Let us consider a body with mass
in a gravitational force field
exerted by the origin and directed always from the body towards the origin. Set the plane through the origin and the velocity
vector
of the body. Apparently, the body is forced to move constantly in this plane, i.e. there is a question of a planar motion. We want to derive the trajectory of the body.
Equip the plane of the motion with a polar coordinate system
and denote the position vector
of the body by
. Then the velocity vector is
![]() |
(1) |
Because the gravitational force on the body is exerted along the position vector, its moment is 0 and therefore the angular momentum
of the body is constant; thus its magnitude is a constant,
whence
![]() |
(2) |
This equation may be revised to
i.e.
where
is a constant. We introduce still an auxiliary angle
![]() |
(3) |
whence, by (2),
This means that
![]() |
(4) |
The result (4) shows that the trajectory of the body in the gravitational field
of one point-like sink is always a conic section whose focus contains the sink causing the field.
As for the type of the conic, the most interesting one is an ellipse. It occurs when
. This condition is easily seen to be equivalent with a negative total energy
of the body.
One can say that any planet revolves around the Sun along an ellipse having the Sun in one of its foci -- this is Kepler's first law.
As of this snapshot date, this entry was owned by pahio.