THE CANONICAL EQUATIONS 355 



As a last example, it may be noticed that a ship will have a free period 

 of rolling about its vertical position. If it is in a rolling sea, the waves 

 which meet it will apply external forces which may be regarded as approxi- 

 mately periodic. If the period of the waves happens to coincide with that 

 of the ship, the ship will roll heavily even though the waves may be com- 

 paratively small. This danger can be remedied by altering the course of 

 the ship and so causing it to meet the waves at a different interval. Another 

 way is to give the ship a list by spreading canvas, and so causing it to 

 oscillate about a different position of equilibrium, about which the periods 

 of free vibrations are different. 



THE CANONICAL EQUATIONS 



287. If lt 2 , are Lagrangian coordinates of any system, the 

 kinetic energy T is a quadratic function of 6 lt 6 2 , 3 , . Let the 

 corresponding momenta be u lt u 2 , , u n , these being given by 



u^ = ^?> etc. (200) 



30 i 



Now let us introduce a function T 1 , defined by 

 T' = U& 4- uj z + ---- T, 



so that T 1 is a function of u lt u 2 , - , 1} 2) , 6 V 2 , ; and 

 u lt u z , - are of course functions of 6 lt 2 , , lt Z , 



On differentiation of T f we have 



dT' = u + ud0 + 



and this, by equation (200), reduces to 



dT' = e.dn, + i du 2 +...- S dd l --d0 2 ---- . (201) 



