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Central Force Field

The subject of this lecture is a theory behind practical problems of space travel around some gravitating mass or a movement of an electron around an electrically charged nucleus, or other types of movement in a central force field, where the force at any point A is always directed towards or away from some central point O and its magnitude is the same at all points that lie on the same distance from the center O.

Consider an inertial reference frame in our three-dimensional space with origin at point O and a force field F defined in all points with the following conditions satisfied:
(a) for any point A in our space the vector of force F(A) at this point is always colinear with position vector r=OA;
(b) for any two points A and B lying on the same distance from the center O vectors F(A) and F(B) have the same magnitude and both directed either towards or away from the center O.

Obviously, from condition (b) follows that all force vectors at all points lying on the same distance around center O have the same magnitude and all of them are directed the same way relatively to center O - either towards it or away from it.

Defined in such way, the force field is called central force field.

The vector of central force at any point A is defined by position vector r=OA and can be expressed as a scalar f(r) that depends only on the distance r=|r| of a point A from the origin of coordinates (positive or negative to differentiate between the directions towards or away from center O) multiplied by the vector of position r.
F(A) = F(r) = f(r)·r.