2ii.] LAGRANGE'S EQUATIONS. H3 



V. Lagrange s Form of the Equations of Motion. 



210. It has been shown in Arts. 180, 181, how the equations 

 of motion on a fixed curve can be made to depend on a single 

 variable q, and in Art. 191 how the motion on a fixed surface 

 can be expressed by means of two variables q v q^. By apply- 

 ing this idea, and by introducing the kinetic energy T and its 

 derivatives, the equations of motion of a particle with or without 

 conditions can be put into a remarkably compact form, which 

 was first devised by Lagrange for the general equations of 

 motion of a system of n particles (comp. Arts. 385-394). We 

 proceed to establish these equations, first for the case of motion 

 on a variable curve, then for motion on a variable surface, and 

 finally for a free particle. 



211. Particle Subject to Two Conditions. As shown in Art. 

 194 (comp. Art. 192), the equations of motion in Cartesian 

 co-ordinates can be written in the form 



mX = X+ \<f) x 



my = F+x<k + ^,, (i) 



mz = Z 4- \<f) z + A^, 

 if the equations of condition are 



<(*, y, 2, /)=o, -f(>:, y, z, t}=Q. (2) 



The single variable q, that determines the position of the 

 particle on the curve, is called the Lagrangian, or generalized, 

 co-ordinate of the particle. The Cartesian co-ordinates, x, y, z, 

 are functions of the Lagrangian co-ordinate q, and of the time 

 /, say 



*=/i('. <?)> y=A(*> ?)> *=/*(*> ?) (3) 



To introduce q in the place of x, y, z, we shall need the 

 derivatives x> j/, z. The first of the equations (3) gives 



PART III 8 



