108 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



coefficients of the remaining 3n k arbitrary variations being put 

 equal to zero with these k give the differential equations 



From the 3n equations 23) we may eliminate the It multipliers 

 Z 1; A 2 , . . . A* and obtain 3w k equations of motion , which is the 

 number of degrees of freedom of the system. 



The equations 23) are known as Lagrange's differential equations 

 in the first form. They can evidently be deduced from equations 16) 

 of Chapter III by d'Alembert's principle , replacing X r by 



d*x r 

 X r r dt z > etc. 



36. Generalized Coordinates. Lagrange's Equations. In 



many investigations in dynamics where constraints are introduced, 

 instead of denoting the positions of particles by rectangular coordinates 

 (not all of which are independent) it is advantageous to specify the 

 positions by means of certain parameters whose number is just equal 

 to the number of degrees of freedom of the system, so that they 

 are all independent variables. For instance if a particle is constrained 

 to move on the surface of a sphere of radius I, we may specify its 

 position by giving its longitude cp and colatitude #, as in 23. 

 These are two independent variables. 



The potential energy depending only on position will be expressed 

 in terms of cp and #. The kinetic energy will depend upon the 

 expression for the length of the arc of the path in terms of cp and #. 

 Now we have, if / be the radius of the sphere, 



Dividing by dt 2 and writing #' = -j-> (p' = -~> we have 

 24) T = y m P (#' 2 + sin 2 & y 1 2 ). 



The parameters # and cp are coordinates of the point, since when 

 they are known the position of the point is fully specified. Their 

 time -derivatives &', <p' being time -rates of change of coordinates may 

 be termed velocities, and when they together with & and (p are 

 known, the velocity of the particle may be calculated. The kinetic 



