SOME GEOMETRIC INVESTIGATIONS ON THE GENERAL 

 PROBLEM OF DYNAMICS. 



By Joseph Lipka. 



Received January 20, 1920. Presented February 11, 1920. 



§L Physical Interpertations. 1) Kinetics. Consider the mo- 

 tion of a system in a conservative field of force from one position 

 Po to another Pi, with the sum of its kinetic and potential energies 

 equal to a given constant; the system will move naturally or unguided 

 along the path for which the action is least. This is known as Jacobi's 

 principle of least action. Action is defined as twice the time-integral 

 of the kinetic energy T, i. e. 



J'' (Pi) 

 (Po) 



Action = / 2Tdt. 



If U is the potential energy, W is the work function (negative poten- 

 tial), h is the given constant of energy, v is the velocity of the system, 

 then, for a unit mass, we have the energy equation 



T + U = h, or I «2 _ jy = /,^ or v'' = 2(W -f- A), 



and 



I 2Tdt= I v^ dt = I vvdt^J v ds = J V2 (TF + h) ds. 



The totality of trajectories for all initial conditions are thus deter- 

 mined according to the principle of least action by 



J -"(Pi) riPi) 



V ds = I V 2 (ir + h) ds = minimum. 

 (Po) J(Po) 



2) Brackistochroncs. If the conservative system described above 

 moves not according to the principle of least action but so that the 

 time of motion from Po to Pi is least, the path is called a brachisto- 

 chrone, and the totality of trajectories for all initial conditions are 

 thus determined by 



PiPO riPOfis r^^') ds 



Time = I dt = I — = / 



J(Po) J(Po) V ♦>'(Po) 



V2(jr + /o 



= minimum. 



