constant): The force affects the path, r(t), only indirectly, mediated via the velocity, v
(t) (see also (Suisky 2009, Sect. 4.1)). For this, let us examine the hodograph first.
4.3 The Hodograph
The fact, that the hodograph is plane (despite of motions through the origin), can
most easily be derived from the following consideration.
For each initial velocity, v(0), there is a plane containing this vector and the origin
being the position of the force center, see Fig. 4.2. By the symmetry of the force
field, there is nowhere on this plane a net force perpendicular to it, see Fig. 4.3.
Hence, the velocity vector, v(t), will stay in this plane for all times.
The most general central force reads
F ¼ f r
ð Þ
r
r
Fig. 4.2 Plane defined by
the central force (origin),
initial position, r(0), and
initial velocity, v(0)
Fig. 4.3 Vanishing net
force perpendicular to the
plane of initial conditions
4 “Equat Causa Effectum”
41
(t) (see also (Suisky 2009, Sect. 4.1)). For this, let us examine the hodograph first.
4.3 The Hodograph
The fact, that the hodograph is plane (despite of motions through the origin), can
most easily be derived from the following consideration.
For each initial velocity, v(0), there is a plane containing this vector and the origin
being the position of the force center, see Fig. 4.2. By the symmetry of the force
field, there is nowhere on this plane a net force perpendicular to it, see Fig. 4.3.
Hence, the velocity vector, v(t), will stay in this plane for all times.
The most general central force reads
F ¼ f r
ð Þ
r
r
Fig. 4.2 Plane defined by
the central force (origin),
initial position, r(0), and
initial velocity, v(0)
Fig. 4.3 Vanishing net
force perpendicular to the
plane of initial conditions
4 “Equat Causa Effectum”
41