160-164] GENERAL PRINCIPLES. 147 



162. Conservative Field. As in Article 80 let m be the 

 mass of the particle, x, y, z its coordinates at time t, X, Y, Z the 

 resolved parts of the forces acting upon it ; then, if the field of 

 force is conservative, Xdx + Ydy + Zdz is the complete differential 

 of a function W, so that 



The equations of motion of the particle are 

 dW dW dW 



~, =%. = W &amp;gt; 

 and they possess the integral (Article 151) 



ira (tf 2 4- f + 2 2 ) = W+ const. 



If v is the magnitude of the velocity of the particle at time t, 

 this equation can be written 



mv z = W + const. 



This equation applies to all the particular cases discussed in 

 Chapter IV. 



163. Conservation of Linear Momentum. Suppose the 

 axis x is a direction in which there is no resolved part of the 

 force acting on the particle. The equation of motion by resolution 

 parallel to the axis x is mx = 0, and it follows that x is constant 

 throughout the motion, or the resolved velocity in any direction in 

 which the resolved part of the force vanishes is constant. This is 

 a special case of the general principle considered in Article 111. 



164. Conservation of Angular Momentum. Suppose 

 the axis z is a line about which the forces acting on the particle 

 have no moment. Then we have 



xY-yX = Q. 

 Hence multiplying the equations 



mx = X, my = Y 

 respectively by y and x, and subtracting, we have 



m (xy yx) = 0, 



and this equation possesses the integral 



xy yx = const. 



This is a special case of the general principle considered in 

 Article 112. 



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