CONTEMPORARY ADVANCES IN PHYSICS 657 



member, introduce a symbol V to designate the value of U at the 

 locality where at any moment the particle actually is, multiplied by 

 + e; this is the potential-energy-function of the particle, and the 

 right-hand member of (2) is its rate of change. Therefore: 



T + V = constant = E. . (3) 



The constant E is (by definition) the energy. As the behavior of the 

 particle depends upon the field, the ensemble of particle and field 

 should be considered as one entity, the system, of which kinetic energy 

 T and potential-energy-function V and total energy E are properties. 

 To bring out the next feature, I take the still more specific case of a 

 particle of charge e and mass m moving in the inverse-square central 

 field of a "nucleus," an immobile point-charge equal in magnitude and 

 opposite in sign to the electron-charge. Using Cartesian coordinates 

 with the origin at the nucleus, we have F = — e^j-^x^ + ^ + S"; 

 using polar coordinates,^ we have F = — e^lr. It is obvious that 

 polar coordinates permit a much simpler expression for F than do 

 Cartesians; on the other hand, they entail a distinctly more com- 

 plicated expression for T. The proper choice of coordinates is often 

 a vital question. For a few paragraphs I will carry along the reasoning 

 in both coordinate-systems. The underlying equation (3) becomes, 

 in the one and in the other: 



\m{x" + 2/" + ^") — e^l-^x^ + 3'" + s" = E, (4a) 



In these equations, we have the potential-energy-function expressed 

 as a function of the coordinates (x, y, z or r, d, cp) and the kinetic energy 

 expressed in terms of the coordinates and the velocities {x, y, z or r, d, 

 (f). It is desirable to express the kinetic energy in terms of the co- 

 ordinates and the momenta. We have already met the momenta in 

 Cartesian coordinates, the quantities mx, my, ?nz. It is obvious that 

 they are the derivatives of the expression for the kinetic energy with 

 respect to the velocities, always in Cartesian coordinates: 



p^ = dT/dx; Py = dT/dy; p, = dT/dz. (5) 



The momenta in any other coordinate-system are defined in the same 



'The equations of transformation are: x = r sin 9 cos 0, y = r sin sin <^, 

 2 = r cos d. 



