motion in central-force field
Let us consider a body with in a gravitational force field (http://planetmath.org/VectorField) 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.
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,
This equation may be revised to
is a constant. We introduce still an auxiliary angle such that
Differentiation of the first of these equations implies
whence, by (2),
This means that , where the constant is determined by the initial conditions. We can then solve from the first of the equations (3), obtaining
By the http://planetmath.org/node/11724parent entry, the result (4) shows that the trajectory of the body in the gravitational field (http://planetmath.org/VectorField) of one point-like sink is always a conic section whose focus the sink causing the field.
As for the of the conic, the most interesting one is an ellipse. It occurs, by the
http://planetmath.org/node/11724parent entry, 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.
- 1 Я. Б. Зельдович & А. Д. Мышкис: Элементы прикладной математики. Издательство ‘‘Наука’’. Москва (1976).
|Title||motion in central-force field|
|Date of creation||2013-03-22 18:52:41|
|Last modified on||2013-03-22 18:52:41|
|Last modified by||pahio (2872)|
|Synonym||Kepler’s first law|