1918-19.] 
On Hamilton’s Principle. 
113 
IX. — On Hamilton’s Principle and the Modified Function in 
Analytical Dynamics. By G. H. Livens, M.A. Communicated 
by Professor E. T. Whittaker, F.R.S. 
(MS. received February 1, 1919. Read March 3, 1919.) 
1. The following note may be of some interest as helping in the 
elucidation of the rather complex analytical questions involved in the 
derivation of the modified Lagrangian function for a dynamical system. 
The results derived also have some bearing on the various questions 
involved in the transformation theory based on Hamilton’s equations of 
motion. The discussion is given for the simplest type of system, but it can 
be easily generalised to the less restricted cases covered by the results. 
We suppose that the configuration of the system is completely defined 
by ^-generalised co-ordinates q v q 2 , . . . q n , the velocities in which may be 
denoted by q v q 2 , . . . q n . If then L denotes the complete Lagrangian 
function for the system expressed directly in terms of these co-ordinates 
and velocities, then we know that the motion is completely determined by 
the conditions that the integral 
ly 
taken between fixed time-limits is stationary. 
This is the ordinary form of Hamilton’s principle, but it involves in any 
case a complete knowledge of the constitution of the system, because, before 
it can be applied, it is necessary to know the exact values of the kinetic and 
potential energies expressed properly in terms of the chosen co-ordinates 
and their velocities. As, however, we have frequently to deal with systems 
whose ultimate constitution is either partly or wholly unknown, it is 
necessary to establish, along the lines laid down by Bouth, a modified form 
of the principles allowing for this ignorance of the constitution of the 
systems with which we have to deal. The modification was effected by 
Routh himself for the Lagrangian equations, and by Larmor for the 
Hamiltonian principle, the result obtained in both cases being practically 
equivalent to the statement that the ordinary equations may be used if the 
energy in all ignored co-ordinates is treated as potential energy. 
2. In forming the variation of the integral 
we proceed by varying the co-ordinates arbitrarily and then calculating 
VOL. XXXIX. 
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