Analytical dynamics, being a synopsis of leading topics in the analytical theory of dynamics with numerous examples and selections from Newton's Principia and other sources, by Arthur S. Hathaway.

(10) F==D2r-r(DO)2, is the radial component of acceleration.
(11).G=r-1D(r2DO) is the transverse component of acceleration.
The triangle OPT is the areal velocity, and if U denote
the radius vector area OAP, v the speed Ds, and p the perpendicular ON, on PT, the magnitude of this area, or the areal speed is
expressed in the forms,
(12) DU —/2r'2DO 2pv.
Note. v is found in polar coordinates by the right triangle
PST.
(13) vdv=fds= —Fdr+Grd6 by projecting F and G upon PT,
to find f.
Polar coordinates are particularly useful when considering
plane motions in central fields. Since the whole acceleration is
radial, this gives at once G=O, i. e., D (r2DO) =0. Hence,
(14) r2DO=pv — H, a constant, (in a central field).
This expresses, of course, that the radius vector describes,equal areas in equal times, as heretofore shown [DU==-H, U=
JHt+c]. By this equaticn, which is dt=r2d0/H, we may eliminate dt from the differential equations.. Let a small cap D stand
for d/dO; Then D become everywhere (Hl/r2)D.
i. e., Dr=(H/r2) Mr=-HD(1/r), etc. The most important.equations thus holding for a central field, with r and 0 the only
variables involved are,
(10') F — (Hu)2(D2U+U), (u standing for 1/r)
(12') pv=-H, whence vdv=(H/p)d(H/p).
(13') vdv=fds=Fdr.
EXAMPLES IV
1. Find the laws of the central fields in which a particle will
describe the following curves, and determine the starting conditions that it may do so, in particular the curve of zero speed.
(a) The hyperbolic spiral, r==a.
(b) The equiangular spiral p=r sin c. [(12') and (13')]
(c) The logarithmic spiral log (r/a)=0/c. [The same as
(b)].
(d) The spiral in which r/a is the n'th power of sec (0/n).
(e) r=a sec nO, and r ---a sech nO.
(f) The spiral of Archimedes, r=a0, and the rose, r=-asin20
(f) The. cissoid r=asinOtan0; the lemniscate (r/a) 2
eos20.

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