658 BELL SYSTEM TECHNICAL JOURNAL 



way; first the kinetic energy is expressed as a function of the velocities, 

 then dififerentiated with respect to these. In polar coordinates 



pr = dT/df = mf; pe = dT/dd = mr^d; 

 p^ = dT/dip = mr- sin^ d- ip. 



Expressing in the equations (4a) and (4b) the kinetic energy in terms 

 of the coordinates and momenta, we have 



2^ (/>.' + Pv' + P.') - e'Hx' + / + z' = E, (7a) 



Whenever in any problem the kinetic energy and the potential energy 

 of the system are given as functions of coordinates and momenta, the 

 problem is prepared for treatment by the methods of classical mechanics. 



To make the next step, we consider the function L = T — V, the 

 difference between the kinetic energy and the potential-energy-function 

 of the particle, a function of the particle as it travels along its path 

 in the force-field: 



L=T-V=2T-E (8) 



and the time-integral of this function 



W = fLdt = flTdt - EL (9) 



Into the expression for W, insert explicitly the expression for kinetic 

 energy in Cartesian or in polar (or in any other) coordinates : 



W = mf{x^ + 2/- + z'-)dt - Et = mf{xdx -f ydy + zdz) - Et, (10a) 



(10b) 



W = mfif^ + r'^^ _^ f. sin2 d-'ip^)dt - Et 



= mj'{fdr + r^ddd -\- r" sin^ 6- lpd(p) — Et. 



From all of this it follows that 



p^ = dWIdx, py = dWidy, p^ = dW/dz, (Ua) 



pr = dW/dr, pe = dW/dO, p^ = dW/d^, (lib) 



and in general, the momenta belonging to any coordinate-system are the 

 derivatives of the function W with respect to the coordinates. 



Into the fundamental equation (7a) substitute these expressions 

 for the momenta, and obtain : 



2m 



dWY , /dwy- , /dw 



dx / ~^ \ dv I ~^ \ dz 



+ V{x, y,z) =E (12) 



