119 
1918-19.] 
On Hamilton’s Principle. 
where (O r fi 2 , . . . W) are functions of the variables (q v q s 
Qi, Q 2 , . . .Q n , t). From these equations we have immediately 
,Pr 
dq r __ 
dt 
v p d Qr , 8W , y , <» 1 . aw 
^ r .7/ + ^ S a/ ,// 
0£ 
3Q g _ 
0£ dt 
so that we have 
IT, V « fyrm ir I V P riW 
where 
K = H - ! 
dt " " 0 £ 
and the equations of motion are thus derived by varying the integral 
IX- ^ 
drv 
The last term in this integral is irrelevant to the problem, as it integrates 
out to terms at the time-limits, and therefore the equations of motion are 
0K c?Q r 
0P r dt ’ 
0K 
0Qr 
dY, 
dt 
Thus, if the transformation of co-ordinates is a contact transformation, the 
Hamiltonian form of the equations of motion for any system is conserved. 
These are a few of the more important results of the transformation 
theory in dynamics : the remainder can be derived equally readily if the 
principles underlying the above discussion are kept in view ; but it does 
not seem necessary to develop the discussion any further in the present 
place. It may, however, serve some purpose to conclude by emphasising the 
fact that the two types of equation, the Lagrangian and the Hamiltonian, 
can both be derived from the same integral by the variational method, using 
as the modified Lagrangian function of the system the expression 
L ~ 2A Ur 
and treating as independent variables the co-ordinates q r and either the 
velocities q r or their momenta p r , or any suitable functions of these. 
(. Issued separately July 3, 1919.) 
