MOTION ON A SURFACE FOR ANY POSITIONAL FIELD OF 



FORCE. 



By Joseph Lipka. 



Received Nov. 8, 1920. Presented Nov. 8, 1920. 



§ 1. Introduction. 



The first part of the paper presents a study of the geometric proper- 

 ties of the system of trajectories generated by the motion of a particle 

 on any constraining surface (spread of two dimensions) under any 

 positional forces. The complete characteristic properties are derived. ^ 

 Starting at any point on the surface in a given direction and with a 

 given speed, a unique trajectory is generated, and the complete set of 

 trajectories forms a triply infinite system of curves corresponding 

 uniquely to a given field of force. 



Through a given point and in a given direction, there pass <»^ 

 trajectories. We associate with these trajectories the ooi curves 

 obtained by orthogonal projection into the tangent plane to the sur- 

 face at 0. The first two properties derived deal with the bicircular 

 quartic which is the locus of the foci of the osculating parabolas of the 

 associated system. 



A second set of geometric properties is derived by considering the 

 CO 2 trajectories through a point on the surface, and the directions 

 through in which the trajectories hyperosculate (have 4-pbint 

 contact with) their corresponding geodesic circles of curvature. It is 

 found that the hyperosculation property holds for only one trajectory 

 in each direction, and that the corresponding locus of the centers of 

 geodesic curvature is a conic. 



A final set of properties is then derived showing the relations exist- 

 ing at a point between the geodesic curvatures of the trajectories and 

 the lines of force, and any isothermal net of curves on the surface. It 

 is shown that the entire five properties are characteristic of the system 



1 For a study of the corresponding problem in a plane and in ordinary 3- 

 space, see Edward Kasner, "The trajectories of dynamics," Trans. Am. Math. 

 Soc, 7 (1906), pp. 401-424; also "Dynamical Trajectories: the motion of a par- 

 ticle in an arbitrary field of force," Trans. Am. Math. Soc, 8 (1907), pp. 135- 

 158. The results of these 2 papers are summarized in Professor Kasner's 

 Princeton Colloqium Lectures on the differential geometric aspects of dy- 

 namics, chapt. I. 



