33 lines
1.8 KiB
Markdown
33 lines
1.8 KiB
Markdown
---
|
|
title: Motion in a central field
|
|
---
|
|
|
|
On reducing the two-body problem to one of the motion of a single body,
|
|
we arrive at the problem of determining the motion of a single particle in an
|
|
external field such that its potential energy depends only on the distance r
|
|
from some fixed point. This is called a central field. The force acting on the
|
|
particle is F = du(r)/dr = - (dU/dr)r/r; its magnitude is likewise a func-
|
|
tion of r only, and its direction is everywhere that of the radius vector.
|
|
As has already been shown in §9, the angular momentum of any system
|
|
relative to the centre of such a field is conserved. The angular momentum of a
|
|
single particle is M = rxp. Since M is perpendicular to r, the constancy of
|
|
M shows that, throughout the motion, the radius vector of the particle lies
|
|
in the plane perpendicular to M.
|
|
Thus the path of a particle in a central field lies in one plane. Using polar
|
|
co-ordinates r, in that plane, we can write the Lagrangian as
|
|
(14.1)
|
|
see (4.5). This function does not involve the co-ordinate explicitly. Any
|
|
generalised co-ordinate qi which does not appear explicitly in the Lagrangian
|
|
is said to be cyclic. For such a co-ordinate we have, by Lagrange's equation,
|
|
(d/dt) aL/dqi = aL/dqi = 0, so that the corresponding generalised momen-
|
|
tum Pi = aL/dqi is an integral of the motion. This leads to a considerable
|
|
simplification of the problem of integrating the equations of motion when
|
|
there are cyclic co-ordinates.
|
|
In the present case, the generalised momentum is the same as
|
|
the angular momentum M z = M (see (9.6)), and we return to the known law
|
|
of conservation of angular momentum:
|
|
M = mr2o = constant. =
|
|
(14.2)
|
|
This law has a simple geometrical interpretation in the plane motion of a single
|
|
particle in a central field. The expression 1/2 . rdo is the area of the sector
|
|
bounded by two neighbouring radius vectors and an element of the path
|