64, 65] HAMILTON'S PRINCIPLE. 125 



But H is a function only of the ps and qs, so that the left-hand 

 member is j- ; and since H is equal to T + W it represents the 



(MI 



total energy. Also P 8 dq s is the work done by the external impressed 

 force component P s in the displacement dq S) so that the right- 

 hand side is the time-rate at which the external forces do work on 

 the system, or the activity of the external forces. The equation 



(6) 



Wl/ ]_ \AJ\J 



is accordingly sometimes called the Equation of Activity, while if 

 there are no external forces, but only conservative ones, we have 

 the equation of Conservation of Energy, 



0, H=T+W= const. 



A case of frequent occurrence is that where there are non- 

 conservative forces proportional to the first powers of the velocities 

 q', so that any P s =tc s q 8 '. We may then form a function F 

 which is also a homogeneous quadratic function of the velocities 



i dq s ' ' 



and since in this case 



F represents one-half the time-rate of loss, or dissipation of energy. 

 F is called the Dissipation Function. It was introduced by Lord 

 Rayleigh *, and, like the other function used above, is of use in the 

 theory of electric currents. 



65. Transformation of Routh and Helmholtz. We 



shall in general find Lagrange's form of the equations of motion 

 more convenient than those of Hamilton. An intermediate form, 

 introduced by Routh f, and afterwards by Helmholtz J, is of great 

 importance. 



* Proceedings London Mathematical Society, June, 1873. 



t Bouth. Stability of a given State of Motion, p. 61. Rigid Dynamics, i. p. 318. 



Helmholtz. Ueber die physikalische Bedeutung des Princips der kleinsten 



Wirkung. Borchardt's Jour, fur Math. Bd. 100, 1886. Wissensch. Abh. in. p. 203. 



