576 Prof. R, C, Tolman on the Relativity Theory ; 

 The Expression for Kinetic Energy. 



Before proceeding to a derivation of the principle of least 

 action, we shall find it desirable to consider the expression 

 for the kinetic energy of a system of particles. If F is the 

 force acting on a particle, we have for its increase in kinetic 

 energy 



dK=F . <7r = F . ~dt = Y .qdt. 

 dt * 



Substituting for F the equation of definition (1), we have 



7T- d*L 7, dm 7 



d 1\. — m -J- .qdt + — q . q dt, 



= mq . dq + (q . q)dm, 

 = mqdq-j-q 2 dm. 

 Introducing the expression for the mass of a moving body 



m= ° .= - , equation (2), integrating and evaluating the 



^/l — q 2 jc 2 



constant of integration with the help of the relation # = 0, 



K — 0, m = m , we obtain the familiar relation 



~K — mc 2 — m c 2 (5) 



For a system of particles the kinetic energy will be the 

 sum of the kinetic energies of the individual particles 



~K = t(mc 2 -m c 2 ) ■ . (6) 



The Function T. 



Our immediate purpose in considering the kinetic energy 

 of a system of particles is to point out that it is not a 

 suitable quantity for use in defining the Lagrangian 

 function. 



In the classical mechanics the Lagrangian function is 

 defined as the difference between the kinetic and potential 

 energies of the system. The value of this definition, how- 

 ever, arises from the fact that in the classical mechanics the 

 derivative with respect to the velocity of the kinetic energy 

 of a particle is equal to its momentum. This is not so in 

 non -Newtonian mechanics, since we have from equation (5) 



dK d , 2 d / ?n \ 



dq dq y ^V 1 "^ ' 



l-f/c 2 > 



which only in the case of low velocities becomes equal to mq 

 as in Newtonian mechanics. 



