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.

s-c=-acos(nt+b), or asin(nt+b'), or Acos(nt)+Bsin(nt)
[equivalent forms.]
D2s+n2 (s-c) O.
The center is s=c. The radius or amplitude is a==/ ( A2+B2)
The angular speed is n, the period, 27r/n. The phase angle is
b=b' —7r=tan- (B/A).
(e) Hyperbolic Motion. The "harmonic" functions above
replaced by "hyperbolic" functions. This includes geometric motion, and is in general the compound (sum) of two geometric motions of reciprocal ratios, as in the second form.
s —c=Acosh(nt) +Bsinh(nt), or aexp(nt)+bexp(-nt) &c.
D2S=n2 (s-C).
(f) Harmonic Motion of Geometrically Decreasing Amplitude.
s-c-a exp (1 — t).cos(nt+b).
D2s+kDs+ (n2+1/4k2) (s-c)=O.
Note. To prove that the integral equation includes every
solution of the differential equation, let s=x.exp( —lkt) +c,
whence x is any solution of D2x-+n2x-O.
FIELDS AND MEDIUMS
In nature, a moving point is associated with matter, and because of the inertia property of matter, cannot be conceived as in
itself changing its state of rest or of uniform motion in a straight
line. In this view of the motion of a point, it will be called a
particle. The motion of a particle is simply the geometrical motion of a point, and its velocity and acceleration are geometrical
quantities only, but we regard the acceleration, or change of
velocity per-unit time, as being produced by some cause outside the
particle.
Thus, a particle near the surface of the earth is given an
acceleration g, downward, wherever it is placed, whose effect is only
anulled by some other cause which gives the particle the opposite
acceleration, so that the resultant acceleration of the particle is
zero.
A space in which a particle is given an acceleration which
varies only as its position varies, will be called a field. In a field
the acceleration, in direction and magnitude, is the same at the
same point at all times and becomes the acceleration of the particle
when it arrives at that point.
In a constant field, the acceleration is the same in magnitude
and direction at all points of the field. It will be suggestive to call
its direction downward, and to illustrate the magnitude, g, by the
numerical value 32, or 981, in the foot second, or centimeter
second system.

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