37] . EFFECTIVE FORCE COMPONENTS. 119 



If we had begun with d'Alembert's principle we should evidently 

 have gone through precisely the same process that we have here 

 followed, and assuming all the displacements dx, 8y, dz to he virtual, 

 all the dg's would have been independent, so that from the trans- 

 formed equation 56) would have followed the individual equations 47). 

 This was in fact the mode of deduction followed by Lagrange. 



We have a noteworthy difference between generalized and 

 rectangular coordinates, in that the effective force -component is not 



dp 

 generally equal to the time -derivative of the momentum -~> but 



dt 

 dT 

 contains in addition the term This we may accordingly call 



the non-momental part of the effective force. Thus in general, even 

 though the momentum p s is unchanging, a force P s must be impressed 



dT 

 in order to balance the kinetic reaction -~ - As an example, let us 



take the case of polar coordinates in a plane. We then have for a 

 single particle, for the coordinates q lf q z the distance r from the 

 origin, and the angle q> subtended by the radius vector and a fixed 

 radius. The kinetic energy is 



from which we have the momenta, 



dT 



= 



dT , 



Thus if the momentum p r is constant, which is the case when the 

 radial velocity r' is constant, we still have to impress a radial com- 

 ponent of force 



The kinetic reaction P r = mrcp' 2 is called the centrifugal force, a 

 name to which it is as much entitled as any sort of reaction is to 

 the term force. 



By analogy we might in general call the non-momental parts of 

 the reversed effective forces or forces of inertia the centrifugal forces 

 of the system. These non-momental parts may be absent for some 

 coordinates. For instance in the present example (p does not appear 

 in the kinetic energy, but only its velocity (p'. We have then 



\ /j-j 



- = 0. so that force need be impressed to change <p only to change 



o (p 



the momentum p v . Accordingly if no such force is impressed, the 

 momentum p is conserved. Thus in the case of a central force, 

 the momentum p^ = mr 2 cp' is constant. But this is the theorem of 

 areas, or of conservation of moment of momentum. In fact we see 



