The Least-Action Principle: Theory of Cosmological Solutions and the   Radial-Velocity Action
  Formulating the equations of motion for cosmological bodies (such as galaxies) in an integral, rather than differential, form has several advantages. Using an integral the mathematical instability at early times is avoided and the boundary conditions of the integral correspond closely with available data. Here it is shown that such a least-action calculation for a number of bodies interacting by gravity has a finite number of solutions, possibly only one. Characteristics of the different possible solutions are explored. The results are extended to cover the motion of a continuous fluid. A method to generalize an action to use velocities, instead of positions, in boundary conditions, is given, which reduces in particular cases to those given by Giavalisco et al. (1993) and Schmoldt & Saha (1998). The velocity boundary condition is shown to have no effect on the number of solutions. 
Alan B. Whiting