CHAPTER II 



GENERAL DYNAMICAL PRINCIPLES 



17. In general the configuration of a dynamical system can be expressed 

 in terms of Lagrangian coordinates 



ft, ft, ft,...ft>... ........................... (1), 



while its motion at any instant can be specified in terms of the corresponding 

 velocities 



ft, ft, ft, ... O n .............................. (2). 



The potential energy W will be a function of the coordinates of position 

 only, say 



^=/(ft, ft, ... W ) ........................... (3), 



while the kinetic energy T will be a function both of the coordinates of 

 position and of the velocities, say 



T = F(0 l , & ... e n , e,, e 2 , ... e n ) .................. (4), 



and this function will be quadratic in the velocities ft, ft. fti- 

 The equations of motion will be the Lagrangian equations 

 d dT dT dW 



where F^ F 2 , ... F n are the "generalised forces" applied from outside. 



In a number of cosmogonical problems, we shall be concerned with the 

 motion of astronomical masses, and the equations determining this motion 

 will be equations (5) or some appropriate special form of these equations. 

 But in a much greater number of cosmogonical problems we shall be con- 

 cerned with astronomical masses which are either in a state of equilibrium 

 or whose motion is so slow that their kinetic energy is negligible. For such 

 configurations, putting T=0, equations (5) reduce to 



dW ?>w dW 



8ft =' W 2 = ' 8ft = ' etc ...................... (6) ' 



These may be regarded either as equations of equilibrium or as equations 

 determining the configuration of a very slowly changing mass. Regarded 

 as equations in ft, ft, ft, ..., the equations will have a number of solutions of 

 which a typical one may be taken to be 



ft = ^, ft = 2 ,etc ............................ (7). 



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