers of gravity of Earth and the satellite is activated when the satellite is at position

. This force can be chosen to point either toward or away from Earth.

acts on the satellite for a finite amount of time. During this time the position of the satellite is calculated from equations (3) and (4) using equation (6) for

. When the force

is turned off,

and

are evaluated at that time and the new orbit is calculated using equation (5).

[1] K. R. Symon,

*Mechanics*, 3rd. ed., Reading, MA: Addison–Wesley, 1971.