© 1952 by Oxford University Press
HAMILTON'S PRINCIPLE IN RELATION TO NON-HOLONOMIC MECHANICAL SYSTEMS
( Dept. of Math., The University of Adelaide S. Australia )
The behaviour of Hamilton's principle in non-holonomic systems has been investigated in two ways. H. Hertz (1), adopting generally accepted methods of the calculus of variations, concluded that though there may be natural paths which satisfy the principle, there is an infinity of paths generated by the principle which are not natural. Hölder (2), on the other hand, using a method which is incompatible with the generally accepted formalism of the calculus of variations, was led to the conclusion that Hamilton's principle is satisfied in non-holonomic systems.
Hertz was a little unfortunate in his choice of an example; in the only instance of a non-holonomic system to which he refers, all the natural paths do in fact satisfy Hamilton's principle. This may have contributed to the frequent acceptance of Hölder's incorrect theory, even today. In view of the misunderstandings which exist, it seems desirable to restate Hertz's investigation, and to demonstrate that there are natural paths in non-holonomic systems which do not satisfy Hamilton's principle; that is to say, natural paths for which 
L dt has not a stationary value.