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title: Motion in a central field
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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