Light weight shafts reduce bearing forces and allow the use of smaller bearings, seals, and supports. The Minimum Principle is used to set up the problem of determining the minimum weight shaft for a specified critical speed of given order. Specific examples include simply supported shafts and cantilevered shafts with and without a disk. The designs are obtained from the solution to a nonlinear multi-point boundary-value-problem. Minimum weight configurations represent a standard against which other designs, such as stepped shafts, can be compared.