Dynamics of a
60 58.
When
Particle
the central acceleration is
when
a single-valued function
a function of the always the same at the same distance), every o.pse-line divides the orbit into two equal and similar portions and thus there can only he tivo apse-distances.
of the distance
and
distance only
Let
{i.e.
ABC be
apses A, B, and
the acceleration is
is
a portion of the path having three consecutive
C and
let
be the
c
centre of force.
Let
V
be the velocity of the particle at B. Then, if the velocity
/
of the particle were reversed at B,
would describe the path BPA. For, as the acceleration depends on the distance from only, the velocity, by equations (1) and (3) of Art. 53, would depend only on the distance from and not on the
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/
/
/
^'^JXP\
/
/
it
V
^n^
./
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-^
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direction of the motion.
Again the
original particle starting from
from
B
and the reversed
B
with equal velocity F, must describe similar paths. For the equations (1) and (3) of Art. 53, which do not depend on the direction of motion, shew that the value of r and d at any time t for the first particle {i.e. OP' and Z BOP') are equal to the same quantities at the same time t particle, starting
for
the second particle
{i.e.
OP
and Z BOP).
BPA are exactly the same by being rotated about the line OB, would give the other. Hence, since A and G are the points where the radius vector is perpendicular to the tangent, we have OA = 00. Hence the curves BP'C and
either,
Similarly, if
OB
and
OD
D
equal,
were the next apse after
and
C,
we should have
so on.
Thus there are only two different apse-distances. The angle between any two consecutive apsidal distances
is
called the apsidal angle.
59.
When
power of the
the central acceleration varies as some integral /i^i'^ it is easily seen analytically that
distance, say
there are at most two apsidal distances.