151, 152] EQUATION OF ENERGY. 135 



whether the system is conservative or not, but it is not always an 

 integral of the equations of motion. In order that it may be so, 

 we must be able to calculate the work done without solving the 

 equations of motion, and, except for conservative systems, this is 

 generally impossible. 



152. Positional and Motional Forces. We may imagine 

 cases in which the forces acting on a particle have definite magni- 

 tudes, directions, and senses determined by the positions of the 

 particles of the system and of other .systems, or, in other words, 

 where the forces are one-valued functions of the geometrical 

 quantities defining the position. Such forces are said to be posi- 

 tional. 



We can imagine other cases in which the forces in any 

 positions depend on the velocities in those positions or on the 

 paths by which those positions are reached. Such forces are said 

 to be motional. 



The characteristic of positional forces is that the work done by 

 them as the system passes, by any set of paths, from a position A 

 to a position B is numerically equal to the work done in passing, 

 by the same set of paths, from the position B to the position A, 

 but has the opposite sign. 



Now suppose a system moves under the action of positional 

 forces only. Let it pass by one set of paths from the position A 

 to the position B, and by another set of paths from the position B 

 to the position A. If the system is conservative no work is done, 

 and the kinetic energy at the end of the cycle is the same as that 

 at the beginning. If the system is not conservative the paths can 

 be adjusted so that the work done in some cycle is positive : the 

 kinetic energy at the end of the cycle is greater than at the 

 beginning. By repeatedly performing the cycle the system can 

 continually acquire kinetic energy. Such a system would constitute 

 a " perpetual motion." 



In Nature no perpetual motion has ever been found, and it is 

 therefore in accordance with observed facts to subject the forces of 

 Rational Mechanics to the restriction that positional forces are 

 always conservative. 



Motional forces on the other hand are generally non-conserva- 

 tive. For example, the friction between two rough bodies is 



