68 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



U t = Ux8 . . . and 



where the affix denotes the value of the coordinate at the time t . 

 The equation of energy then is 



24) T ti -T,. = U tl -U ia . 



The function U is called the force -function, and its negative 

 W= U is called the Potential Energy of the system. Inserting 

 W in 24) we have 



25) T tl + W tl = T,, + W ta . 



The sum of the kinetic and potential energies of a system possessing 

 a force -function depending only on the coordinates is the same for 

 all instants of time. This is the Principle of Conservation of Energy. 



Systems for which the conditions 23) are satisfied are accord- 

 ingly called conservative systems. 



The potential energy, being defined by its derivatives, contains 

 an arbitrary constant. The functions T and W have one essential 



d x r dy r dz r 

 difference, namely, T contains only the velocities, ^r\ ~^r> ~j^' ' ' > 



while W does not contain the velocities, but only the coordinates. 

 One important consequence of the equation of Conservation of Energy 

 is that if at any time in the course of a motion, all the points of 

 the system pass simultaneously through positions that they have 

 occupied at a previous instant, the kinetic energy will be the same 

 as at that instant, irrespective of the directions in which the particles 

 may be moving, for T -f W is constant during the whole motion, 

 and W depends only on the coordinates, consequently when all the 

 coordinates resume their former values, the kinetic energy does the 

 same. 



In other words, the work done on the system has been stored up 

 or conserved, to the amount TF, and may be got out again by bringing 

 the system back to its former configuration. 



For instance, a particle thrown vertically upward, or a pendulum 

 swinging, have the same velocity when passing a given point whether 

 rising or falling. 



As an example, consider a particle acted upon by gravity. We have 



26) X = 0, F=0, Z=-mg, 



so that U= mgz -f const. 

 The equation of energy is 



27) m (v* - t> ) = 



or the velocity depends only on the vertical height fallen. Accord- 

 ingly a particle, descending from a point A to another I?, constrained 



