115 
1918-19.] On Hamilton’s Principle. 
system is determined by the usual equations. We can therefore now alter 
these co-ordinates by substituting for them others whose values of course 
depend in some way on those replaced, but which will in themselves still 
be sufficient to determine the configuration and motion of the system : this 
is the intrinsic advantage of the variational principle. Let us, for example, 
replace the first m velocities by their corresponding momenta and regard 
these latter, with the remaining velocities and all the geometrical co- 
ordinates, as the independent co-ordinates of the system. The substitution 
is effected by solving the m equations 
9L -i 0 
— = r=l, 2, . . . m 
°q r 
for the velocities q 1} q 2 , . . . q m , thereby determining them as functions 
explicitly of the other velocities, all the co-ordinates and the momenta 
Pi, P 2 ’ • • • Pm, and then substituting these values in L and the first m 
n 
terms of the series y 
7—1 
The variation of the integral, modified in the manner specified, can now 
be obtained in the usual way, but it leads to a different set of equations. 
Firstly, as regards the co-ordinates representing the momenta, we have for 
each of them two equations of the type 
_ 0 _ 
dp, 
>7'=1 
dqs = Q 
dt 
L - y \p r q r 
/ l -l 
The summation in both cases need only be taken as far as the m th term, 
for all the terms beyond are explicitly independent of the independent 
co-ordinates and momenta. 
For the remaining co-ordinates the equations are of the type 
in which the sum 2 must now be restricted to the first m terms as given. 
o 
The four equations thus derived are equivalent to those usually given 
for this type of system, and are identical with those given by Routh. The 
function 
L ,= L- 
is usually called Routh s modified function, and so far as the motion in the 
