General Dynamical Principles. 573- 



relative motion. In the present article we shall derive for 

 a system of particles the important principles of the con- 

 servation of momentum and of the conservation of moment 

 of momentum. We shall further derive a principle corre- 

 sponding to that of least action which applies in ordinary 

 mechanics, and show that it leads to generalized equations 

 of motion in the Lagrangian and Hamiltonian (canonical) 

 forms. We shall find, however, that the Lagrangian 

 function is not to be taken equal to the difference between, 

 the kinetic and potential energies of the system as in the 

 classical mechanics, and the generalized momenta used in 

 the modified Hamiltonian equations are not to be defined as 

 the partial differential of the kinetic energy with respect to 

 the generalized velocities as in the older mechanics. 



It is hoped that this presentation of the principle of least 

 action and its consequences, in the modified form necessitated 

 by the principle of relativity, will assist in that careful 

 scrutiny which now has to be given to all the results of the 

 older classical mechanics. In ihe present article we shall 

 illustrate the use of the generalized equations by the de- 

 rivation of a principle corresponding to that of vis cira in 

 ordinary mechanics. In a following article the ideas here 

 developed will be used in a consideration of the important 

 problem of the equi-partition of energy. 



This derivation of the principle of least action from the 

 laws of motion is also of particular interest, since writers on 

 relativity * have made considerable use of the principle in 

 the fields of pure dynamics, electromagnetics, and thermo- 

 dynamics, without presenting any derivation in the only 

 field — that of pure dynamics — where such can be obtained/ 



The Conservation of Momentum. 



In accordance with the laws of motion, the force acting on 

 a particle is defined by the equation f 



_ <l , . da dm 



T=- dtim) = m - H+ < ldr . . . . (1) 



where q is the velocity of the particle and its mass m is 

 given by the equation J 



m=— J^ (2) 



* Planck, Ann. d. Physik, xxvi. p. 1 (1908) : Herglotz, Ann. d. Physik, 

 xxxvi. p. 493 (1911) ; de WisniewsM, Ann. d. Physik, xl. p. 668 (1913). 

 See also Lane, Das Relativittitsprivcip, Braunschweig, 1913. 



t Throughout the article we shall indicate vectors by heavy Clarendon 

 type, the inner product by ', and the outer product by X.* The mag- 

 nitude of the vector a will be indicated by | a j or a. 



\ See Tolniau, Phil. Mag. xxiii. p. 375 (1912). 



