n. i] MATHEMATICAL INTRODUCTION 17
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represent the state of the system at any particular instant of
time.
The equation of motion may be written
d 2 q
dt 2
+ ~q = o
m
2 : 2
or
d 2 q
dt 2
+ co 2 q = o
2 13
where co 2 = —. For our present purpose the solution of the
m
equation may be written in the form
q — a cos {cot — e) 2:4
where the constants of integration a and e determine the
amplitude and the phase of the vibration, respectively. The
quantity co = 2nv, and is the number of complete vibrations
in 2n seconds. It is often convenient to have a special
name for this quantity, and following Albert Campbell, it may
be called the pulsatance of the motion. The French call it
“ pulsation.” The momentum p, may be written
, dq
P = m-~
ai
— maa)sm{(ot — e)
2:5
We may easily deduce the relation between p and q from the
two equations 2 : 4 and 2 : 5 in the form
p 2 1
(maco) 2 a 2
2 : 6
This result gives us a very convenient geometrical representation
of the state of the resonator. We take q and p to represent
rectangular Cartesian co-ordinates in a q -p plane (the so-called
“ state ’’-plane or “ phase ’’-plane). Each point of this plane
may be regarded as corresponding to some assigned momentary
condition of the resonator.
The equation between p and q may be written in the form
q^ p 2
a 2 b 2
2:7
(where b = maco) and represents an ellipse. For different
amplitudes we have a number of similar and similarly situated
ellipses.
Consider next the energy of the system. It will be partly
kinetic energy, T = > an d partly potential energy, V.
The latter is equal to the work done against the restoring force
