It would seem that the Action integral for the world sheets of string theory are presently justified as a higher dimensional version. But I wonder if this formulation would be a natural description if we were first given the geometry of a world-sheet. Suppose we understood the necessity of a world-sheet geometry. What would be the most natural mathematical description of what is happening with a world-sheet? We have a sheet that is growing and expanding in time. It adds more surface area with time. This might suggest that we integrate this area from one string to the next. And String theory poses that the Action Integral is protortional to the surface area of the string. But if the motion of the world-sheet is not arbitrary, then it seems there must be a function at every point on the world-sheet which forms the basis for why the string is there and where it will go in the future. This function would give a wieght to every differential area, or at least a weight for every differential length on a string. This weighting function would determine how the string would oscillate and what direction it will have a tendency to go. Is there any functions along the string or world-sheet that serves this purpose? Does the tension serve this function? Or maybe there is a potential at each point along a string or world-sheet. I have a justification for a type of world-sheet geometry, a growing surface in space-time as nothing more than a growing "events" in sample space. See here at: http://www.sirus.com/users/mjake/StringTh.html#consider [Broken] But I am not sure of the mathematics to descript it? Is this too easy and of course it is the Action integral that describes the situation? Or is it not quite as simple as that? It would be so nice to develop some sort of Action integral for this and then apply symmetries to come up with the same sort of physical laws from Euler-Lagrange equations. But I don't want to simply jump to that conclusion. Your help would be greatly appreciated. Thanks.