MATHEMATICAL INTRODUCTION 
19 
{uq* 
is 
ut. 
d 
»or. 
^cetW 
j u ^ 
Xu» 
IT or ^ i- 
s ^e stable c®| 
^ an integral nn^ 
must be an inte 
’ding to the ¡re 
by a series of <2 
y pair of consar 
The representar 
of the elliptic ci 
gions of equal 1 
absorbs energy, ii 
: to another. 
:om the ellipse! 
Her ellipse; ini 
rger ellipse, fe 
place in miiliflc 
ere employed ® 
jvnamics in ^ 
id velocity oil 
losition of a poifi 
dimensions, ft 
11. 2] 
present case is the simplest of all because it is assumed that 
the directions of vibration of the resonators are perpendicular 
to a fixed plane, and that the resonators themselves are fixed ; 
thus two co-ordinates are sufficient to determine the condition 
of a resonator and the phase-space is of two dimensions only. 
2. The Fundamental Equations of Classical Dynamics 
The case of the simple resonator just considered may be 
used to illustrate certain dynamical principles of fundamental 
importance. The kinetic energy is, as we have seen, 
t — 2 
Hi) 
It may also be written T = \P^-$y because^» — ; or it may 
CvC CtO 
£2 
be written T = —. 
2 m 
We notice that 
j* Tdt = J pdq 
2:9 
The latter integral is to be taken along the boundary curve. 
The expression on the left is an important quantity called the 
characteristic function, or the action of the system in passing 
from its position at a given instant to its position at a subsequent 
instant. That is, to find the action we integrate the kinetic 
energy with regard to the time between two assigned instants, 
or if we prefer it, we integrate pdq between the two conditions 
corresponding to those instants. 
The equations of classical dynamics may be summarized in 
the Principle of Least Action, according to which the actual 
motion of a dynamical system is such that the action is a minimum. 
Here it is assumed that the action is capable of continuous 
variation. 
A Conservative System is such that if, after the system has 
undergone any series of changes it is brought back in any manner 
to its original state, the whole work done by external agents on 
the system is equal to the whole work done by the system in 
overcoming external forces. Such a system is distinguished from 
a Dissipative System in which the energy available for work 
becomes gradually degraded to less available forms by frictional 
agencies. (Clerk Maxwell, Matter and Motion, Chapter V.) 
In a conservative system the potential energy, V, is the 
work done by the external forces in a displacement of the system 
from any assigned configuration to the standard configuration. 
3