118 G. F. Becker — Maximum Dissipativity. 



The theorem of greatest action admits of a further important 

 extension. If we suppose a material point revovling about 

 an attracting center and if, during the revolution, the position 

 of the center of attraction is suddenly changed without alter- 

 ing its distance from the material point, this will immediately 

 begin a new orbit without change in its total energy and the 

 action on the new orbit will be the same as on the old one. 

 The path, however, will not in this case return upon itself. By 

 pursuing this train, of reasoning it becomes clear that the action 

 becomes an absolute maximum twice in every period for which 

 the integral curvature amounts to 360° and at the expiration of 

 which the original velocity is recovered. In certain symmetri- 

 cal cases of motion a starting point may be so selected that the 

 action becomes an absolute maximum once in every 180° of 

 integral curvature. The interposition of a straight path upon 

 which there is no change in kinetic energy, between two arcs, 

 affects the proposition only by lengthening the period between 

 successive maxima. 



The two most important cases of motion for the present dis- 

 cussion are that of a particle of an elastic solid which is in vi- 

 bration and that of a gas molecule in a confined space. In the 

 former the particle vibrates on some stable curvilinear path 

 about a position of equilibrium ; for it is well known that a 

 simple rectilinear vibration is an unstable motion which must 

 pass over into an orbit or circuit when disturbed never so little. 

 A gas molecule also pursues a stable path, for if it is rebound- 

 ing against parallel walls and traversing a zig-zag course in a 

 given plane, and if the angle is changed by an infinitesimal 

 quantity, the disturbed path cuts the undisturbed one at finite 

 intervals and without more than an infinitesimal digression.. 

 It is clear from what precedes that the action in both these 

 cases becomes the greatest possible twice for every complete 

 change of direction of 360° accompanied by the resumption of 

 the original velocity. 



The meaning of the statement that the action of a particle or 

 a system is a maximum or a minimum for a given period is 

 most readily appreciated geometrically. A natural motion may 

 be supposed to begin with the same energy as a guided motion 

 at any instant of time. Then if we represent the value of the 

 energy of the real motion by a full line and that of the guided 



motion by a dotted line, the abscissa representing the time, 

 as in the accompanying diagrams, it is clear that the statement 



