45] NORMAL COORDINATES. 163 



We further notice that, since the different periods depending 

 upon v are derived from the roots of an algebraic equation, they 

 are not in general commensurable, so that the motion is not as a 

 whole generally periodic. For instance in the case of Lissajous's 

 curves described in 19, unless the two periods are commensurable 

 the curve will never close. In the case, however, of the spherical 

 pendulum performing small oscillations the periods of the two co- 

 ordinates were equal, so that the path became a closed curve, an 

 ellipse. 



There is one set of coordinates of peculiar importance. For 

 simplicity let us suppose there is no dissipation, FQ. Let us 

 make a linear transformation with constant coefficients, putting 



& = yn 9>i + fia 9>a H f- nn<pn, 



& = y*i 9i + r 22 9 2 + + r*n<p n , 



75) 



Differentiating by t the #"s are obtained from the qp f 's by the same 

 substitution. It is shown by algebra that we may determine the 

 coefficients y in such a way that the two quadratic functions T 

 and W are simultaneously transformed to sums of squares, the product 

 terms being absent. Supposing this done we have 



76) = 



Then we have 

 and our differential equations 63) are 



nn\ *V i^ 



and the integrals, 



78) <p r = A r c( 



In other words, each coordinate <p appears in its own differential 

 equation entirely separate from the rest, and performs a harmonic 

 vibration independent of the * others with its own period. The qp's 

 are called normal or principal coordinates. The #'s being linear func- 

 tions of the 9?'s describe compound harmonic oscillations. A^yibration 

 in which all the normal coordinates but one vanish is called a normal 

 vibration. The effect of this on the g's is, if every <p equals zero 



except (p s , to make 



11* 



