The Least‐Action Principle: Theory of Cosmological Solutions and the Radial Velocity Action

Formulating the equations of motion for cosmological bodies (such asgalaxies) in an integral, rather than differential, form has severaladvantages. Using an integral the mathematical instability at early times isavoided and the boundary conditions of the integral correspond closely withavailable data. Here it is shown that such a least-action calculation for anumber of bodies interacting by gravity has a finite number of solutions,possibly only one. Characteristics of the different possible solutions areexplored. The results are extended to cover the motion of a continuous fluid. Amethod to generalize an action to use velocities, instead of positions, inboundary conditions, is given, which reduces in particular cases to those givenby Giavalisco et al. (1993) and Schmoldt & Saha (1998). The velocity boundarycondition is shown to have no effect on the number of solutions.Comment: 28 pages, 1 ps figure; accepted for publication in the Astrophysical Journa