20 THE QUANTUM [n. 3 
The function L = T — V is called the Lagrangian function, or 
the kinetic potential. 
The integral S = J L dt = j (T — V) dt was called by Hamilton 
the principal function. If the system move by a varied course, 
but in the same time, from the one given position to the other, 
then S is a minimum. 
The total energy considered as a function of q and p is the 
Hamiltonian function H, so that we have 
H = T + V 2: io 
where V is the potential energy, \iiq i in the present case. We 
may then write 
0H_0V 0H = 0T 
dq dq ’ dp dp 
The Hamiltonian or canonical form of the equations of motion 
is given by 
dq = 0H dp = _0H 
dt dp ’ dt dq 
(See Whittaker’s Analytical Dynamics, Chapter X, p. 258.) 
These symmetrical equations are very general, and the results 
given above in the simple case of a resonator are typical of 
those which obtain in any dynamical system of a conservative 
type. We may notice that the impulse co-ordinate p is given 
by means of the equation 
2:13 
where q is an abbreviation for This is taken as a general 
dt 
definition of the impulse co-ordinate for any system, it being 
assumed that the kinetic energy T is expressed as a function of 
P and q. 
The Hamiltonian method will be required in the generaliza 
tions of the quantum theory to be discussed later. 
3. The Quantum of Action 
In the earlier presentation of Planck’s theory the quantum 
appears as an element of energy, and might almost be termed 
an atom of energy. But such phraseology may suggest an 
incorrect analogy, for the properties of an atom are generally 
regarded as invariable, whilst the quantum of energy is propor 
tional to the frequency—a fact which is merely assumed and